Abstract
Let \(X\) be a compact generalized Sasakian CR manifold of dimension \(2n-1\), \(n\geqslant 2\), and let \(L\) be a generalized Sasakian CR line bundle over \(X\) equipped with a rigid semi-positive Hermitian fiber metric \(h^L\). In this paper, we prove that if \(h^L\) is positive at some point of \(X\) and conditions \(Y(0)\) and \(Y(1)\) hold at each point of \(X\), then \(L\) is big.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and statement of the main results
Let \(X\) be a compact CR manifold of dimension \(2n-1\), \(n\ge 2\). When \(X\) is strongly pseudoconvex and dimension of \(X\) is greater than five, a classical theorem of Boutet de Monvel [3] asserts that \(X\) can be globally CR embedded into \(\mathbb {C}^N\), for some \(N\in {\mathbb {N}}\). For a strongly pseudoconvex CR manifold of dimension greater than five, the dimension of the kernel of the tangential Cauchy–Riemmann operator \(\overline{\partial }_b\) is infinite and we can find many CR functions to embed \(X\) into complex space. When the Levi form of \(X\) has negative eigenvalues, the dimension of the kernel of \(\overline{\partial }_b\) is finite and could be zero and in general, \(X\) can not be globally CR embedded into complex space. Inspired by Kodaira, we introduced in [9] (see also [12]) the idea of embedding CR manifolds by means of CR sections of tensor powers \(L^k\) of a CR line bundle \(L\rightarrow X\). If the dimension of the space \(H^0_b(X,L^k)\) of CR sections of \(L^k\) is large, when \(k\rightarrow \infty \), one should find many CR sections to embed \(X\) into projective space. In analogy to the Kodaira embedding theorem, it is natural to ask if \(X\) can be globally embedded into projective space when it carries a CR line bundle with positive curvature? To understand this question, it is crucial to be able to know if \(\mathrm{dim\,}H^0_b(X,L^k)\sim k^n\), for \(k\) large? The following conjecture was implicit in [12, p.47-48]
Conjecture 1.1
If \(L\) is positive and the Levi form of \(X\) has at least two negative and two positive eigenvalues\(,\) then
for \(k\) large.
The difficulty of this conjecture comes from the presence of positive eigenvalues of the curvature of the line bundle and negative eigenvalues of the Levi form of \(X\) and this causes the associated Kohn Laplacian to have no semi-classical spectral gap. This problem is also closely related to the fact that in the global \(L^2\)-estimates for the \(\overline{\partial }_b\)-operator of Kohn–Hörmander, there is a curvature term from the line bundle as well from the boundary and, in general, it is very difficult to control the sign of the total curvature contribution.
In complex geometry, Demailly’s holomorphic Morse inequalities [6] handled the corresponding analytical difficulties in a new way. Inspired by Demailly, we established analogs of the holomorphic Morse inequalities of Demailly for CR manifolds (see [9, Theorem 1.8])
Theorem 1.2
(Hsiao–Marinescu, 2009) We assume that the Levi form of \(X\) has at least two negative and two positive eigenvalues. Then\(,\) as \(k\rightarrow \infty ,\)
where \(M^\phi _x\) is the associated curvature of \(L\) at \(x\in X\) \((\)see Definition 1.9), \(H^1_b(X, L^k)\) denotes the first \(\overline{\partial }_b\) cohomology group with values in \(L^k,\) \(\mathrm{d}v_X(x)\) is the volume form on \(X,\) \({\mathcal L}_x\) denotes the Levi form of \(X\) at \(x\in X,\) and for \(x\in X,\) \(q=0,1,\)
From (1.1), we see that if
then \(L\) is big that is \(\mathrm{dim\,}H^0_b(X,L^k)\sim k^n\). This is a very general criterion and it is desirable to refine it in some cases where (1.3) is not easy to verify. The problem still comes from the presence of positive eigenvalues of \(M^\phi _x\) and negative eigenvalues of \({\mathcal L}_x\).
For the better understanding, let’s see a simple example. We consider compact analogs of the Heisenberg group \(H_n\). Let \(\lambda _1,\ldots ,\lambda _{n-1}\) be given non-zero integers. We assume that \(\lambda _1<0,\ldots ,\lambda _{n_-}<0,\lambda _{n_-+1}>0,\ldots ,\lambda _{n-1}>0\). Let \({\fancyscript{C}}H_n=(\mathbb {C}^{n-1}\times \mathbb {R})/_\sim \), where \((z, \theta )\sim (\widetilde{z}, \widetilde{\theta })\) if
We can check that \(\sim \) is an equivalence relation and \({\fancyscript{C}}H_n\) is a compact manifold of dimension \(2n-1\). The equivalence class of \((z, \theta )\in \mathbb {C}^{n-1}\times \mathbb {R}\) is denoted by \([(z, \theta )]\). For a given point \(p=[(z, \theta )]\), we define the CR structure \(T^{1, 0}_p{\fancyscript{C}}H_n\) of \({\fancyscript{C}}H_n\) to be the space spanned by \(\left\{ \frac{\partial }{\partial z_j}+i\lambda _j\overline{z}_j\frac{\partial }{\partial \theta },\ \ j=1,\ldots ,n-1\right\} \). Then, \(({\fancyscript{C}}H_n,T^{1,0}_p{\fancyscript{C}}H_n)\) is a compact CR manifold of dimension \(2n-1\). With a suitable choose of a Hermitian metric on the complexified tangent bundle of \({\fancyscript{C}}H_n\), the Levi form of \({\fancyscript{C}}H_n\) at \(p\in {\fancyscript{C}}H_n\) is given by \({\mathcal L}_p=\sum ^{n-1}_{j=1}\lambda _j\mathrm{d}z_j\wedge \mathrm{d}\overline{z}_j\). Let \(L=(\mathbb {C}^{n-1}\times \mathbb {R}\times \mathbb {C})/_{\equiv }\) where \((z,\theta ,\eta )\equiv (\widetilde{z}, \widetilde{\theta }, \widetilde{\eta })\) if
where \(\mu _{j,t}=\mu _{t,j}\), \(j, t=1,\ldots ,n-1\), are given integers. We can check that \(\equiv \) is an equivalence relation and \(L\) is a CR line bundle over \({\fancyscript{C}}H_n\). For \((z, \theta , \eta )\in \mathbb {C}^{n-1}\times \mathbb {R}\times \mathbb {C}\), we denote \([(z, \theta , \eta )]\) its equivalence class. Take the pointwise norm \( \big |[(z, \theta , \eta )]\big |^2_{h^L}:=\left| \eta \right| ^2\exp \big (-\textstyle \sum ^{n-1}_{j,t=1}\mu _{j,t}z_j\overline{z}_t\big ) \) on \(L\). Then, the associated curvature of \(L\) is given by \(M^\phi _x=\sum ^{n-1}_{j,t=1}\mu _{j,t}\mathrm{d}z_j\wedge \mathrm{d}\overline{z}_t,\ \ \forall x\in {\fancyscript{C}}H_n\). In this simple example, Conjecture 1.1 becomes
Question 1.3
If \(n_-\ge 2\), \(n-1-n_-\ge 2\), and the matrix \(\left( \mu _{j,t}\right) ^{n-1}_{j,t=1}\) is positive definite, then \(\mathrm{dim\,}H^0_b({\fancyscript{C}}H_n,L^k)\sim k^n\)?
If \(\mu _{j,t}=\left| \lambda _j\right| \delta _{j,t}\), \(j,t=1,\ldots ,n-1\), and \(n_-\ge 2\), \(n-1-n_-\ge 2\), where \(\delta _{j,t}=1\) if \(j=t\), \(\delta _{j,t}=0\) if \(j\ne t\), then it is easy to see that \(\mathbb {R}_{\phi (x),1}=\emptyset \), where \(\mathbb {R}_{\phi (x),1}\) is given by (1.2). Combining this observation with Morse inequalities for CR manifolds [see (1.1)], we get
Theorem 1.4
If \(n_-\ge 2,\) \(n-1-n_-\ge 2,\) and \(\mu _{j,t}=\left| \lambda _j\right| \delta _{j,t},\) \(j,t=1,\ldots ,n-1,\) then \(\mathrm{dim\,}H^0_b({\fancyscript{C}}H_n,L^k)\sim k^n.\)
The assumptions in Theorem 1.4 are somehow restrictive. It is clear that we cannot go much further from Morse inequalities.Using Morse inequalities to approach Conjecture 1.1, we always have to impose extra conditions linking the Levi form and the curvature of the line bundle \(L\). Similar problems also appear in the works of Marinescu [12, 13] and Berman [2] where they studied the \(\overline{\partial }\)-Neumann cohomology groups associated to a high power of a given holomorphic line bundle on a compact complex manifold with boundary. To get many holomorphic sections, they also have to assume that, close to the boundary, the curvature of the line bundle is adapted to the Levi form of the boundary. In this work, by carefully studying semi-classical behavior of microlocal Fourier transforms of the extremal functions for the spaces of lower energy forms of the associated Kohn Laplacian, we could solve Conjecture 1.1 under rigidity conditions on \(X\) and \(L\) without any extra condition linking the Levi form of \(X\) and the curvature of \(L\). As an application, we solve Question 1.3 completely. The proof of our main result presents a new way to overcome the analytic difficulty mentioned in the discussion after Conjecture 1.1 under rigidity conditions. Using this new method, it is possible to remove the assumptions linking the curvatures of the line bundle and the boundary in the works of Marinescu [12, 13] and Berman [2] under rigidity conditions on the boundary and the line bundle.
The rigidity conditions we used in this work are inspired by the work of Baouendi–Rothschild–Treves [1]. They introduced rigidity condition on CR structure and proved that such a manifold can always be locally CR embedded in complex space as a generic submanifold. From their work, rigidity condition on CR structure seems suitable for our purpose. Initially, it is reasonable to first assume that \(X\) can be locally embedded and study global embeddability of \(X\). We can expect that the curvature of the line bundle and its transition functions have to satisfy some rigidity conditions (see Definition 1.7 and Definition 1.12). Moreover, with these geometric conditions, it is possible to establish a micolocal asymptotic expansion of the Szegö kernel and extend Kodaira embedding theorem to this situation.
The geometric objects introduced in this paper form large classes of CR manifolds and CR line bundles. We hope that these geometric objects will be interesting for CR geometers and will be useful in CR geometry.
1.1 Some standard notations
We shall use the following notations: \(\mathbb {R}\) is the set of real numbers, \(\overline{\mathbb {R}}_+:=\left\{ x\in \mathbb {R};\, x\ge 0\right\} \), \({\mathbb {N}}=\left\{ 1,2,\ldots \right\} \), \({\mathbb {N}}_0={\mathbb {N}}\bigcup \left\{ 0\right\} \). An element \(\alpha =(\alpha _1,\ldots ,\alpha _n)\) of \({\mathbb {N}}_0^n\) will be called a multiindex and the length of \(\alpha \) is: \(\left| \alpha \right| =\alpha _1+\cdots +\alpha _n\). We write \(x^\alpha =x_1^{\alpha _1}\cdots x^{\alpha _n}_n\), \(x=(x_1,\ldots ,x_n)\), \(\partial ^\alpha _x=\partial ^{\alpha _1}_{x_1}\cdots \partial ^{\alpha _n}_{x_n}\), \(\partial _{x_j}=\frac{\partial }{\partial x_j}\), \(D^\alpha _x=D^{\alpha _1}_{x_1}\cdots D^{\alpha _n}_{x_n}\), \(D_x=\frac{1}{i}\partial _x\), \(D_{x_j}=\frac{1}{i}\partial _{x_j}\). Let \(z=(z_1,\ldots ,z_n)\), \(z_j=x_{2j-1}+ix_{2j}\), \(j=1,\ldots ,n\), be coordinates of \(\mathbb {C}^n\). We write \(z^\alpha =z_1^{\alpha _1}\cdots z^{\alpha _n}_n\), \(\overline{z}^\alpha =\overline{z}_1^{\alpha _1}\cdots \overline{z}^{\alpha _n}_n\), \(\frac{\partial ^{\left| \alpha \right| }}{\partial z^\alpha }=\partial ^\alpha _z=\partial ^{\alpha _1}_{z_1}\cdots \partial ^{\alpha _n}_{z_n}\), \(\partial _{z_j}= \frac{\partial }{\partial z_j}=\frac{1}{2}\left( \frac{\partial }{\partial x_{2j-1}}-i\frac{\partial }{\partial x_{2j}}\right) \), \(j=1,\ldots ,n\). \(\frac{\partial ^{\left| \alpha \right| }}{\partial \overline{z}^\alpha }=\partial ^\alpha _{\overline{z}}=\partial ^{\alpha _1}_{\overline{z}_1}\cdots \partial ^{\alpha _n}_{\overline{z}_n}\), \(\partial _{\overline{z}_j}= \frac{\partial }{\partial \overline{z}_j}=\frac{1}{2}\left( \frac{\partial }{\partial x_{2j-1}}+i\frac{\partial }{\partial x_{2j}}\right) \), \(j=1,\ldots ,n\).
Let \(\Omega \) be a \(C^\infty \) paracompact manifold. We let \(T\Omega \) and \(T^*\Omega \) denote the tangent bundle of \(\Omega \) and the cotangent bundle of \(\Omega \), respectively. The complexified tangent bundle of \(\Omega \) and the complexified cotangent bundle of \(\Omega \) will be denoted by \(\mathbb {C}T\Omega \) and \(\mathbb {C}T^*\Omega \), respectively. We write \(\langle \,\cdot ,\cdot \,\rangle \) to denote the pointwise duality between \(T\Omega \) and \(T^*\Omega \). We extend \(\langle \,\cdot ,\cdot \,\rangle \) bilinearly to \(\mathbb {C}T\Omega \times \mathbb {C}T^*\Omega \). Let \(E\) be a \(C^\infty \) vector bundle over \(\Omega \). The fiber of \(E\) at \(x\in \Omega \) will be denoted by \(E_x\). Let \(F\) be another vector bundle over \(\Omega \). We write \(E\boxtimes F\) to denote the vector bundle over \(\Omega \times \Omega \) with fiber over \((x, y)\in \Omega \times \Omega \) consisting of the linear maps from \(E_x\) to \(F_y\).
1.2 Generalized Sasakian CR manifolds and generalized Sasakian CR line bundles
Let \((X,T^{1,0}X)\) be a CR manifold of dimension \(2n-1\), \(n\geqslant 2\), where \(T^{1,0}X\) is a CR structure of \(X\). That is, \(T^{1,0}X\) is a complex \(n-1\) dimensional subbundle of the complexified tangent bundle \(\mathbb {C}TX\), satisfying \(T^{1,0}X\bigcap T^{0,1}X=\left\{ 0\right\} \), where \(T^{0,1}X=\overline{T^{1,0}X}\), and \([{\mathcal {V}},{\mathcal {V}}]\subset {\mathcal {V}}\), where \({\mathcal {V}}=C^\infty (X,T^{1,0}X)\). In this section, we denote \(Y:=X\times \mathbb {R}\) and we write \(t\) to denote the standard coordinate of \(\mathbb {R}\). We need
Definition 1.5
We say that \((X,T^{1,0}X)\) is a generalized Sasakian CR manifold if there exists an integrable almost complex structure \(J:TY\rightarrow TY\), \(\mathbb {C}TY\rightarrow \mathbb {C}TY\), such that \(Ju=iu\), \(\forall u\in T^{1,0}X\).
Let \((X,T^{1,0}X)\) be a CR manifold of dimension \(2n-1\), \(n\geqslant 2\), and let \(J:TY\rightarrow TY\), \(\mathbb {C}TY\rightarrow \mathbb {C}TY\), be an almost complex structure. We say that \(J\) is a canonical complex structure on \(Y\) if \(J\) is integrable and \(Ju=iu\), \(\forall u\in T^{1,0}X\). Thus, \((X,T^{1,0}X)\) is a generalized Sasakian CR manifold if and only if there exists a canonical complex structure on \(Y\).
