Embeddings into Projective Space

Abstract

A complex torus is an abelian variety if and only if it admits a holomorphic embedding into some projective space. Hence a general complex torus does not admit a projective embedding. We will show in this chapter that if (X, H) is a nondegenerate complex torus of dimension g and index k, then X admits a differentiable embedding into projective space which is holomorphic in g — k variables and antiholomorphic in k variables. For this choose a line bundle L with first Chern class 3H. The vector space H^k(X,L) is the only nonvanishing cohomology group of L. It may be considered as the vector space of harmonic forms of bidegree (g — k, k) with values in L. Choosing a suitable metric of L, these forms yield the embedding X → ℙ _N . This embedding depends on the choice of a k-dimensional subvector space V of V = T₀X on which the hermitian form H is negative definite. This embedding comes out of the proof of the Riemann-Roch Theorem of [CAV], Chapter 3. It goes back to a trick of Wirtinger [Wi]: A suitable change of the complex structure of X defines in a canonical way a line bundle M which is positive definite and satisfies h^k(L) = h⁰(M). As we learned from R. R. Simha, this approach appears already in the work of Matsushima (see [Ma]).