Let \((X,T^{1,0}X)\) be a generalized Sasakian CR manifold and let \(J:TY\rightarrow TY\), \(\mathbb {C}TY\rightarrow \mathbb {C}TY\) be any canonical complex structure on \(Y\). From the Newlander–Nirenberg theorem, \(J\) defines a complex structure \(T^{1,0}Y\supset T^{1,0}X\). Put \(T=J\frac{\partial }{\partial t}\). Then, \(T\in C^\infty (X,TX)\), \(T\) is a global real vector field on \(X\). Since \(J\) is integrable, it is easy to see that
Conversely, let \((X,T^{1,0}X)\) be a CR manifold of dimension \(2n-1\), \(n\geqslant 2\). We assume that there exists a global real vector field \(T\in C^\infty (X,\mathbb {C}TX)\) such that (1.4) hold. Then, one can define a canonical complex structure on \(Y\) by the rule:
Thus, \((X,T^{1,0}X)\) is a generalized Sasakian CR manifold if and only if there exists a global real vector field \(T\in C^\infty (X,\mathbb {C}TX)\) such that (1.4) hold. We call \(T\) a rigid global real vector field.
Let’s see some examples
Example 1.6
(I) Let \(M\) be an open subset with \(C^\infty \) boundary \(\partial M\) of a complex manifold \(M'\) of dimension \(n\). If for every \(x_0\in \partial M\), we can find local holomorphic coordinates \((z_1,\ldots ,z_n)\) defined in some neighborhood of \(x_0\), such that near \(x_0\), \(\partial M\) is given by the equation
then \(\partial M\) is a generalized Sasakian CR manifold of dimension \(2n-1\).
(II) Let \(M\) be a complex manifold and \((E,h^E)\) be a holomorphic Hermitian line bundle on \(M\), where the Hermitian fiber metric on \(E\) is denoted by \(h^E\). Let \((E^*,h^{E^*})\) be the dual bundle of \(E\). We denote
The domain \(G\) is called Grauert tube associated to \(E\). It is easy to see that \(\partial G\) is a generalized Sasakian CR manifold.
(III) The hypersurface
is a generalized Sasakian CR manifolds, where \(\lambda _j\in \mathbb {R}\), \(j=0,1,\ldots ,n\), \(R\in \mathbb {R}\).
(IV) Heisenberg groups and compact Heisenberg groups (see Sect. 9.1) are generalized Sasakian CR manifolds.
From now on, we assume that \((X,T^{1,0}X)\) is a compact generalized Sasakian CR manifold and we let \(\pi :Y\rightarrow X\) denote the standard projection.
Definition 1.7
Let \(L\) be a complex line bundle over \(X\). \((L,J)\) is a generalized Sasakian CR line bundle over \(X\), where \(J\) is a canonical complex structure on \(Y\) if the pull back \(\pi ^*L\) is a holomorphic line bundle over \(Y\) with respect to \(J\).
We need
Definition 1.8
Let \(T\in C^\infty (X,TX)\) be a rigid global real vector field on \(X\). Let \(U\Subset X\) be an open set. A function \(u\in C^\infty (U)\) is said to be a \(T\)-rigid CR function on \(U\) if \(Tu=0\) and \(Zu=0\) for all \(Z\in C^\infty (U,T^{0,1}X)\).
From now on, we let \((L,J)\) be a generalized Sasakian CR line bundle over \(X\) and we fix \(T=J\frac{\partial }{\partial t}\). \(T\) is a rigid global real vector field. Since \(\pi ^*L\) is a holomorphic line bundle over \(Y\) with respect to the canonical complex structure \(J\) on \(Y\), it is easy to see that \(X\) can be covered with open sets \(U_j\) with trivializing sections \(s_j\), \(j=1,2,\ldots \), such that the corresponding transition functions are \(T\)-rigid CR functions. In this paper, when trivializing sections \(s\) are used, we will assume that they are of this special form.
Fix a Hermitian fiber metric \(h^L\) on \(L\) and we will denote by \(\phi \) the local weights of the Hermitian metric \(h^L\). More precisely, if \(s\) is a local trivializing section of \(L\) on an open subset \(D\subset X\), then the local weight of \(h^L\) with respect to \(s\) is the function \(\phi \in C^\infty (D,\mathbb {R})\) for which
We write \(h^{\pi ^*L}\) to denote the pull back of \(h^L\) by the projection \(\pi \). Then, \(h^{\pi ^*L}\) is a Hermitian fiber metric on the holomorphic line bundle \(\pi ^*L\). Let \(R^{\pi ^*L}\) be the canonical curvature induced by \(h^{\pi ^*L}\). Let \(\overline{\partial }_J\) and \(\partial _J\) be the \((0,1)\) and \((1,0)\) part of the exterior differential operator \(d\) on functions with respect to \(J\). If \(s\) is a local trivializing section of \(L\) on an open subset \(D\subset X\), \(\left| s\right| ^2_{h^L}=\mathrm{e}^{-\phi (x)}\), then
We need
Definition 1.9
For \(p\in X\), we define the Hermitian quadratic form \(M^\phi _p\) on \(T^{1,0}_pX\) by
Remark 1.10
Let \(s\) be a local trivializing section of \(L\) on an open subset \(D\subset X\) and \(\phi \) the corresponding local weight as in (1.5). Let \(\overline{\partial }_b\) denote the tangential Cauchy–Riemann operator on functions (see [4, Chapter 7]). It is not difficult to see that for every \(p\in D\), we have
where \(d\) is the usual exterior derivative and \(\overline{\partial _b\phi }=\overline{\partial }_b\overline{\phi }\).
For \(p\in X\), let \({\mathcal L}_p\) be the Levi form (with respect to \(T\)) at \(p\) (see Definition 1.14, for the precise meaning).
Definition 1.11
We say that \(h^L\) is positive at \(x_0\in X\) if the Hermitian quadratic form \(M^\phi _{x_0}\) is positive, \(h^L\) is semi-positive if there is a positive constant \(\delta >0\) such that for every \(x\in X\) and \(s\in [-\delta , \delta ]\), the Hermitian quadratic form \(M^\phi _x+2s{\mathcal L}_x\) is semi-positive.
Since the transition functions are \(T\)-rigid CR functions, we can check that \(T\phi \) is a well-defined global smooth function on \(X\).
Definition 1.12
\(h^L\) is said to be a \(T\)-rigid Hermitian fiber metric on \((L,J)\) if
where \(\phi \) denotes the corresponding local weight as in (1.5).
Note that the constant \(C\) in (1.9) can be non-zero. (See Sect. 9.1).
Definition 1.13
We say that \((L,J,h^L)\) is a rigid generalized Sasakian CR line bundle over X if \((L,J)\) is a generalized Sasakian CR line bundle over \(X\) and \(h^L\) is a \(T\)-rigid Hermitian fiber metric on \((L,J)\), \(T=J\frac{\partial }{\partial t}\).
1.3 Hermitian CR geometry and the main results
Fix a smooth Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(\mathbb {C}TX\) so that \(T^{1,0}X\) is pointwise orthogonal to \(T^{0,1}X\), \(T\) is pointwise orthogonal to \(T^{1,0}X\oplus T^{0,1}X\), \(\langle T|T\rangle :=\left\| T\right\| ^2=1\) and \(\langle u|v\rangle \) is real if \(u\), \(v\) are real tangent vectors.
Define
\(T^{*1,0}X\) and \(T^{*0,1}X\) are subbundles of the complexified cotangent bundle \(\mathbb {C}T^*X\). Define the vector bundle of \((0, q)\) forms of \(X\) by \(\Lambda ^{0, q}T^*X:=\Lambda ^{q}T^{*0,1}X\). Let \(D\subset X\) be an open set. Let \(\Omega ^{0,q}(D)\) denote the space of smooth sections of \(\Lambda ^{0,q}T^*X\) over \(D\). Similarly, if \(E\) is a vector bundle over \(D\), then we let \(\Omega ^{0,q}(D, E)\) denote the space of smooth sections of \(\Lambda ^{0,q}T^*X\otimes E\) over \(D\). Let \(\Omega ^{0,q}_0(D, E)\) be the subspace of \(\Omega ^{0,q}(D, E)\) whose elements have compact support in \(D\). Let
be the tangential Cauchy–Riemann operator (see [4, Chapter 7]).
The Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(\mathbb {C}TX\) induces, by duality, a Hermitian metric on \(\mathbb {C}T^*X\) and also on \(\Lambda ^{0, q}T^*X\) the bundle of \((0, q)\) forms of \(X\). We shall also denote all these induced metrics by \(\langle \,\cdot \,|\,\cdot \,\rangle \). For \(f\in \Omega ^{0,q}(X)\), we denote the pointwise norm \(\left| f(x)\right| ^2:=\langle f(x)|f(x)\rangle \). Locally, there is a real \(1\)-form \(\omega _0\) of length one which is orthogonal to \(T^{*1,0}X\oplus T^{*0,1}X\). The form \(\omega _0\) is unique up to the choice of sign. We choose \(\omega _0\) so that \(\langle T, \omega _0\rangle =-1\). Therefore, \(\omega _0\) is uniquely determined. We call \(\omega _0\) the uniquely determined global real \(1\)-form. We have the pointwise orthogonal decompositions:
We recall
Definition 1.14
For \(p\in X\), the Levi form \({\mathcal L}_p\) is the Hermitian quadratic form on \(T^{1,0}_pX\) defined as follows. For any \(U,\ V\in T^{1,0}_pX\), pick \({\mathcal {U}},{\mathcal {V}}\in C^\infty (X, T^{1,0}X)\) such that \({\mathcal {U}}(p)=U\), \({\mathcal {V}}(p)=V\). Set
where \(\big [{\mathcal {U}},\overline{{\mathcal {V}}}\,\big ]={\mathcal {U}}\ \overline{{\mathcal {V}}}-\overline{{\mathcal {V}}}\ {\mathcal {U}}\) denotes the commutator of \({\mathcal {U}}\) and \(\overline{{\mathcal {V}}}\). Note that \({\mathcal L}_p\) does not depend on the choices of \({\mathcal {U}}\) and \({\mathcal {V}}\).
Since \({\mathcal L}_p\) is a Hermitian form there is a local orthonormal basis \(\{{\mathcal {U}}_1,\ldots ,{\mathcal {U}}_{n-1}\}\) of \(T^{1,0}X\) with respect to \(\langle \,\cdot \,|\,\cdot \,\rangle \) such that \({\mathcal L}_p\) is diagonal in this basis, \({\mathcal L}_p({\mathcal {U}}_j,\overline{{\mathcal {U}}}_t)=\delta _{j,t}\lambda _j(p)\), \(j,t=1,\ldots ,n-1\), \(\delta _{j,t}=1\) if \(j=t\), \(\delta _{j,t}=0\) if \(j\ne t\), \(\lambda _j(p)\in \mathbb {R}\), \(j=1,\ldots ,n-1\). The diagonal entries \(\{\lambda _1(p),\ldots ,\lambda _{n-1}(p)\}\) are called the eigenvalues of the Levi form at \(p\in X\) with respect to \(\langle \,\cdot \,|\,\cdot \,\rangle \).
Given \(q\in \{0,\ldots ,n-1\}\), the Levi form is said to satisfy condition \(Y(q)\) at \(p\in X\), if \({\mathcal L}_p\) has at least either \(\max {(q+1, n-q)}\) eigenvalues of the same sign or \(\min {(q+1,n-q)}\) pairs of eigenvalues with opposite signs. Note that the sign of the eigenvalues does not depend on the choice of the metric \(\langle \,\cdot \,|\,\cdot \,\rangle \).
Let \(L^k\), \(k>0\), be the \(k\)-th tensor power of the line bundle \(L\). We write \(\overline{\partial }_{b,k}\) to denote the tangential Cauchy–Riemann operator acting on forms with values in \(L^k\), defined locally by:
where \(s\) is a local trivialization of \(L\) on an open subset \(D\subset X\) and \(u\in \Omega ^{0,q}(D)\). We obtain a \(\overline{\partial }_{b,k}\)-complex \((\Omega ^{0,\bullet }(X, L^k),\overline{\partial }_{b,k})\) with cohomology
We assume that \(X\) is compact and \(Y(0)\) holds. By [11, 7.6-7.8], [7, 5.4.11-12], [4, Props. 8.4.8-9] and [8, Chapter 6], condition \(Y(0)\) implies that \(\dim H^0_b(X,L^k)<\infty \).
Our main result is the following
Theorem 1.15
Let \((X,T^{1,0}X)\) be a compact generalized Sasakian CR manifold of dimension \(2n-1,\) \(n\geqslant 2\) and let \((L,J,h^L)\) be a rigid generalized Sasakian CR line bundle over \(X\). Assume that \(h^L\) is semi-positive and positive at some point of \(X\). Suppose conditions \(Y(0)\) and \(Y(1)\) hold at each point of \(X\). Then\(,\) for \(k\) large\(,\) there is a constant \(c>0\) independent of \(k,\) such that
It should be mentioned that the Levi curvature assumptions in Theorem 1.15 are a bit more general than the ones in Conjecture 1.1.
Remark 1.16
It should be mentioned that Theorem 1.15 implies the famous Grauert–Riemenschneider conjecture in complex geometry. Let \(M\) be a compact complex manifold of complex dimension \(n\) and let \(E\rightarrow M\) be a holomorphic line bundle with a Hermitian fiber metric \(h^E\). Let \(R^E\) denotes the canonical curvature on \(E\) induced by \(h^E\). We assume that \(R^E\) is semi-positive and positive at some point of \(M\). Then, Grauert–Riemenschneider conjecture claims that \(L\) is big, that is, \(\mathrm{dim\,}H^0(M, E^k)\sim k^n\), where \(H^0(M,E^k)\) denotes the space of global holomorphic sections of \(E^k\) the \(k\)-th power of \(E\). This conjecture was first solved by Siu [14]. Let’s see how to obtain this conjecture from Theorem 1.15. With the notations used above, let \((\widetilde{X},T^{1,0}\widetilde{X})\) be a compact generalized Sasakian CR manifold of dimension \(2m-1\), \(m\geqslant 2\), such that the Levi form of \(\widetilde{X}\) has at least two negative and two positive eigenvalues and let \((\widetilde{L},\widetilde{J},h^{\widetilde{L}})\) be a rigid generalized Sasakian CR line bundle over \(\widetilde{X}\) with \(h^{\widetilde{L}}\) is positive at every point of \(\widetilde{X}\). We can find such \((\widetilde{X},T^{1,0}\widetilde{X})\) and \((\widetilde{L},\widetilde{J},h^{\widetilde{L}})\) (see Sect. 9). Consider \(X=M\oplus \widetilde{X}\), \(T^{1,0}X:=T^{1,0}M\oplus T^{1,0}\widetilde{X}\), where \(T^{1,0}M\) denotes the holomorphic tangent bundle of \(M\). Then, \((X,T^{1,0}X)\) is a compact generalized Sasakian CR manifold of dimension \(2(m+n)-1\) and the Levi form of \(X\) has at least two negative and two positive eigenvalues. Thus, conditions \(Y(0)\) and \(Y(1)\) hold at each point of \(X\). Put \(L:=E\otimes \widetilde{L}\). Then, \(L\) is a complex line bundle over \(X\). Let \(J\) be the canonical complex structure on \(X\times \mathbb {R}\) induced by \(\widetilde{J}\) and the complex structure on \(M\). It is obviously that \((L,J)\) is a generalized Sasakian CR line bundle over \(X\). Put \(h^{L}=h^{E}\otimes h^{\widetilde{L}}\). Then, \(h^L\) is a Hermitian fiber metric on \(L\) and \((L,J,h^L)\) is a rigid generalized Sasakian CR line bundle over \(X\). Moreover, it is easy to check that \(h^L\) is semi-positive and positive at some point of \(X\). From Theorem 1.15, we conclude that for \(k\) large, there is a constant \(C_0>0\) such that
We notice that \(\mathrm{dim\,}H^0_b(X,L^k)=\mathrm{dim\,}H^0(M,E^k)\times \mathrm{dim\,}H^0_b(\widetilde{X},\widetilde{L}^k)\) and it is well known that there is a constant \(C_1>0\) such that \(\mathrm{dim\,}H^0_b(\widetilde{X},\widetilde{L}^k)\le C_1k^{m}\) (see [9, Theorem 1.5]). Combining this observation and (1.15), we conclude that there is a constant \(c>0\) such that \(\mathrm{dim\,}H^0(M,E^k)\ge ck^{n}\).
We investigate Theorem 1.15 on generalized torus CR manifolds. Let \(\Phi ^t(x)\) be the \(T\)-flow. That is, \(\Phi ^t(x)\) is a differentiable mapping:
\(I\) is an open interval in \(\mathbb {R}\), \(0\in I\), such that \(\Phi ^0(x)=x\), \(\forall x\in X\), and \(\frac{\mathrm{d}\Phi ^t(x)}{\mathrm{d}t}=T(\Phi ^t(x))\). We need
Definition 1.17
We say that \((X,T^{1,0}X)\) is a generalized torus CR manifold if there is a constant \(\gamma _0>0\) such that \(\Phi ^t(x)\) is well defined, \(\forall \left| t\right| \le \gamma _0\), \(\forall x\in X\), and \(\Phi ^{\gamma _0}(x)=x\) for every \(x\in X\).
Definition 1.18
We say that \((L,J)\) is an admissible generalized Sasakian CR line bundle over a compact generalized torus CR manifold \(X\) if we can find an open covering \(\left\{ U_j\right\} ^N_{j=1}\) of \(X\) such that \(L\) is trivial on \(U_j\), for each \(j\), and
for each \(j\), where \(\gamma _0>0\) is as in Definition 1.17.
Let \((L,J)\) be an admissible generalized Sasakian CR line bundle over a compact generalized torus CR manifold \((X,T^{1,0}X)\). Take any Hermitian fiber metric \(h^{L}\) on \(L\) and let \(\phi \) denotes the corresponding local weight as in (1.5). Let \(h^{L}_1\) be the Hermitian fiber metric on \(L\) locally given by \(\left| s\right| ^2_{h^{L}_1}=\mathrm{e}^{-\phi _1}\), where \(\phi _1=\frac{1}{\gamma _0}\int ^{\gamma _0}_0\phi (\Phi ^t(x))\mathrm{d}t\), \(\gamma _0>0\) is as in Definition 1.17, \(s\) is a local trivializing section of \(L\) with the special form in Definition 1.18. It is easy to check that \(h^{L}_1\) is well defined and \(T\phi _1=0\). Thus, \((L,J,h^{L}_1)\) is a rigid generalized Sasakian CR line bundle over \((X,T^{1,0}X)\). Moreover, we can show that if \(M^\phi _x\) is positive on \(X\) then \(M^{\phi _1}_x\) is positive on \(X\) (see Proposition 3.3, for the proof). Combining this with Theorem 1.15, we obtain
Theorem 1.19
Let \((X,T^{1,0}X)\) be a compact generalized torus CR manifold of dimension \(2n-1,\) \(n\geqslant 2\) and let \((L,J)\) be an admissible generalized Sasakian CR line bundle over \(X\) with a Hermitian fiber metric \(h^L\). We assume that \(h^L\) is positive on \(X\) and conditions \(Y(0)\) and \(Y(1)\) hold at each point of \(X.\) Then\(,\) for \(k\) large\(,\) there is a constant \(c>0\) independent of \(k,\) such that
1.4 The outline of the proof of Theorem 1.15
Let \(\Box ^{(q)}_{b,k}\) denote the Kohn Laplacian with values in \(L^k\) (see Sect. 2). Fix \(q=0,1,\ldots ,n-1\). We assume that \(Y(q)\) holds. It is well known that \(\Box ^{(q)}_{b,k}\) has a discrete spectrum, each eigenvalues occurs with finite multiplicity and all eigenforms are smooth and \(\mathrm{Ker\,}\Box ^{(q)}_{b,k}:=\fancyscript{H}^q_b(X,L^k)\cong H^q_b(X,L^k)\). For \(\lambda \ge 0\), let \(\fancyscript{H}^q_{b,\le \lambda }(X,L^k)\) denote the space spanned by the eigenforms of \(\Box ^{(q)}_{b,k}\) whose eigenvalues are bounded by \(\lambda \). Now, we assume that \(Y(0)\) and \(Y(1)\) hold and \((L,h^L)\) is semi-positive and positive at some point of \(X\). Take \(\delta _0>0\) be a small constant so that \(M^\phi _x+2s{\mathcal L}_x\ge 0\), \(\forall \left| s\right| \le \delta _0\), \(\forall x\in X\). Take \(\psi (\eta )\in C^\infty _0(]-\delta _0,\delta _0[,\overline{\mathbb {R}}_+)\) so that \(\psi (\eta )=1\) if \(-\frac{\delta _0}{2}\le \eta \le \frac{\delta _0}{2}\). Take \(\chi (t)\in C^\infty _0(]-2,2[,\overline{\mathbb {R}}_+)\) so that \(0\le \chi (t)\le 1\) and \(\chi (t)=1\) if \(-1\le t\le 1\) and \(\chi (-t)=\chi (t)\) for all \(t\in \mathbb {R}\). Fix \(M>0\). Under the rigidity assumptions in Theorem 1.15, we can construct global continuous operators \(Q^{(0)}_{M,k}:C^\infty (X,L^k)\rightarrow C^\infty (X,L^k)\) and \(Q^{(1)}_{M,k}:\Omega ^{0,1}(X,L^k)\rightarrow \Omega ^{0,1}(X,L^k)\) such that
and \(Q^{(0)}_{M,k}\), \(Q^{(1)}_{M,k}\) are formally given by the following. Let \(s\) be a local section of \(L\) on \(D\subset X\), \(\left| s\right| ^2_{h^L}=\mathrm{e}^{-\phi }\), and let \(\Phi ^t(x)\) be the \(T\)-flow. Then,
where \(f=s^k\widetilde{f}\in C^\infty _0(D,L^k)\), \(g=s^k\widetilde{g}\in \Omega ^{0,1}_0(D,L^k)\). (See Sect. 5, for the precise definitions of the operators \(Q^{(0)}_{M,k}\), \(Q^{(1)}_{M,k}\).) Let \(\langle \,\cdot \,|\,\cdot \,\rangle _{h^{L^k}}\) denote the Hermitian metric on \(\Lambda ^{0,q}T^*X\otimes L^k\) induced by \(h^{L}\) and \(\langle \,\cdot \,|\,\cdot \,\rangle \). Let \(\mathrm{d}v_X=\mathrm{d}v_X(x)\) be the volume form on \(X\) induced by \(\langle \,\cdot \,|\,\cdot \,\rangle \) and let \((\,\cdot \,|\,\cdot \,)_{h^{L^k}}\) be the \(L^2\) inner product on \(\Omega ^{0,q}(X,L^k)\) induced by \(\langle \,\cdot \,|\,\cdot \,\rangle _{h^{L^k}}\) and \(\mathrm{d}v_X\). For \(\lambda \ge 0\), define
where \(f_j(x)\in C^\infty (X, L^k)\), \(j=1,\ldots ,m_k\), is an orthonormal frame for the space \(\fancyscript{H}^0_{b,\,\le \lambda }(X, L^k)\) with respect to \((\,\cdot \,|\,\cdot \,)_{h^{L^k}}\), \(g_j(x)\in \Omega ^{0,1}(X, L^k)\), \(j=1,\ldots ,p_k\), is an orthonormal frame for \(\fancyscript{H}^1_{b,\,\le \lambda }(X, L^k)\) with respect to \((\,\cdot \,|\,\cdot \,)_{h^{L^k}}\). It is straightforward to see that the definitions (1.18) are independent of the choices of orthonormal frames. The point of our proof is that there exists a sequence \(\nu _k>0\) with \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty \), such that
and
where \(\mathbb {R}_{x,0}\) is given by (2.15) and \({1\!\!1}_{\mathbb {R}_{x,0}}(\xi )=1\) if \(\xi \in \mathbb {R}_{x,0}\), \({1\!\!1}_{\mathbb {R}_{x,0}}(\xi )=0\) if \(\xi \notin \mathbb {R}_{x,0}\) and \(C_1>0\) is a constant independent of \(k\) and \(M\).
From (1.22), we can apply Lebesgue dominate theorem and Fatou’s lemma and we get using (1.20) and (1.21),
where \(C_2>0\) is a constant independent of \(M\) and \(k\).
Let \(f_{1,k},f_{2,k},\ldots ,f_{d_k,k}\) be an orthonormal basis for \(\fancyscript{H}^0_b(X,L^k)\), where \(d_{k}=\mathrm{dim\,} \fancyscript{H}^0_{b}(X,L^k)\). Let \(\widetilde{f}_{1,k},\widetilde{f}_{2,k},\ldots ,\widetilde{f}_{n_k,k}\) be an orthonormal basis for the space \(\fancyscript{H}^0_{b,0<\mu \le k\nu _k} (X,L^k)\). From (1.23) and (1.18), we see that if \(M\) is large enough, then
for \(k\) large. From (1.16) and (1.18), it is not difficult to check that
It is well known (see [9, Theorem 1.4]) that
From (1.27), (1.24), (1.26) and (1.25), it is straightforward to see that if \(M\) is large enough, then
for \(k\) large. Moreover, it is straightforward to see that there is a constant \(C_M>0\) independent of \(k\) such that \(\left| \int _X\langle Q^{(0)}_{M,k}u|u\rangle _{h^{L^k}}(x)\mathrm{d}v_X(x)\right| \le C_M\int _X\langle u|u\rangle _{h^{L^k}}(x)\mathrm{d}v_X(x)\), for all \(u\in C^\infty (X,L^k)\). Combining this with (1.28), we have
Theorem 1.15 follows.
The paper is organized as follows. In Sect. 2, we review the results in [9] about the asymptotic behavior of the Szegö kernel for lower energy forms to prove (1.19), (1.20) and (1.22). We introduce the extremal function for the space of lower energy forms with respect to a given continuous operator and relate it to the function \(Q^{(1)}_{M,k}\varPi ^{(1)}_{k,\le \lambda }\overline{Q^{(1)}_{M,k}}\) (see Lemma 2.2). This result will be used in the proof of (1.21). In Sect. 3, we introduce canonical coordinates on generalized Sasakian CR manifolds and prove that locally we can always find canonical coordinates and local section such that the corresponding local weight has a simple form (see Proposition 3.2). Canonical coordinates will be used in the constructions of the operators \(Q^{(0)}_{M,k}\) and \(Q^{(1)}_{M,k}\) and Proposition 3.2 will be used in Sect. 4 and the proofs of (1.19), (1.20) and (1.21). In Sect. 4, we modify the scaling technique developed in [9] and [10] to establish the semi-classical Kohn estimates (see Propositions 4.2) and a result about the asymptotic behavior of a sequence of forms with small energy (see Proposition 4.3). These results play important roles in the proofs of (1.19), (1.20) and (1.21). In Sect. 5, we construct the operators \(Q^{(0)}_{M,k}\) and \(Q^{(1)}_{M,k}\). In Sect. 6, we prove (1.22), (1.19), (1.20) and (1.23). In Sect. 7, we prove (1.21) and (1.24). In Sect. 8, we first prove the inequality (1.26) and then we complete the proof of Theorem 1.15. In Sect. 9, we exemplify our main result in two concrete examples, one of a quotient of the Heisenberg group and the other of a Grauert tube over the torus.
2 Szegö kernels for lower energy forms
We will use the same notations as Sect. 1. From now on, we assume that \((L,J,h^L)\) is a rigid generalized Sasakian CR line bundle over \(X\).
The Hermitian fiber metric on \(L\) induces a Hermitian fiber metric on \(L^k\) that we shall denote by \(h^{L^k}\). If \(s\) is a local trivializing section of \(L\) then \(s^k\) is a local trivializing section of \(L^k\). The Hermitian metrics \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(\Lambda ^{0,q}T^*X\) and \(h^{L^k}\) induce Hermitian metrics on \(\Lambda ^{0,q}T^*X\otimes L^k\). We shall denote these induced metrics by \(\langle \,\cdot \,|\,\cdot \,\rangle _{h^{L^k}}\). For \(f\in \Omega ^{0,q}(X, L^k)\), we denote the pointwise norm \(\left| f(x)\right| ^2_{h^{L^k}}:=\langle f(x)|f(x)\rangle _{h^{L^k}}\). As (1.13), let
denote the tangential Cauchy–Riemann operator acting on forms with values in \(L^k\). We denote by \(\mathrm{d}v_X=\mathrm{d}v_X(x)\) the volume form on \(X\) induced by the fixed Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(\mathbb {C}TX\). Then, we get natural global \(L^2\) inner products \((\ |\ )_{h^{L^k}}\), \((\ |\ )\) on \(\Omega ^{0,q}(X, L^k)\) and \(\Omega ^{0,q}(X)\), respectively. We denote by \(L^2_{(0,q)}(X, L^k)\) the completion of \(\Omega ^{0,q}(X, L^k)\) with respect to \((\ |\ )_{h^{L^k}}\). For \(f\in \Omega ^{0,q}(X,L^k)\), we denote \(\left\| f\right\| ^2_{h^{L^k}}:=(f\ |\ f)_{h^{L^k}}\). Similarly, for \(f\in \Omega ^{0,q}(X)\), we denote \(\left\| f\right\| ^2:=(f\ |\ f)\). Let
be the formal adjoint of \(\overline{\partial }_{b,k}\) with respect to \((\ |\ )_{h^{L^k}}\). The Kohn Laplacian with values in \(L^k\) is given by
We extend \(\overline{\partial }_{b,k}\) to \(L^2_{(0,r)}(X,L^k)\), \(r=0,1,\ldots ,n-1\), by
where \(\mathrm{Dom\,}\overline{\partial }_{b,k}:=\{u\in L^2_{(0,r)}(X, L^k);\, \overline{\partial }_{b,k}u\in L^2_{(0,r+1)}(X, L^k)\}\), where for any \(u\in L^2_{(0,r)}(X,L^k)\), \(\overline{\partial }_{b,k} u\) is defined in the sense of distribution. We also write
to denote the Hilbert space adjoint of \(\overline{\partial }_{b,k}\) in the \(L^2\) space with respect to \((\ |\ )_{h^{L^k}}\). Let \(\Box ^{(q)}_{b,k}\) also denote the Gaffney extension of the Kohn Laplacian given by
and \(\Box ^{(q)}_{b,k}s=\overline{\partial }_{b,k}\overline{\partial }^{\,*}_{b,k}s+\overline{\partial }^{\,*}_{b,k}\overline{\partial }_{b,k}s\) for \(s\in \mathrm{Dom\,}\Box ^{(q)}_{b,k}\). We notice that \(\Box ^{(q)}_{b,k}\) is a positive self-adjoint operator. For a Borel set \(B\subset \mathbb {R}\), we denote by \(E(B)\) the spectral projection of \(\Box ^{(q)}_{b,k}\) corresponding to the set \(B\), where \(E\) is the spectral measure of \(\Box ^{(q)}_{b,k}\) (see [5, section 2], for the precise meanings of spectral projection and spectral measure). We notice that the spectrum of \(\Box ^{(q)}_{b,k}\) is contained in \(\overline{\mathbb {R}}_+\). For \(\lambda \ge 0\), we set
It is well known (see [5, section 2]) that for all \(\lambda >0\),
and
For \(\lambda =0\), we denote
Now, fix \(q\in \left\{ 0,1,\ldots ,n-1\right\} \) and until further notice we assume that \(Y(q)\) holds. By [11, 7.6-7.8], [7, 5.4.11-12], [4, Props. 8.4.8-9] and [8, Chapter 6], we know that \(\Box ^{(q)}_{b,k}\) is hypoelliptic, has compact resolvent, the strong Hodge decomposition holds and \(\Box ^{(q)}_{b,k}\) has a discrete spectrum, each eigenvalues occurs with finite multiplicity and all eigenforms are smooth. Hence, for any \(\lambda \ge 0\),
Let \(g_j(x)\in \Omega ^{0,q}(X, L^k)\), \(j=1,\ldots ,d_k\), \(d_k=\mathrm{dim\,}\fancyscript{H}^q_{b,\,\leqslant \lambda }(X,L^k)\), be any orthonormal frame for the space \(\fancyscript{H}_{b,\,\le \lambda }^q(X, L^k)\) with respect to \((\,\cdot \,|\,\cdot \,)_{h^{L^k}}\). The Szegö kernel function \(\varPi ^{(q)}_{k,\le \lambda }(x)\) of the space \(\fancyscript{H}^q_{b,\leqslant \lambda }(X,L^k)\) is given by
Let
be a continuous operator. We define
It is straightforward to see that the definitions (2.12), (2.13) and (2.14) are independent of the choices of orthonormal frame \(g_j\), \(j=1,\ldots ,d_k\).
For \(q=0,1,\ldots ,n-1\) and \(x\in X\), set
where \(M^\phi _x\) is given by (1.8) and the eigenvalues of the Hermitian quadratic form \(M^\phi _x+2s{\mathcal L}_x\), \(s\in \mathbb {R}\), are calculated with respect to the Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \). It is not difficult to see that if \(Y(q)\) holds at each point of \(X\) then there is a constant \(C>0\) such that
Denote by \(\det (M^\phi _x+2s{\mathcal L}_x)\) the product of all the eigenvalues of \(M^\phi _x+2s{\mathcal L}_x\). Assuming (2.16) holds, the function
is well defined. Since \(M^\phi _x\) and \({\mathcal L}_x\) are continuous functions of \(x\in X\), we conclude that the function (2.17) is continuous.
The following is well known (see [9, Theorem 1.6])
Theorem 2.1
Assume that condition \(Y(q)\) holds at each point of \(X\). Then\(,\) for any sequence \(\nu _k>0\) with \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty ,\) there is a constant \(C_0>0\) independent of \(k,\) such that
for all \(x\in X\). Moreover\(,\) there is a sequence \(\mu _k>0,\) \(\mu _k\rightarrow 0\) as \(k\rightarrow \infty ,\) such that for any sequence \(\nu _k>0\) with \(\lim _{k\rightarrow \infty }\frac{\mu _k}{\nu _k}=0\) and \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty ,\) we have
for all \(x\in X.\)
We introduce some notations. For \(p\in X\), we can choose a smooth orthonormal frame \(e_1,\ldots ,e_{n-1}\) of \(T^{*0,1}X\) over a neighborhood \(U\) of \(p\). We say that a multiindex \(J=(j_1,\ldots ,j_q)\in \{1,\ldots ,n-1\}^q\) has length \(q\) and write \(\left| J\right| =q\). We say that \(J\) is strictly increasing if \(1\leqslant j_1<j_2<\cdots <j_q\leqslant n-1\). For \(J=(j_1,\ldots ,j_q)\) we define \(e_J:=e_{j_1}\wedge \cdots \wedge e_{j_q}\). Then, \(\left\{ e_J;\, \left| J\right| =q, J\hbox { strictly increasing}\right\} \) is an orthonormal frame for \(\Lambda ^{0,q}T^*X\) over \(U\).
For \(f\in \Omega ^{0,q}(X, L^k)\), we may write
where \(\sum '\) means that the summation is performed only over strictly increasing multiindices. We call \(f_J\) the component of \(f\) along \(e_J\). It will be clear from the context what frame is being used. For \(q>0\), the extremal function \(S^{(q)}_{k,\le \lambda ,J}\) for the space \(\fancyscript{H}^q_{b,\le \lambda }(X,L^k)\) along the direction \(e_J\) is defined by
where \(\alpha _J\) denotes the component of \(\alpha \) along \(e_J\). Let
be a continuous operator. For \(\left| J\right| =q\), \(J\) is strictly increasing, we define
where \((A\alpha )_J\) denotes the component of \(A\alpha \) along \(e_J\). Similarly, when \(q=0\), we define
We need the following
Lemma 2.2
Fix \(\lambda \ge 0\). Let \(A:\Omega ^{0,q}(X,L^k)\rightarrow \Omega ^{0,q}(X,L^k)\) be a continuous operator. For every local orthonormal frame
of \(\Lambda ^{0,q}T^*X\) over an open set \(U\subset X,\) we have when \(q>0,\)
for every \(y\in U\).
Similarly\(,\) when \(q=0,\) we have
for every \(y\in U\).
We remind that \(A\varPi ^{(q)}_{k,\le \lambda }\overline{A}\) is given by (2.14).
Proof
Let \((f_j)_{j=1,\ldots ,d_k}\) be an orthonormal frame for the space \(\fancyscript{H}_{b,\le \lambda }^q(X, L^k)\). Let \(s\) be a local section of \(L\) on \(U\), \(\left| s\right| ^2_{h^L}=\mathrm{e}^{-\phi }\). On \(U\), we write
On \(U\), we write
where
It is easy to see that \((A\varPi ^{(q)}_{k,\le \lambda ,J}\overline{A})(y)\) is independent of the choice of the orthonormal frame \((f_j)_{j=1,\ldots ,d_k}\). Take \(\alpha \in \fancyscript{H}^q_{b,\le \lambda }(X, L^k)\) of unit norm. Since \(\alpha \) is contained in an orthonormal base, obviously \(|(A\alpha )_J(y)|^2_{h^{L^k}}\leqslant (A\varPi ^{(q)}_{k,\le \lambda ,J}\overline{A})(y)\), where \((A\alpha )_J\) denotes the component of \(A\alpha \) along \(e_J\). Thus,
Fix a point \(p\in U\) and a strictly increasing multiindex \(J\) with \(\left| J\right| =q\). We may assume that \(\sum ^{d_k}_{j=1}\left| \widetilde{g}_{j,J}(p)\right| ^2\ne 0\). Put
We can easily check that \(u\in \fancyscript{H}^q_{b,\le \lambda }(X, L^k)\) and \(\left\| u\right\| _{h^{L^k}}=1\). Hence,
therefore,
From this and (2.26), we conclude that \(A\varPi ^{(q)}_{k,\le \lambda ,J}\overline{A}=AS^{(q)}_{k,\le \lambda ,J}\overline{A}\) for all strictly increasing multiindices \(J\) with \(\left| J\right| =q\). Combining this with (2.25), (2.23) follows.
The proof of (2.24) is the same. The lemma follows. \(\square \)
3 Canonical coordinates of generalized Sasakian CR manifolds
In this work, we need the following beautiful result due to Baouendi–Rothschild–Treves [1, section1]
Theorem 3.1
We recall that we work with the assumption that \((X,T^{1,0}X)\) is a generalized Sasakian CR manifold and we fix a rigid global real vector field \(T=J\frac{\partial }{\partial t}\). For every point \(x_0\in X,\) there exists local coordinates \(x=(x_1,\ldots ,x_{2n-1})=(z,\theta )=(z_1,\ldots ,z_{n-1},\theta ),\) \(z_j=x_{2j-1}+ix_{2j},\) \(j=1,\ldots ,n-1,\) \(\theta =x_{2n-1},\) defined in some small neighborhood \(U\) of \(x_0\) such that
where \(Z_j(x),\) \(j=1,\ldots ,n-1,\) form a basis of \(T^{1,0}_xX,\) for each \(x\in U,\) and \(\varphi (z)\in C^\infty (U,\mathbb {R})\) independent of \(\theta .\)
Let \(x=(x_1,\ldots ,x_{2n-1})\) be local coordinates of \(X\) defined in some open set in \(X\). In this paper, when we write \(x=(x_1,\ldots ,x_{2n-1})=(z,\theta )\) we mean that \(z=(z_1,\ldots ,z_{n-1})\), \(z_j=x_{2j-1}+ix_{2j}\), \(j=1,\ldots ,n-1\), \(\theta =x_{2n-1}\). We call \(x\) canonical coordinates if \(x\) satisfies (3.1).
We also need
Proposition 3.2
For a given point \(p\in X\), we can find canonical coordinates \(x=(x_1,\ldots ,x_{2n-1})=(z,\theta )\) and local section \(s\), \(\left| s\right| ^2_{h^L}=\mathrm{e}^{-\phi }\), defined in some small neighborhood \(D\) of \(p\) such that
where \(Z_1(x),\ldots , Z_{n-1}(x)\) form a basis of \(T^{1, 0}_xX\) varying smoothly with \(x\) in a neighborhood of \(p\), \(\lambda _1,\ldots ,\lambda _{n-1}\) are the eigenvalues of \({\mathcal L}_p\) with respect to \(\langle \,\cdot \,|\,\cdot \,\rangle \), \(\beta \in \mathbb {R}\), \(\mu _{j,t}\in \mathbb {C}\), \(\mu _{j,t}=\overline{\mu _{t,j}}\), \(j,t=1,\ldots ,n-1\).
Proof
Fix \(p\in X\). Let \(\widetilde{x}=(\widetilde{x}_1,\ldots ,\widetilde{x}_{2n-1})=(\widetilde{z},\widetilde{\theta })=(\widetilde{z}_1,\ldots ,\widetilde{z}_{n-1},\widetilde{\theta })\), \(\widetilde{z}_j=\widetilde{x}_{2j-1}+i\widetilde{x}_{2j}\), \(j=1,\ldots ,n-1\), \(\widetilde{\theta }=\widetilde{x}_{2n-1}\) be canonical coordinates of \(X\) defined in some small neighborhood \(D\) of \(p\). We have
where \(Z_j(\widetilde{x})\), \(j=1,\ldots ,n-1\), form a basis of \(T^{1,0}_{\widetilde{x}}X\), for each \(\widetilde{x}\in D\), and \(\widetilde{\varphi }(\widetilde{z})\in C^\infty (D,\mathbb {R})\) independent of \(\widetilde{\theta }\). It is easy to see that we can take \(\widetilde{x}\) so that \(\widetilde{x}(p)=0\). Near \(p\), we write
where \(a\in \mathbb {C}\), \(\alpha _j\in \mathbb {C}\), \(j=1,\ldots ,n-1\). Let \({\hat{z}}=\widetilde{z}\), \(\hat{\theta }=\widetilde{\theta }-\sum ^{n-1}_{j=1}(i\alpha _j\widetilde{z}_j-i\overline{\alpha }_j\overline{\widetilde{z}_j})\). Then, \(({\hat{z}},\hat{\theta })\) form canonical coordinates of \(X\) near \(p\) and we can check that
From (3.5), (3.4) and (3.3), we see that
where
Thus, \(\frac{\partial }{\partial {\hat{z}}_1},\ldots ,\frac{\partial }{\partial {\hat{z}}_{n-1}}\) is a basis of \(T^{1,0}_pX\). By taking some linear transformation, we can take \({\hat{z}}\) so that \(\frac{\partial }{\partial {\hat{z}}_j}\), \(j=1,\ldots ,n-1\), is an orthonormal frame for \(T^{1,0}_pX\) and the Levi form is diagonal at \(p\) with respect to \(\frac{\partial }{\partial {\hat{z}}_j}\), \(j=1,\ldots ,n-1\). We write
where \(\beta _{j,t}\in \mathbb {C}\), \(\gamma _{j,t}\in \mathbb {C}\), \(\gamma _{j,t}=\overline{\gamma _{t,j}}\), \(j,t=1,\ldots ,n-1\). Since the Levi form is diagonal at \(p\) with respect to \(\frac{\partial }{\partial {\hat{z}}_j}\), \(j=1,\ldots ,n-1\), we can check that
where \(\lambda _1,\ldots ,\lambda _{n-1}\) are the eigenvalues of \({\mathcal L}_p\) with respect to \(\langle \,\cdot \,|\,\cdot \,\rangle \). Let \(z={\hat{z}}\), \(\theta =\hat{\theta }-\sum ^{n-1}_{j,t=1}i(\beta _{j,t}{\hat{z}}_j{\hat{z}}_t-\overline{\beta }_{j,t}\overline{{\hat{z}}_j}\,\overline{{\hat{z}}_t})\). Then, \((z,\theta )\) form canonical coordinates of \(X\) near \(p\) and we can check that
From (3.6), (3.7), (3.8) and (3.9), we can check that
Since \(\frac{\partial }{\partial z_j}\), \(j=1,\ldots ,n-1\), is an orthonormal frame of \(T^{1,0}_pX\), we conclude that \(x=(z,\theta )\) satisfies the first three properties in (3.2).
Let \(\hat{s}\) be a local section defined in some neighborhood of \(p\), \(\left| \hat{s}\right| ^2_{h^L}=\mathrm{e}^{-\hat{\phi }}\). Near \(p\), we write
where \(c\in \mathbb {R}\), \(\beta \in \mathbb {R}\) and \(a_j\in \mathbb {C}\), \(j=1,\ldots ,n-1\). Let
Then, \(g(z)\) is a rigid CR function. We may replace \(\hat{s}\) by \(g\hat{s}:=\widetilde{s}\). We have
From (3.11), we can check that
Combining this with (3.12) and (3.10), we conclude that
Near \(p\), we write
where \(c_{j,t}\in \mathbb {C}\), \(\mu _{j,t}\in \mathbb {C}\), \(\mu _{j,t}=\overline{\mu _{t,j}}\), \(j,t=1,\ldots ,n-1\). Let
Then, \(g_1(z)\) is a rigid CR function. We may replace \(\widetilde{s}\) by \(g_1\widetilde{s}:=s\). We have
From (3.14), we can check that
Combining this with (3.15) and (3.13), we conclude that
The proposition follows. \(\square \)
Proposition 3.3
We assume that \(X\) is a generalized torus CR manifold and \((L,J)\) is an admissible generalized Sasakian CR line bundle over \(X\) \((\)see Definition 1.17 and Definition 1.18). Let \(\phi \) and \(\phi _1\) be as in the discussion after Definition 1.18. If \(M^{\phi }_x\) is positive on \(X,\) then \(M^{\phi _1}_x\) is positive on \(X.\)
Proof
Let \(\left\{ W_1\subset W'_1,\ldots ,W_N\subset W'_N\right\} \) be open sets of \(X\) such that \(X=\bigcup ^N_{j=1}W_j\) and there exist canonical coordinates on \(W'_j\), for each \(j\) and there is a constant \(\epsilon _0>0\) such that for each \(x\in X\), \(\Phi ^t(x)\) is well defined, \(\forall \left| t\right| \le \epsilon _0\), and
for each \(j\), where \(\Phi ^t(x)\) is the \(T\)-flow. Fix \(t_0\in [-\epsilon _0,\epsilon _0]\). Put \(\widetilde{\phi }(x)=\phi (\Phi ^{t_0}x)\). It is obviously that \(\widetilde{\phi }(x)\) also define a Hermitian fiber metric on \(L\). Using canonical coordinates (3.1), we can check that
Thus,
Similarly, fix \(t_1\in [-\epsilon _0,\epsilon _0]\) and put \(\hat{\phi }(x)=\widetilde{\phi }(\Phi ^{t_1}x)=\phi (\Phi ^{t_0+t_1}x)\). We have
Continuing in this way, we obtain for any \(t\in [0,\gamma _0]\), \(M^{\phi (\Phi ^t(x))}_{x}=M^{\phi }_{\Phi ^t(x)}\), \(\forall x\in X\), where \(\gamma _0\) is as in Definition 1.17. Thus,
The proposition follows. \(\square \)
4 The scaling technique
In this section, we modify the scaling technique developed in [9] and [10] to prove (1.19), (1.20) and (1.21).
Fix a point \(p\in X\). Let \(x=(x_1,\ldots ,x_{2n-1})=(z,\theta )\) be canonical coordinates of \(X\) defined in some small neighborhood \(D\) of \(p\) and let \(s\) be a local section of \(L\) on \(D\), \(\left| s\right| ^2_{h^L}=\mathrm{e}^{-\phi }\). We take \(x\) and \(s\) so that
where \(Z_1(x),\ldots , Z_{n-1}(x)\) form a basis of \(T^{1, 0}_xX\) varying smoothly with \(x\) in a neighborhood of \(p\), \(\lambda _1,\ldots ,\lambda _{n-1}\) are the eigenvalues of \({\mathcal L}_p\) with respect to \(\langle \,\cdot \,|\,\cdot \,\rangle \), \(\beta \in \mathbb {R}\), \(\mu _{j,k}\in \mathbb {C}\), \(\mu _{j,t}=\overline{\mu _{t,j}}\), \(j,t=1,\ldots ,n-1\). By Proposition 3.2, this is always possible. Fix \(q\in \left\{ 0,1,\ldots ,n-1\right\} \). In this section, we work on \((0,q)\) forms and we work with this local coordinates \(x=(z,\theta )\).
Let \((\ |\ )_{k\phi }\) be the inner product on the space \(\Omega ^{0,q}_0(D)\) defined as follows:
where \(f, g\in \Omega ^{0,q}_0(D)\). Let \(\overline{\partial }^{\,*,k\phi }_b:\Omega ^{0,q+1}(D)\rightarrow \Omega ^{0,q}(D)\) be the formal adjoint of \(\overline{\partial }_b\) with respect to \(( \ |\ )_{k\phi }\). Put
Let \(u\in \Omega ^{0,q}(D, L^k)\). Then, there exists \(\widetilde{u}\in \Omega ^{0,q}(D)\) such that \(u=s^k\widetilde{u}\) and we have
Let \(U_1(z,\theta ),\ldots ,U_{n-1}(z,\theta )\) be an orthonormal frame of \(T^{1,0}_{(z,\theta )}X\) varying smoothly with \((z,\theta )\) in a neighborhood of \(p\). We take \(U_1,\ldots ,U_{n-1}\) so that \(U_j(0,0)=\frac{\partial }{\partial z_j}\), \(j=1,\ldots ,n-1\). Put
where \(a_{j,t}\in C^\infty \), \(j,t=1,\ldots ,n-1\), \(Z_1,\ldots ,Z_{n-1}\) are as in (4.1). Then, we have
Let \((e_j(z, \theta ))_{j=1,\ldots ,n-1}\) denote the basis of \(T^{*0,1}_{(z,\theta )}X\), dual to \((\overline{U}_j(z,\theta ))_{j=1,\ldots ,n-1}\). If \(w\in T^{*0,1}_zX\), let \((w\wedge )^*: \Lambda ^{0,q+1}T^*_zX\rightarrow \Lambda ^{0,q}T^*_zX,\ q\geqslant 0\), be the adjoint of the left exterior multiplication \(w\wedge : \Lambda ^{0,q}T^*_zX\rightarrow \Lambda ^{0,q+1}T^*_zX\), \(u\mapsto w\wedge u\):
for all \(u\in \Lambda ^{0,q}T^*_zX\), \(v\in \Lambda ^{0,q+1}T^*_zX\). Notice that \((w\wedge )^*\) depends \(\mathbb {C}\)-anti-linearly on \(w\). It is easy to see that
and correspondingly
where \(\overline{U}^{\,*,k\phi }_j\) is the formal adjoint of \(\overline{U}_j\) with respect to \((\ |\ )_{k\phi }\), \(j=1,\ldots ,n-1\). We can check that for \(j=1,\ldots ,n-1\),
where \(s_j\in C^\infty (D)\), \(s_j\) is independent of \(k\), \(j=1,\ldots ,n-1\).
For \(r>0\), let \(D_r=\left\{ x=(z, \theta )\in \mathbb {R}^{2n-1};\, \left| x_j\right| <r,\ j=1,\ldots ,2n-1\right\} \). Let \(F_k\) be the scaling map: \(F_k(z, \theta )=(\frac{z}{\sqrt{k}}, \frac{\theta }{k})\). From now on, we assume that \(k\) is large enough so that \(F_k(D_{\log k})\subset D\). We define the scaled bundle \(F^*_k\Lambda ^{0,q}T^*X\) on \(D_{\log k}\) to be the bundle whose fiber at \((z, \theta )\in D_{\log k}\) is
We take the Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle _{F^*_k}\) on \(F^*_k\Lambda ^{0,q}T^*X\) so that at each point \((z, \theta )\in D_{\log k}\),
is an orthonormal basis for \(F^*_k\Lambda ^{0,q}T^*_{(z,\theta )}X\). For \(r>0\), let \(F^*_k\Omega ^{0,q}(D_r)\) denote the space of smooth sections of \(F^*_k\Lambda ^{0,q}T^*X\) over \(D_r\). Let \(F^*_k\Omega ^{0,q}_0(D_r)\) be the subspace of \(F^*_k\Omega ^{0,q}(D_r)\) whose elements have compact support in \(D_r\). Given \(f\in \Omega ^{0,q}(F_k(D_{\log k}))\), we write \(f=\sum '_{\left| J\right| =q}f_Je_J\). We define the scaled form \(F_k^*f\in F^*_k\Omega ^{0,q}(D_{\log k})\) by:
Let \(P\) be a partial differential operator of order one on \(F_k(D_{\log k})\) with \(C^\infty \) coefficients. We write \(P=a(z, \theta )\frac{\partial }{\partial \theta }+\sum ^{2n-2}_{j=1}a_j(z, \theta )\frac{\partial }{\partial x_j}\), \(a, a_j\in C^\infty (F_k(D_{\log k}))\), \(j=1,\ldots ,2n-2\). The partial differential operator \(P_{(k)}\) on \(D_{\log k}\) is given by
Let \(f\in C^\infty (F_k(D_{\log k}))\). We can check that
The scaled differential operator \(\overline{\partial }_{b,(k)}:F^*_k\Omega ^{0,q}(D_{\log k})\rightarrow F^*_k\Omega ^{0,q+1}(D_{\log k})\) is given by (compare to the formula (4.6) for \(\overline{\partial }_b\)):
From (4.6) and (4.11), we can check that if \(f\in \Omega ^{0,q}(F_k(D_{\log k}))\), then
Let \((\ |\ )_{kF^*_k\phi }\) be the inner product on the space \(F^*_k\Omega ^{0,q}_0(D_{\log k})\) defined as follows:
where \(\mathrm{d}v_X=m\mathrm{d}v(z)\mathrm{d}\theta \) is the volume form, \(\mathrm{d}v(z)=2^{n-1}\mathrm{d}x_1\cdots \mathrm{d}x_{2n-2}\). Note that \(m(0,0)=1\). Let \(\overline{\partial }^{\,*,k\phi }_{b,(k)}:F^*_k\Omega ^{0,q+1}(D_{\log k})\rightarrow F^*_k\Omega ^{0,q}(D_{\log k})\) be the formal adjoint of \(\overline{\partial }_{b,(k)}\) with respect to \((\ |\ )_{kF^*_k\phi }\). Then, we can check that [compare the formula for \(\overline{\partial }^{\,*,k\phi }_b\), see (4.7) and (4.8)]
where \(s_j\in C^\infty (D_{\log k})\), \(j=1,\ldots ,n-1\), are independent of \(k\). We also have
We define now the scaled Kohn Laplacian:
From (4.13) and (4.15), we see that if \(f\in \Omega ^{0,q}(F_k(D_{\log k}))\), then
From (4.3), (4.4) and (4.1), we can check that
on \(D_{\log k}\), where \(\epsilon _k\) is a sequence tending to zero with \(k\rightarrow \infty \) and \(Z_{j,k}\) is a first order differential operator and all the derivatives of the coefficients of \(Z_{j,k}\) are uniformly bounded in \(k\) on \(D_{\log k}\), \(j=1,\ldots ,n-1\). Similarly, from (4.3), (4.4) and (4.1), it is straightforward to see that
on \(D_{\log k}\), where \(\delta _k\) is a sequence tending to zero with \(k\rightarrow \infty \) and \(V_{t,k}\) is a first order differential operator and all the derivatives of the coefficients of \(V_{t,k}\) are uniformly bounded in \(k\) on \(D_{\log k}\), \(t=1,\ldots ,n-1\). From (4.19), (4.18) and (4.16), (4.14), (4.12), it is straightforward to obtain the following
Proposition 4.1
We have that
on \(D_{\log k},\) where \(\varepsilon _k\) is a sequence tending to zero with \(k\rightarrow \infty ,\) \(P_k\) is a second order differential operator and all the derivatives of the coefficients of \(P_k\) are uniformly bounded in \(k\) on \(D_{\log k}.\)
Let \(D\subset D_{\log k}\) be an open set and let \(W^s_{kF^*_k\phi }(D,F^*_k\Lambda ^{0, q}T^*X)\), \(s\in {\mathbb {N}}_0\), denote the Sobolev space of order \(s\) of sections of \(F^*_k\Lambda ^{0,q}T^*X\) over \(D\) with respect to the weight \(\mathrm{e}^{-kF^*_k\phi }\). The Sobolev norm on this space is given by
where \(u=\sum '_{\left| J\right| =q}u_Je_J\big (\frac{z}{\sqrt{k}},\frac{\theta }{k}\big )\in W^s_{kF^*_k\phi } (D,F^*_k\Lambda ^{0,q}T^*X)\) and \(m\) is the volume form. If \(s=0\), we write \(\left\| \cdot \right\| _{kF^*_k\phi ,D}\) to denote \(\left\| \cdot \right\| _{kF^*_k\phi ,0,D}\). The following is well known (see [9, Proposition 2.4 and Lemma 2.5])
Proposition 4.2
Assume that \(Y(q)\) holds at each point of \(X\). For every \(r>0\) with \(D_{2r}\subset D_{\log k}\) and \(s\in {\mathbb {N}}_0,\) there are constants \(C_{r,s}>0,\) \(C_r>0,\) \(C_{r,s}\) and \(C_r\) are independent of \(k,\) such that
and
We pause and introduce some notations. We identify \(\mathbb {R}^{2n-1}\) with the Heisenberg group \(H_n:=\mathbb {C}^{n-1}\times \mathbb {R}\). We also write \((z, \theta )\) to denote the coordinates of \(H_n\), \(z=(z_1,\ldots ,z_{n-1})\in \mathbb {C}^{n-1}\), \(z_j=x_{2j-1}+ix_{2j}\), \(j=1,\ldots ,n-1\), and \(\theta \in \mathbb {R}\). Then,
are orthonormal bases for the bundles \(T^{1, 0}H_n\) and \(\mathbb {C}TH_n\), respectively. Then,
is the basis of \(\mathbb {C}T^*H_n\) which is dual to \(\{U_{j,H_n},\overline{U}_{j,H_n}, -T;j=1,\ldots ,n-1\}\). We take the Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on \(\Lambda ^{0,q}T^*H_n\) such that
is an orthonormal basis of \(\Lambda ^{0,q}T^*H_n\). The Cauchy–Riemann operator \(\overline{\partial }_{b,H_n}\) on \(H_n\) is given by
Put \(\phi _0(z, \theta )=\beta \theta +\sum ^{n-1}_{j,t=1}\mu _{j,\,t}\overline{z}_jz_t\in C^\infty (H_n,\mathbb {R})\), where \(\beta \) and \(\mu _{j,\,t}\), \(j,t=1,\ldots ,n-1\), are as in (4.1). Note that
Let \((\ |\ )_{\phi _0}\) be the inner product on \(\Omega ^{0,q}_0(H_n)\) defined as follows:
where \(\mathrm{d}v(z)=2^{n-1}\mathrm{d}x_1\mathrm{d}x_2\cdots \mathrm{d}x_{2n-2}\). Let \(\overline{\partial }^{\,*,\phi _0}_{b,H_n}:\Omega ^{0,q+1}(H_n)\rightarrow \Omega ^{0,q}(H_n)\) be the formal adjoint of \(\overline{\partial }_{b,H_n}\) with respect to \((\ |\ )_{\phi _0}\). We have
where
The Kohn Laplacian on \(H_n\) is given by
From (4.23), (4.25) and (4.26), we can check that
Now, we can prove
Proposition 4.3
Assume that \(Y(q)\) holds at each point of \(X\). For each \(k,\) let \(\alpha _k\in F^*_k\Omega ^{0,q}(D_{\log k})\). We assume that \(\left\| \alpha _k\right\| _{kF^*_k\phi ,D_{\log k}}\leqslant 1\) for each \(,k\) and there is a sequence \(\nu _k>0,\) \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty ,\) such that for each \(k,\)
Identify \(\alpha _k\) with a form on \(H_n\) by extending it with zero and write
Then\(,\) there is a subsequence \(\left\{ \alpha _{k_j}\right\} \) of \(\left\{ \alpha _k\right\} \) such that for each strictly increasing multiindex \(J,\) \(\left| J\right| =q,\) \(\alpha _{k_j,\,J}\) converges uniformly with all its derivatives on any compact subset of \(H_n\) to a smooth function \(\alpha _J.\) Furthermore\(,\) if we put \(\alpha =\sum '_{\left| J\right| =q}\alpha _J\mathrm{d}\overline{z}_J,\) then \(\Box ^{(q)}_{b,H_n}\alpha =0.\)
Proof
From (4.21) and using induction, we get for any \(r>0\) and for every \(s\in {\mathbb {N}}_0\), there is a constant \(C_{r,s}>0\) independent of \(k\), such that
for \(k\) large, where \(\widetilde{C}_{r,s}>0\) is independent of \(k\). Fix a strictly increasing multiindex \(J\), \(\left| J\right| =q\), and \(r>0\). Combining (4.29) with Rellich’s compactness theorem [15, p. 281], we conclude that there is a subsequence of \(\left\{ \alpha _{k,J}\right\} \), which converges in all Sobolev spaces \(W^s(D_r)\) for \(s>0\). From the Sobolev embedding theorem [15, p. 170], we see that the sequence converges in all \(C^l(D_r)\), \(l\geqslant 0\), \(l\in {\mathbb {Z}}\), locally uniformly. Choosing a diagonal sequence, with respect to a sequence of \(D_r\) exhausting \(H_n\), we get a subsequence \(\left\{ \alpha _{k_j,J}\right\} \) of \(\left\{ \alpha _{k,J}\right\} \) such that \(\alpha _{k_j,J}\) converges uniformly with all derivatives on any compact subset of \(H_n\) to a smooth function \(\alpha _J\).
Let \(J'\) be another strictly increasing multiindex, \(\left| J'\right| =q\). We can repeat the procedure above and get a subsequence \(\left\{ \alpha _{k_{j_s},J'}\right\} \) of \(\left\{ \alpha _{k_j,J'}\right\} \) such that \(\alpha _{k_{j_s},J'}\) converges uniformly with all derivatives on any compact subset of \(H_n\) to a smooth function \(\alpha _{J'}\). Continuing in this way, we get the first statement of the proposition.
Now, we prove the second statement of the proposition. Let \(P=(p_1,\ldots ,p_q)\), \(R=(r_1,\ldots ,r_q)\) be multiindices, \(\left| P\right| =\left| R\right| =q\). Define
For \(j, t=1,\ldots ,n-1\), define
We may assume that \(\alpha _{k,J}\) converges uniformly with all derivatives on any compact subset of \(H_n\) to a smooth function \(\alpha _J\), for all strictly increasing \(J\), \(\left| J\right| =q\). As (4.29), we have for any \(r>0\) and for every \(s\in {\mathbb {N}}_0\), there is a constant \(C_{r,s}>0\) independent of \(k\), such that
Put
Combining (4.30) with Sobolev embedding theorem [15, p. 170], we conclude that
From the explicit formula of \(\Box ^{(q)}_{b,k\phi ,(k)}\) (see Proposition 4.1), it is not difficult to see that for all strictly increasing \(J\), \(\left| J\right| =q\), we have
on \(D_{\log k}\), where \(\epsilon _k\) is a sequence tending to zero with \(k\rightarrow \infty \) , \(P_{k,J}\) is a second order differential operator and all the derivatives of the coefficients of \(P_{k,J}\) are uniformly bounded in \(k\) on \(D_{\log k}\) and \(\beta _{k,J}\) is as in (4.31). By letting \(k\rightarrow \infty \) in (4.32) we get
on \(H_n\), for all strictly increasing \(J\), \(\left| J\right| =q\). From this and the explicit formula of \(\Box ^{(q)}_{b,H_n}\) [see (4.28)], we conclude that \(\Box ^{(q)}_{b,H_n}\alpha =0\). The proposition follows. \(\square \)
5 The operators \(Q^{(0)}_{M,k}\) and \(Q^{(1)}_{M,k}\)
From now on, we assume that \(h^L\) is semi-positive on \(X\) and positive at some point of \(X\) and conditions \(Y(0)\) and \(Y(1)\) hold at each point of \(X\).
Take \(\delta _0>0\) be a small constant so that
Take \(\psi (\eta )\in C^\infty _0(]-\delta _0,\delta _0[,\overline{\mathbb {R}}_+)\) so that \(\psi (\eta )=1\) if \(-\frac{\delta _0}{2}\le \eta \le \frac{\delta _0}{2}\). Let \(\hat{\psi }(t)=\int \mathrm{e}^{-it\eta }\psi (\eta )\mathrm{d}\eta \) be the Fourier transform of \(\psi \). Put
Let \(E>0\) be a small constant so that
Fix \(M>0\) be a large constant so that
and
where \({1\!\!1}_{\mathbb {R}_{x,1}}(\xi )=1\) if \(\xi \in \mathbb {R}_{x,1}\), \({1\!\!1}_{\mathbb {R}_{x,1}}(\xi )=0\) if \(\xi \notin \mathbb {R}_{x,1}\). Take \(\chi (t)\in C^\infty _0(]-2,2[,\overline{\mathbb {R}}_+)\) so that \(0\le \chi (t)\le 1\) and \(\chi (t)=1\) if \(-1\le t\le 1\) and \(\chi (-t)=\chi (t)\), for all \(t\in \mathbb {R}\). Put
As before, let \(\Phi ^t(x)\) be the \(T\)-flow. The operator \(Q^{(0)}_{M,k}\) is a continuous operator \(C^\infty (X,L^k)\rightarrow C^\infty (X,L^k)\) defined as follows. Let \(u\in C^\infty (X,L^k)\). Let \(D\Subset D'\Subset X\) be open sets of \(X\) and let \(s\) be a local section of \(L\) on \(D'\), \(\left| s\right| ^2_{h^L}=\mathrm{e}^{-\phi }\). On \(D'\), we write \(u=s^k\widetilde{u}\), \(\widetilde{u}\in C^\infty (D')\). Then,
We first notice that for \(k\) large, \(\Phi ^{\frac{t}{k}}(x)\) is well defined for all \(t\in \mathrm{Supp\,}\chi _M\), every \(x\in X\) and \(\Phi ^{\frac{t}{k}}(x)\in D'\) for all \(t\in \mathrm{Supp\,}\chi _M\), every \(x\in D\). We may assume that \(\Phi ^{\frac{t}{k}}(x)\) is well defined for all \(t\in \mathrm{Supp\,}\chi _M\), every \(x\in X\) and \(\Phi ^{\frac{t}{k}}(x)\in D'\) for all \(t\in \mathrm{Supp\,}\chi _M\), every \(x\in D\). Now, we check that the definition above is independent of the choice of local sections. Let \(\hat{s}\) be another local section of \(L\) on \(D'\), \(\left| \hat{s}\right| ^2_{h^L}=\mathrm{e}^{-\hat{\phi }}\). Then, we have \(\hat{s}=gs\) for some non-zero rigid CR function \(g\). We can check that
Let \(u\in C^\infty (X,L^k)\). On \(D\), we write \(u=s^k\widetilde{u}=\hat{s}^k\hat{u}\). We have
From (5.8) and (5.9), we can check that
Since \(Tg=0\), we have \((\left| g\right| ^kg^{-k})(\Phi ^{\frac{t}{k}}x)=(\left| g\right| ^kg^{-k})(x)\) for all \(t\in \mathrm{Supp\,}\chi _M\), \(x\in D\). From this observation and (5.10), it is easy to see that
Furthermore, we can check that
Combining this with (5.11), we obtain
Thus, the definition of \(Q^{(0)}_{M,k}\) is well defined.
We consider \((0,1)\) forms. The operator \(Q^{(1)}_{M,k}\) is a continuous operator
defined as follows. Let \(D\) be an open set of \(X\). We assume that there exist canonical coordinates \(x\) defined in some neighborhood \(W\) of \(\overline{D}\) and \(L\) is trivial on \(W\). Let \(\psi (\eta )\) and \(\chi _M\) be as in (5.7). For \(k\) large, we have
Let \(s\) be a local section of \(L\) on \(W\), \(\left| s\right| ^2_{h^L}=\mathrm{e}^{-\phi }\). Let \(x=(x_1,\ldots ,x_{2n-1})=(z,\theta )\) be canonical coordinates on \(W\). Then,
where \(Z_j(x)\), \(j=1,\ldots ,n-1\), form a basis of \(T^{1,0}_xX\), for each \(x\in D\), and \(\varphi (z)\in C^\infty (D,\mathbb {R})\) independent of \(\theta \). We can check that \(\mathrm{d}\overline{z}_j\), \(j=1,\ldots ,n-1\), is the basis of \(T^{*0,1}X\), dual to \(\overline{Z}_j\), \(j=1,\ldots ,n-1\). Let \(u\in \Omega ^{0,1}(X,L^k)\). On \(W\), we write
Then,
As before, we can show that the definition (5.13) is independent of the choices of local sections. Now, we check that the definition (5.13) is independent of the choice of canonical coordinates. Let \(y=(y_1,\ldots ,y_{2n-1})=(w,\gamma )\), \(w_j=y_{2j-1}+iy_{2j}\), \(j=1,\ldots ,n-1\), \(\gamma =y_{2n-1}\), be another canonical coordinates on \(W\). Then,
where \(\widetilde{Z}_j(y)\), \(j=1,\ldots ,n-1\), form a basis of \(T^{1,0}_yX\), for each \(y\in D\), and \(\widetilde{\varphi }(w)\in C^\infty (D,\mathbb {R})\) independent of \(\gamma \). From (5.14) and (5.12), it is not difficult to see that on \(W\), we have
where for each \(j=1,\ldots ,n-1\), \(H_j(z)\) is holomorphic. From (5.15), we can check that
From this observation, we have for \(u\in \Omega ^{0,1}(X,L^k)\),
On \(D\), we have \(\Phi ^{\frac{t}{k}}(x)=(z,\frac{t}{k}+\theta )\), \(\Phi ^{\frac{t}{k}}(y)=(w,\frac{t}{k}+\gamma )\) and \(\frac{\partial H_j}{\partial z_l}(\Phi ^{\frac{t}{k}}(z))=\frac{\partial H_j}{\partial z_l}(z)\), \(j,l=1,\ldots ,n-1\), \(t\in \mathrm{Supp\,}\chi _M\). From this observation and (5.17), (5.16), it is straightforward to see that
Thus, the definition (5.13) is independent of the choice of canonical coordinates. The operator \(Q^{(1)}_{M,k}\) is well defined.
Now, we claim that
We work with canonical coordinates \(x=(z,\theta )\) as (5.12). For \(u\in C^\infty (X,L^k)\), we can check that
on \(W\), where \(u=s^k\widetilde{u}\) on \(W\). Combining (5.19) with (5.13), (5.7) and notice that \(\frac{\partial \widetilde{u}}{\partial \theta }(\Phi ^t(x))=\frac{\partial }{\partial \theta }\bigr (\widetilde{u}(\Phi ^t(x))\bigr )\), it is easy to see that
Since \(\overline{\partial }_bT\phi =0\), we have \(\overline{Z}_j\phi (x)=\overline{Z}_j\phi (\Phi ^{\frac{t}{k}}(x))\), \(j=1,\ldots ,n-1\), \(t\in \mathrm{Supp\,}\chi _M\). From this and (5.20), (5.18) follows.
6 The asymptotic behavior of \((Q^{(0)}_{M,k}\varPi ^{(0)}_{k,\le k\nu _k})(x)\)
We will use the same notations as before. We recall that we work with the assumption that \(Y(0)\) and \(Y(1)\) hold at each point of \(X\). We first need
Theorem 6.1
For any sequence \(\nu _k>0\) with \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty ,\) there is a constant \(C>0\) independent of \(k,\) such that
and
for all \(x\in X,\) \(k>0\). Recall that \((Q^{(0)}_{M,k}\varPi ^{(0)}_{k,\le k\nu _k})(x)\) and \((Q^{(0)}_{M,k}\varPi ^{(0)}_{k,\le k\nu _k}\overline{Q^{(0)}_{M,k}})(x)\) are given by (2.13) and (2.14)\(,\) respectively.
Proof
Let \(\nu _k>0\) be any sequence with \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty \). Let \(f_j\in C^\infty (X,L^k)\), \(j=1,\ldots ,d_k\), be an orthonormal frame for \(\fancyscript{H}^0_{b,\le k\nu _k}(X,L^k)\). From (2.24), we see that for each \(x\in X\),
In view of (2.18), we see that there is a constant \(C>0\) independent of \(k\) such that
For \(\alpha \in \fancyscript{H}^0_{b,\le k\nu _k}(X,L^k)\), \(\left\| \alpha \right\| _{h^{L^k}}=1\), we have
where \(C>0\) is a constant independent of \(k\) and \(\alpha \). From (6.5) and (5.7), it is easy to see that there is a constant \(C_1>0\) independent of \(k\) such that
From (6.6) and (6.3), (6.1) follows.
We have
From (6.7), (6.4), (6.3) and (6.1), (6.2) follows. \(\square \)
Fix a point \(p\in X\). Let \(x=(x_1,\ldots ,x_{2n-1})=(z,\theta )\) be canonical coordinates of \(X\) defined in some small neighborhood \(D\) of \(p\) and let \(s\) be a local section of \(L\) on \(D\), \(\left| s\right| ^2_{h^L}=\mathrm{e}^{-\phi }\). We take \(x\) and \(s\) so that (4.1) holds. Until further notice, we work with the local coordinates \(x\) and the local section \(s\) and we will use the same notations as Sect. 4. We identify \(D\) with some open set in \(\mathbb {C}^{n-1}\times \mathbb {R}\). Put
\(u(z,\theta )\in C^\infty (\mathbb {C}^{n-1}\times \mathbb {R})\). We remind that \(\mathbb {R}_{p,0}\) is given by (2.15). Set
where \(\chi _1\in C^\infty \), \(0\le \chi _1\le 1\),
\(\chi _1(z,\theta )=1\) if \(\left| z\right| \le \frac{1}{2}\), \(\left| \theta \right| \le \frac{1}{2}\). We notice that
Thus, for \(k\) large, \(\mathrm{Supp\,}\alpha _k\in D\) and \(\alpha _k\) is well defined. The following is well known (see [9, section 5]).
Proposition 6.2
With the notations used above\(,\) we have
and there is a sequence \(\gamma _k>0,\) independent of the point \(p\) and tending to zero as \(k\rightarrow \infty ,\) such that
We have the following
Proposition 6.3
Let \(\nu _k>0\) be any sequence with \(\lim _{k\rightarrow \infty }\frac{\gamma _k}{\nu _k}=0\) and \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty ,\) where \(\gamma _k\) is as in (6.13). Let \(\alpha _k\) be as in (6.9). Let
Then\(,\)
and
Moreover\(,\) on \(D,\) we put
Fix \(r>0\). Then\(,\) for every \(\varepsilon >0,\) there is a \(k_0>0\) such that for all \(k\ge k_0,\) we have \(F_k(D_{2r})\subset D\) and
In particular\(,\)
Proof
From (2.9), we have
as \(k\rightarrow \infty \). Thus, \(\lim _{k\rightarrow \infty }\left\| \alpha ^2_k\right\| _{h^{L^k}}=0\). Since \(\left\| \alpha _k\right\| _{h^{L^k}}\rightarrow 1\) as \(k\rightarrow \infty \), (6.15) follows.
Now, we prove (6.18). As (6.17), on \(D\), we write \(\alpha ^2_k=s^kk^{\frac{n}{2}}\beta ^2_k\), \(\beta ^2_k\in C^\infty (D)\). From (4.22), we know that
where \(C_r>0\) is independent of \(k\). Now, we have
Moreover, from (4.17), it is easy to can check that for all \(m\in {\mathbb {N}}\),
Here, we used (6.12). Combining (6.20) with (6.21) and (6.22), (6.18) follows.
From (6.18), we deduce
From this and (6.10), (6.16) follows. \(\square \)
Now, we can prove
Theorem 6.4
Let \(\delta _k=\min \left\{ \mu _k,\gamma _k\right\} ,\) where \(\mu _k\) is as in Theorem 2.1 and \(\gamma _k\) is as in (6.13). Let \(\nu _k>0\) be any sequence with \(\lim _{k\rightarrow \infty }\frac{\delta _k}{\nu _k}=0\) and \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty \). Then\(,\)
for all \(x\in X,\) where \(\psi (\eta )\) is as in the discussion after (5.1) and \(\chi _M(t)\) is given by (5.6)\(,\) \(\hat{\psi }(t)=\int \mathrm{e}^{-it\eta }\psi (\eta )\mathrm{d}\eta .\) We remind that \(\mathrm{Supp\,}\psi \bigcap \mathbb {R}_{x,1}=\emptyset ,\) for every \(x\in X.\)
Proof
Let \(\nu _k>0\) be any sequence with \(\lim _{k\rightarrow \infty }\frac{\delta _k}{\nu _k}=0\) and \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty \). Fix a point \(p\in X\). Let \(x=(x_1,\ldots ,x_{2n-1})=(z,\theta )\) be canonical coordinates of \(X\) defined in some small neighborhood \(D\) of \(p\) and let \(s\) be a local section of \(L\) on \(D\), \(\left| s\right| ^2_{h^L}=\mathrm{e}^{-\phi }\). As before we take \(x\) and \(s\) so that (4.1) hold and let \(\alpha ^1_k\in \fancyscript{H}^0_{b,\le k\nu _k}(X,L^k)\) be as in (6.14). We take
to be an orthonormal frame for \(\fancyscript{H}^0_{b,\le k\nu _k}(X,L^k)\). From (6.15), (6.16) and (2.19), we conclude that
Thus,
Now,
From (6.1) and (6.26), we have
Combining (6.28) with (6.27), we conclude that
Let \(\alpha ^2_k\) be as in (6.14). From (6.18) and the definition of \(Q^{(0)}_{M,k}\) [see (5.7)], it is not difficult to see that
Combining (6.30) with (6.29) and (6.15), we deduce
where \(\alpha _k\) is as in (6.9). On \(D\), we put
By the definitions of \(Q^{(0)}_{M,k}\) and \(\alpha _k\) [see (5.7) and (6.9)], we can check that
We notice that \(\frac{k}{2}\phi (0,\frac{t}{k})=\frac{\beta }{2}t+\epsilon _k(t)\), where \(\epsilon _k(t)\rightarrow 0\) as \(k\rightarrow \infty \), uniformly on \(\mathrm{Supp\,}\chi _M\) and \(\chi _1(0,\frac{t}{\sqrt{k}\log k})\rightarrow 1\) as \(k\rightarrow \infty \), uniformly on \(\mathrm{Supp\,}\chi _M\). Combining this observation with (6.33), (6.9) and (6.8), we can check that
where \(\hat{\psi }(t):=\int \mathrm{e}^{-it\eta }\psi (\eta )\mathrm{d}\eta \), \(u\) is as in (6.8). From (6.34) and (6.31), (6.24) follows. We get Theorem 6.4. \(\square \)
We need
Theorem 6.5
Let \(\delta _k>0,\) \(\delta _k\rightarrow 0,\) as \(k\rightarrow \infty ,\) be as in Theorem 6.4 and let \(\nu _k>0\) be any sequence with \(\lim _{k\rightarrow \infty }\frac{\delta _k}{\nu _k}=0\) and \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty .\) Then\(,\) there is a \(k_0>0\) such that for all \(k\ge k_0,\)
Proof
For each \(x\in X\), put
From (6.2), (6.24) and the Lebesgue dominated Theorem, we conclude that
and hence
We first claim that for each \(x\in X\), \(C(x)\) is real. We notice that \(\overline{\hat{\psi }(t)}=\hat{\psi }(-t)\) and \(\chi _M(t)=\chi _M(-t)\). From this observation, we can check that
Thus, \(C(x)\) is real.
Now, we claim that \(\int _XC(x)\mathrm{d}v_X(x)\) is positive and
We have
Here, we used Fourier’s inversion formula. Since \(0\le \chi _M\le 1\) and \(\chi _M=1\) if \(-M\le t\le M\), we have
where \(C_0=\sup _{t\in \mathbb {R}}t^2\left| \hat{\psi }(t)\right| \). Combining (6.40) with (6.39), we get
7 The asymptotic behavior of \(\Big (Q^{(1)}_{M,k}\varPi ^{(1)}_{k,\le k\nu _k}\overline{Q^{(1)}_{M,k}}\Big )(x)\)
We will use the same notations as before. Fix \(p\in X\). Let \(x=(x_1,\ldots ,x_{2n-1})=(z,\theta )\) be canonical coordinates of \(X\) defined in some small neighborhood \(D\) of \(p\) and let \(s\) be a local section of \(L\) on \(D\), \(\left| s\right| ^2_{h^L}=\mathrm{e}^{-\phi }\). We take \(x\) and \(s\) so that (4.1) hold. Until further notice, we work with the local coordinates \(x\) and the local section \(s\). We also write \(t\) to denote the coordinate \(\theta \). We identify \(D\) with some open set in \(H_n=\mathbb {C}^{n-1}\times \mathbb {R}\). Let \(\nu _k>0\) be any sequence with \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty \). We are going to estimate \(\limsup _{k\rightarrow \infty }k^{-n}(Q^{(1)}_{M,k}\varPi ^{(1)}_{k,\le k\nu _k}\overline{Q^{(1)}_{M,k}})(p)\). For the convenience of the reader we recall some notations we used before. Let \(e_j(z,\theta )\), \(j=1,\ldots ,n-1\), denote the basis of \(T^{*(0,1)}X\), dual to \(\overline{U}_j(z,\theta )\), \(j=1,\ldots ,n-1\), where \(U_j\), \(j=1,\ldots ,n-1\), are as in (4.3). For \(f\in \Omega ^{0,1}(X,L^k)\), we write \(f=\sum ^{n-1}_{j=1}f_je_j\), \(f_j\in C^\infty (X,L^k)\), \(j=1,\ldots ,n-1\). We call \(f_j\) the component of \(f\) along \(e_j\). As (2.21), for \(j=1,\ldots ,n-1\), we define
where \((Q^{(1)}_{M,k}\alpha )_j\) denotes the component of \(Q^{(1)}_{M,k}\alpha \) along \(e_j\). From (2.23), we know that
We consider \(H_n\). Let \(\psi (\eta )\) be as in the discussion after (5.1) and let \(\chi _M(t)\) be as in (5.6). The operator \(Q^{(1)}_{M,H_n}\) is a continuous operator \(\Omega ^{0,1}(H_n)\rightarrow \Omega ^{0,1}(H_n)\) defined as follows. Let \(u\in \Omega ^{0,1}(H_n)\). We write \(u=\sum ^{n-1}_{j=1}u_j\mathrm{d}\overline{z}_j\), \(u_j\in C^\infty (H_n)\), \(j=1,\ldots ,n-1\). Then,
We remind that \(\beta \) is as in (4.1). For \(j=1,\ldots ,n-1\), put [compare (7.1)]
where
and
We recall that \(\phi _0\) is as in the discussion after (4.23). We first need
Theorem 7.1
We have
Proof
Fix \(j\in \left\{ 1,2,\ldots ,n-1\right\} \). We claim that
The definition (7.1) of \((Q^{(1)}_{M,k}S^{(1)}_{k,\le k\nu _k,j}\overline{Q^{(1)}_{M,k}})(0)\) yields a sequence
such that \(\left\| \alpha _{k_s}\right\| _{h^{L^{k_s}}}=1\) and
where \((Q^{(1)}_{M,k_s}\alpha _{k_s})_j\) is the component of \(Q^{(1)}_{M,k_s}\alpha _{k_s}\) along \(e_j\). On \(D\), we write
and on \(D_{\log k_s}\), put
We recall that \(F^*_{k_s}\) is the scaling map given by (4.9). It is not difficult to see that
Moreover, from (4.17) and (4.2), it is straightforward to see that
Proposition 4.3 yields a subsequence \(\left\{ \gamma _{k_{s_u}}\right\} \) of \(\left\{ \gamma _{k_s}\right\} \) such that for each \(t\) in the set \(\left\{ 1,2,\ldots ,n-1\right\} \), \(\gamma _{k_{s_u},t}\) converges uniformly with all derivatives on any compact subset of \(H_n\) to a smooth function \(\gamma _t\), where \(\gamma _{k_{s_u},t}\) denotes the component of \(\gamma _{k_{s_u}}\) along \(e_t(\frac{z}{\sqrt{k}},\frac{\theta }{k})\). Set \(\gamma =\sum ^{n-1}_{t=1}\gamma _t\mathrm{d}\overline{z}_t\). Then, we have \(\Box ^{(1)}_{b,H_n}\gamma =0\) and, by (4.24), \(\left\| \gamma \right\| _{\phi _0}\leqslant 1\). Thus,
where
We claim that
We write
Since \(e_t=\mathrm{d}\overline{z}_t+O(\left| (z,\theta )\right| \), \(t=1,\ldots ,n-1\), we conclude that for all \(t=1,\ldots ,n-1\),
Moreover, from the definition of \(Q^{(1)}_{M,k_{s_u}}\) [see (5.13)], it is easy to see that
Combining (7.10) with (7.9), (7.3) and notice that \(-\frac{k}{2}(F^*_k\phi )(0,t)\rightarrow -\frac{\beta }{2}t\), as \(k\rightarrow \infty \), uniformly on \(\mathrm{Supp\,}\chi _M\), (7.8) follows. The claim (7.5) follows from (7.6), (7.7) and (7.8). Finally, (7.5) and (7.2) imply the conclusion of the theorem. \(\square \)
To estimate \(\sum ^{n-1}_{j=1} (Q^{(1)}_{M,H_n}S^{(1)}_{j,H_n}\overline{Q^{(1)}_{M,H_n}})(0)\), we need the some preparation. Put
where \(\mu _{j,t}\), \(j,t=1,\ldots ,n-1\), are as in (4.1). Note that
For \(q=0,1,\ldots ,n-1\), we denote by \(L^2_{(0,q)}(H_n, \Phi _0)\) the completion of \(\Omega _0^{0,q}(H_n)\) with respect to the norm \(\Vert \cdot \Vert _{\Phi _0}\), where
Let \(u(z, \theta )\in \Omega ^{0,1}(H_n)\) with \(\left\| u\right\| _{\phi _0}=1\), \(\Box ^{(1)}_{b,H_n}u=0\). Put \(v(z, \theta )=u(z, \theta )\mathrm{e}^{-\frac{\beta }{2}\theta }\). We have
Choose \(\chi (\theta )\in C^\infty _0(\mathbb {R})\) so that \(\chi (\theta )=1\) when \(\left| \theta \right| <1\) and \(\chi (\theta )=0\) when \(\left| \theta \right| >2\) and set \(\chi _j(\theta )=\chi (\theta /j)\), \(j\in {\mathbb {N}}\). Let
From Parseval’s formula, we have
Thus, there is \(\hat{v}(z, \eta )\in L^2_{(0,1)}(H_n, \Phi _0)\) such that \(\hat{v}_j(z,\eta )\rightarrow \hat{v}(z, \eta )\) in \(L^2_{(0,1)}(H_n, \Phi _0)\). We have
We call \(\hat{v}(z, \eta )\) the Fourier transform of \(v(z, \theta )\) with respect to \(\theta \). Formally,
The following theorem is one of the main technical results in [9] (see [9, section 3], for the proof).
Theorem 7.2
With the notations used above. Let \(u(z, \theta )\in \Omega ^{0,1}(H_n)\) with \(\left\| u\right\| _{\phi _0}=1,\) \(\Box ^{(1)}_{b,H_n}u=0\) and let \(\hat{v}(z, \eta )\in L^2_{(0,1)}(H_n, \Phi _0)\) be the Fourier transform of the function \(u(z,\theta )\mathrm{e}^{-\frac{\beta }{2}\theta }\) with respect to \(\theta \) \((\)see the discussion before (7.14)\().\) Then, for almost all \(\eta \in \mathbb {R},\) we have \(\hat{v}(z,\eta )\) is smooth with respect to \(z\) and
and
for all \(z\in \mathbb {C}^{n-1}\).
Now, we can prove
Proposition 7.3
Let \(u(z, \theta )\in \Omega ^{0,1}(H_n)\) with \(\left\| u\right\| _{\phi _0}=1,\) \(\Box ^{(1)}_{b,H_n}u=0\). We have
where \(E\) is as in (5.3).
Proof
Let \(\varphi \in C^\infty _0(\mathbb {C}^{n-1},\mathbb {R})\) such that \(\int _{\mathbb {C}^{n-1}}\!\varphi (z)\mathrm{d}v(z)=1\), \(\varphi \geqslant 0\), \(\varphi (z)=0\) if \(\left| z\right| >1\). Put \(g_m(z)=m^{2n-2}\varphi (mz)\mathrm{e}^{\Phi _0(z)}\), \(m=1,2,\ldots \). Then, \(\int _{\mathbb {C}^{n-1}}\!g_m(z)\mathrm{e}^{-\Phi _0(z)}\mathrm{d}v(z)=1\) and
Choose \(\chi (t)\in C^\infty _0(\mathbb {R})\) so that \(\chi (t)=1\) when \(\left| t\right| <1\) and \(\chi (t)=0\) when \(\left| t\right| >2\) and set \(\chi _j(t)=\chi (t/j)\), \(j\in {\mathbb {N}}\). For each \(m\), we have
From Parseval’s formula, we can check that for each \(j\),
where \(\hat{v}_j(z,\eta )\) is as in (7.12) and
From (7.19) and (7.18), we obtain for each \(m\),
where \(\hat{v}(z,\eta )\) is as in (7.14). Now,
where
Combining this with (7.21), we have
Since \(\hat{v}(z,\eta )\in L^2_{(0,1)}(H_n,\Phi _0)\), it is easy to see that
From (7.15), we see that \(\hat{v}(z,\eta )=0\) almost everywhere on \(\mathbb {R}\setminus \mathbb {R}_{p,1}\), for every \(z\in \mathbb {C}^{n-1}\). Since \(\mathrm{Supp\,}\psi \bigcap \mathbb {R}_{p,1}=\emptyset \) [see the discussion after (5.1)], we conclude that for each \(m>0\),
From (7.23), (7.24) and Fubini’s theorem, we obtain
for every \(m>0\). From (7.25) and (7.22), we get for each \(m\),
Since \(0\le \chi _M\le 1\) and \(\chi _M=1\) if \(-M\le t\le M\), we have
where \(C_0=\sup _{t\in \mathbb {R}}t^2\left| \hat{\psi }(t)\right| \). Put
From (7.27) and (7.26), we have for each \(m\),
Combining (7.28) with (7.17) and (5.5), we get
where \(E\) is as in (5.3). (7.16) follows. \(\square \)
In view of Proposition 7.3, we have proved that for all \(u(z, \theta )\in \Omega ^{0,1}(H_n)\) with \(\left\| u\right\| _{\phi _0}=1\), \(\Box ^{(1)}_{b,H_n}u=0\), we have
for all \(j=1,\ldots ,n-1\), where \((Q^{(1)}_{M,H_n}u)(0)=\sum ^{n-1}_{j=1}(Q^{(1)}_{M,H_n}u)_j(0)\mathrm{d}\overline{z}_j\) and \(E\) is as in (5.3). Thus, for every \(j=1,\ldots ,n-1\), we have
and
From (7.29) and Theorem 7.1, we obtain the main result of this section
Theorem 7.4
Let \(\nu _k>0\) be any sequence with \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty \). For each \(x\in X,\) we have
where \(E\) is as in (5.3).
The proof of the following theorem is essentially the same as the proof of (6.1). We omit the proof.
Theorem 7.5
For any sequence \(\nu _k>0\) with \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty ,\) there is a constant \(C>0\) independent of \(k,\) such that
Now, we can prove
Theorem 7.6
Let \(\nu _k>0\) be any sequence with \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty \). Then\(,\) there is a \(k_0>0\) such that for all \(k\ge k_0,\)
where \(E\) is as in (5.3).
Proof
In view of Theorem 7.5, \(\sup _k k^{-n}Q^{(1)}_{M,k}\varPi ^{(1)}_{k,\le k\nu _k}\overline{Q^{(1)}_{M,k}})(\cdot )\) is integrable on \(X\). Thus, we can apply Fatou’s lemma and we get using Theorem 7.4:
The theorem follows. \(\square \)
8 The proof of Theorem 1.15
Let \(\delta _k>0\), \(\delta _k\rightarrow \infty \) as \(k\rightarrow \infty \), be as in Theorem 6.4 and let \(\nu _k>0\) be any sequence with \(\lim _{k\rightarrow \infty }\frac{\delta _k}{\nu _k}=0\) and \(\nu _k\rightarrow 0\) as \(k\rightarrow \infty \). Let \(\gamma _{1,k}<\gamma _{2,k}<\cdots <\gamma _{m_k,k}\) be all the distinct non-zero eigenvalues of \(\Box ^{(0)}_{b,k}\) between \(0\) and \(k\nu _k\). Thus, \(\gamma _{1,k}>0\) and \(\gamma _{m_k,k}\le k\nu _k\). We notice that \(\gamma _{j,k}\), \(j=1,\ldots ,m_k\), are also eigenvalues of \(\Box ^{(1)}_{b,k}\). For \(\mu \in \mathbb {R}\), let \(\fancyscript{H}^q_{b,\mu }(X,L^k)\) denote the space spanned by the eigenforms of \(\Box ^{(q)}_{b,k}\) whose eigenvalues are \(\lambda \). For each \(j\in \left\{ 1,\ldots ,m_k\right\} \), let \(f^1_{j,k},f^2_{j,k},\ldots ,f^{d_{j,k}}_{j,k}\) be an orthonormal basis for \(\fancyscript{H}^0_{b,\gamma _{j,k}}(X,L^k)\), where \(d_{j,k}=\mathrm{dim\,}\fancyscript{H}^0_{b,\gamma _{j,k}}(X,L^k)\). Let \(f^1_{0,k},f^2_{0,k},\ldots ,f^{d_{0,k}}_{0,k}\) be an orthonormal basis for \(\fancyscript{H}^0_b(X,L^k)\), where \(d_{0,k}=\mathrm{dim\,}\fancyscript{H}^0_b(X,L^k)\).
Let \(Q^{(0)}_{M,k}\) and \(Q^{(1)}_{M,k}\) be as in (5.7) and (5.13), respectively. By the definition of \((Q^{(0)}_{M,k}\varPi ^{(0)}_{k,\le k\nu _k})(x)\) [see (2.13)], we have
From (8.1) and (6.35), we conclude that
for \(k\) large. For \(j=1,\ldots ,m_k\), we put
For each \(j=1,\ldots ,m_k\),
Hence,
Since \(\left\| g^t_{j,k}\right\| ^2_{h^{L^k}}=\left\| f^t_{j,k}\right\| ^2_{h^{L^k}}=1\), for every \(j\) and \(t\), it is obviously that
Combining this with (8.4) and (8.2), we get
for \(k\) large.
We can check that for each \(j=1,\ldots ,m_k\), \(g^t_{j,k}\), \(t=1,\ldots ,d_{j,k}\) is an orthonormal basis of the space \(\overline{\partial }_{b,k}\fancyscript{H}^0_{b,\gamma _{j,k}}(X,L^k)\subset \fancyscript{H}^1_{b,\gamma _{j,k}}(X,L^k)\). From this observation and the definition of \((Q^{(1)}_{M,k}\varPi ^{(1)}_{k,\le k\nu _k}\overline{Q^{(1)}_{M,k}})(x)\) [see (2.14)], we conclude that
Thus,
Combining (8.7) with (7.32), we get
for \(k\) large, where \(E\) is as in (5.3).
From (2.19) and (2.18), we conclude that
for \(k\) large. It is obviously the case that
Combining this with (8.9), we get
for \(k\) large. From (8.8), (8.10), (8.5) and (5.3), we obtain
Theorem 8.1
Let \(f^1_{0,k},f^2_{0,k},\ldots ,f^{d_{0,k}}_{0,k}\) be an orthonormal basis for \(\fancyscript{H}^0_b(X,L^k),\) where \(d_{0,k}=\mathrm{dim\,}\fancyscript{H}^0_b(X,L^k).\) Then\(,\) for \(k\) large\(,\) we have
The following is straightforward
Lemma 8.2
For \(k\) large\(,\) there is a constant \(C>0\) independent of \(k,\) such that
Proof
Let \(D\Subset D'\Subset D''\Subset X\) be open sets of \(X\) and let \(s\) be a local section of \(L\) on \(D''\). We assume that there exist canonical coordinates \(x=(x_1,\ldots ,x_{2n-1})=(z,\theta )\) on \(D''\). Let \(\chi _M\) be as in (5.6). For \(k\) large, we have
and \(\mathrm{Supp\,}f(\Phi ^{\frac{t}{k}}x)\subset D'\), \(\forall t\in \mathrm{Supp\,}\chi _M\), \(\forall f\in C^\infty _0(D,L^k)\). In canonical coordinates \(x=(z,\theta )\), we have \(\Phi ^{\frac{t}{k}}(x)=(z,\frac{t}{k}+\theta )\). Let \(m(z,\theta )\mathrm{d}v(z)\mathrm{d}\theta \) be the volume form on \(D''\), where \(\mathrm{d}v(z)=2^{n-1}\mathrm{d}x_1\mathrm{d}x_2\ldots \mathrm{d}x_{2n-2}\). Since \(m(z,\theta )\) is strictly positive, for \(k\) large, there is a constant \(C_1>0\) independent of \(k\), such that
Let \(u\in C^\infty _0(D,L^k)\). On \(D''\), we write \(u=s^k\widetilde{u}\), \(\widetilde{u}\in C^\infty _0(D)\). From the definition of \(Q^{(0)}_{M,k}\) [see (5.7)], we can check that for \(k\) large,
where \(\widetilde{C}>0\), \(C>0\) are independent of \(k\) and \(u\) and \(C_1\) is as in (8.12). From (8.13) and using partition of unity, the lemma follows. \(\square \)
Proof of Theorem 1.15
From Lemma 8.2 and (8.11), we see that for \(k\) large,
where \(C>0\) is the constant as in Lemma 8.2 and \(d_{0,k}=\mathrm{dim\,}\fancyscript{H}^0_b(X,L^k)\). Theorem 1.15 follows. \(\square \)
9 Examples
In this section, some examples are collected. The aim is to illustrate the main results in some simple situations.
9.1 Compact Heisenberg groups
Let \(\lambda _1,\ldots ,\lambda _{n-1}\) be given non-zero integers. Let \({\fancyscript{C}}H_n=(\mathbb {C}^{n-1}\times \mathbb {R})/_\sim \), where \((z, \theta )\sim (\widetilde{z}, \widetilde{\theta })\) if
We can check that \(\sim \) is an equivalence relation and \({\fancyscript{C}}H_n\) is a compact manifold of dimension \(2n-1\). The equivalence class of \((z, \theta )\in \mathbb {C}^{n-1}\times \mathbb {R}\) is denoted by \([(z, \theta )]\). For a given point \(p=[(z, \theta )]\), we define \(T^{1, 0}_p{\fancyscript{C}}H_n\) to be the space spanned by
It is easy to see that the definition above is independent of the choice of a representative \((z, \theta )\) for \([(z, \theta )]\). Moreover, we can check that \(T^{1, 0}{\fancyscript{C}}H_n\) is a CR structure and \(T:=\frac{\partial }{\partial \theta }\) is a rigid global real vector field. Thus, \(({\fancyscript{C}}H_n, T^{1, 0}{\fancyscript{C}}H_n)\) is a compact generalized Sasakian CR manifold of dimension \(2n-1\). Let \(J\) denote the canonical complex structure on \(X\times \mathbb {R}\) given by \(J\frac{\partial }{\partial t}=T\), where \(t\) denotes the coordinate of \(\mathbb {R}\). We take a Hermitian metric \(\langle \,\cdot \,|\,\cdot \,\rangle \) on the complexified tangent bundle \(\mathbb {C}T{\fancyscript{C}}H_n\) such that
is an orthonormal basis. The dual basis of the complexified cotangent bundle is
The Levi form \({\mathcal L}_p\) of \({\fancyscript{C}}H_n\) at \(p\in {\fancyscript{C}}H_n\) is given by \({\mathcal L}_p=\sum ^{n-1}_{j=1}\lambda _j\mathrm{d}z_j\wedge \mathrm{d}\overline{z}_j\).
Now, we construct a generalized Sasakian CR line bundle \((L,J)\) over \({\fancyscript{C}}H_n\). Let \(L=(\mathbb {C}^{n-1}\times \mathbb {R}\times \mathbb {C})/_\equiv \) where \((z,\theta ,\eta )\equiv (\widetilde{z}, \widetilde{\theta }, \widetilde{\eta })\) if
where \(\alpha =(\alpha _1,\ldots ,\alpha _{n-1})=\widetilde{z}-z\), \(\mu _{j,t}=\mu _{t,j}\), \(j, t=1,\ldots ,n-1\), are given integers. We can check that \(\equiv \) is an equivalence relation and \((L,J)\) is a generalized Sasakian CR line bundle over \({\fancyscript{C}}H_n\). For \((z, \theta , \eta )\in \mathbb {C}^{n-1}\times \mathbb {R}\times \mathbb {C}\), we denote \([(z, \theta , \eta )]\) its equivalence class. It is straightforward to see that the pointwise norm
is well defined. In local coordinates \((z, \theta , \eta )\), the weight function of this metric is
We can check that \(T\phi =-2\). Thus, \((L,J,h^L)\) is a rigid generalized Sasakian CR line bundle over \({\fancyscript{C}}H_n\). Note that
Thus, \(d(\overline{\partial }_b\phi -\partial _b\phi )=2\sum ^{n-1}_{j,t=1}\mu _{j,t}\mathrm{d}z_j\wedge \mathrm{d}\overline{z}_t\) and for any \(p\in {\fancyscript{C}}H_n\),
From this and Theorem 1.15, we obtain
Theorem 9.1
If the matrix \(\left( \mu _{j,t}\right) ^{n-1}_{j,t=1}\) is positive definite and \(Y(0),\) \(Y(1)\) hold on \({\fancyscript{C}}H_n,\) then for \(k\) large\(,\) there is a constant \(c>0\) independent of \(k,\) such that
9.2 Holomorphic line bundles over a complex torus
Let
be the flat torus. Let \(\lambda =\left( \lambda _{j,t}\right) ^{n}_{j,t=1}\), where \(\lambda _{j,t}=\lambda _{t,j}\), \(j, t=1,\ldots ,n\), are given integers. Let \(L_\lambda \) be the holomorphic line bundle over \(T_n\) with curvature the \((1,1)\)-form \(\Theta _\lambda =\sum ^n_{j,t=1}\lambda _{j,t}\mathrm{d}z_j\wedge \mathrm{d}\overline{z}_t\). More precisely, \(L_\lambda :=(\mathbb {C}^n\times \mathbb {C})/_\sim \), where \((z, \theta )\sim (\widetilde{z}, \widetilde{\theta })\) if
We can check that \(\sim \) is an equivalence relation and \(L_\lambda \) is a holomorphic line bundle over \(T_n\). For \([(z, \theta )]\in L_\lambda \), we define the Hermitian metric by
and it is easy to see that this definition is independent of the choice of a representative \((z, \theta )\) of \([(z, \theta )]\). We denote by \(\phi _\lambda (z)\) the weight of this Hermitian fiber metric. Note that \(\partial \overline{\partial }\phi _\lambda =\Theta _\lambda \).
Let \(L^*_\lambda \) be the dual bundle of \(L_\lambda \) and let \(\left\| \,\cdot \,\right\| _{L^*_\lambda }\) be the norm of \(L^*_\lambda \) induced by the Hermitian fiber metric on \(L_\lambda \). Consider the compact CR manifold of dimension \(2n+1\): \(X=\{v\in L^*_\lambda ;\, \left\| v\right\| _{L^*_\lambda }=1\}\); this is the boundary of the Grauert tube associated to \(L^*_\lambda \). The manifold \(X\) is equipped with a natural \(S^1\)-action. Locally, \(X\) can be represented in local holomorphic coordinates \((z,\eta )\), where \(\eta \) is the fiber coordinate, as the set of all \((z,\eta )\) such that \(\left| \eta \right| ^2\mathrm{e}^{\phi _\lambda (z)}=1\). The \(S^1\)-action on \(X\) is given by \(\mathrm{e}^{i\theta }\circ (z,\eta )=(z,\mathrm{e}^{i\theta }\eta )\), \(\mathrm{e}^{i\theta }\in S^1\), \((z,\eta )\in X\). Let \(T\) be the global real vector field on \(X\) determined by \(Tu(x)=\frac{\partial }{\partial \theta }u(\mathrm{e}^{i\theta }\circ x)\big |_{\theta =0}\), for all \(u\in C^\infty (X)\). We can check that \(T\) is a rigid global real vector field on \(X\). Thus, \(X\) is a compact generalized Sasakian CR manifold of dimension \(2n+1\). Let \(J\) denote the canonical complex structure on \(X\times \mathbb {R}\) given by \(J\frac{\partial }{\partial t}=T\), where \(t\) denotes the coordinate of \(\mathbb {R}\).
Let \(\pi :L^*_\lambda \rightarrow T_n\) be the natural projection from \(L^*_\lambda \) onto \(T_n\). Let \(\mu =\left( \mu _{j,t}\right) ^{n}_{j,t=1}\), where \(\mu _{j,t}=\mu _{t,j}\), \(j, t=1,\ldots ,n\), are given integers. Let \(L_\mu \) be another holomorphic line bundle over \(T_n\) determined by the constant curvature form \(\Theta _\mu =\sum ^n_{j,t=1}\mu _{j,t}\mathrm{d}z_j\wedge \mathrm{d}\overline{z}_t\) as above. The pullback line bundle \(\pi ^*L_\mu \) is a holomorphic line bundle over \(L^*_\lambda \). If we restrict \(\pi ^*L_\mu \) on \(X\), then we can check that \((\pi ^*L_\mu ,J)\) is a generalized Sasakian CR line bundle over \(X\).
The Hermitian fiber metric on \(L_\mu \) induced by \(\phi _\mu \) induces a Hermitian fiber metric on \(\pi ^*L_\mu \) that we shall denote by \(h^{\pi ^*L_\mu }\). We let \(\psi \) to denote the weight of \(h^{\pi ^*L_\mu }\). The part of \(X\) that lies over a fundamental domain of \(T_n\) can be represented in local holomorphic coordinates \((z, \xi )\), where \(\xi \) is the fiber coordinate, as the set of all \((z, \xi )\) such that \(r(z, \xi ):=\left| \xi \right| ^2\exp (\sum ^n_{j,t=1}\lambda _{j,t}z_j\overline{z}_t)-1=0\) and the weight \(\psi \) may be written as \(\psi (z, \xi )=\sum ^n_{j,t=1}\mu _{j,t}z_j\overline{z}_t\). From this we see that \((\pi ^*L_\mu ,J,h^{\pi ^*L_\mu })\) is a rigid generalized Sasakian CR line bundle over \(X\). It is straightforward to check that for any \(p\in X\), we have \(M^\psi _p=\frac{1}{2}d(\overline{\partial }_b\psi -\partial _b\psi )(p)|_{T^{1, 0}X}=\sum ^n_{j,t=1}\mu _{j,t}\mathrm{d}z_j\wedge \mathrm{d}\overline{z}_t\). From this observation and Theorem 1.15, we obtain
Theorem 9.2
If the matrix \(\left( \mu _{j,t}\right) ^{n-1}_{j,t=1}\) is positive definite and \(Y(0),\) \(Y(1)\) hold on \(X,\) then for \(k\) large\(,\) there is a constant \(c>0\) independent of \(k,\) such that
References
Baouendi, M.-S., Rothschild, L.-P., Treves, F.: CR structures with group action and extendability of CR functions. Invent. Math. 83, 359–396 (1985)
Berman, R.: Holomorphic Morse inequalities on manifolds with boundary. Ann. Inst. Fourier (Grenoble) 55(4), 1055–1103 (2005)
Boutet de Monvel, L.: Intégration des équations de Cauchy-Riemann induites formelles, Séminaire Goulaouic-Lions-Schwartz 1974–1975; Équations aux derivées partielles linéaires et non linéaires, Centre Math., École Polytech., Paris, Exp. no. 9, p. 13 (1975)
Chen, S-C., Shaw, M-C.: Partial Differential Equations in Several Complex Variables. AMS/IP Studies in Advanced Mathematics, vol. 19, p. xii. American Mathematical Society (AMS)/International Press, Providence/Somerville (2001)
Davies, E.-B.: Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics, vol. 42, pp. 182. University Press, Cambridge (1995)
Demailly, J.-P.: Champs magnétiques et inegalités de Morse pour la \(d^{\prime \prime }\)-cohomologie. Ann. Inst. Fourier (Grenoble) 35, 189–229 (1985)
Folland, G.B., Kohn, J.J.: The Neumann Problem for the Cauchy–Riemann Complex. Annals of Mathematics Studies. vol. 75. Princeton University Press and University of Tokyo Press, Princeton (1972)
Hsiao, C.-Y.: Projections in several complex variables. Mém. Soc. Math. France, Nouv. Sér 123, 131 p. (2010)
Hsiao, C.-Y., Marinescu, G.: Szegö kernel asymptotics and Morse inequalities on CR manifolds. Math. Z 271, 509–553 (2012)
Hsiao, C.-Y., Marinescu, G.: Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles. Commun. Anal. Geom. 22(1), 1–108 (2014)
Kohn, J.J.: Boundaries of complex manifolds. In: Proceedings of the Conference on Complex Analysis (Minneapolis 1964), pp. 81–94 (1965)
Marinescu, G.: Asymptotic Morse inequalities for pseudoconcave manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23(1), 27–55 (1996)
Marinescu, G.: Existence of holomorphic sections and perturbation of positive line bundles over \(q\)-concave manifolds. Preprint available at arXiv:math.CV/0402041
Siu, Y.T.: A vanishing theorem for semipositive line bundles over non-Kähler manifolds. J. Differ. Geom. 20, 431–452 (1984)
Yosida, K.: Functional Analysis, Reprint of the 6th edn, p. xiv. Springer, Berlin (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
C.-Y. Hsiao is partially supported by the DFG funded Project MA 2469/2-1.
Rights and permissions
About this article
Cite this article
Hsiao, CY. Existence of CR sections for high power of semi-positive generalized Sasakian CR line bundles over generalized Sasakian CR manifolds. Ann Glob Anal Geom 47, 13–62 (2015). https://doi.org/10.1007/s10455-014-9434-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-014-9434-0
Keywords
- Szegö kernel asymptotics
- Bergman kernel asymptotics
- CR manifolds
- CR line bundles
- Complex variables
- CR Grauert–Riemenschneider conjecture