Embeddings into Projective Space

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 Hk(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 hk(L) = h0(M). As we learned from R. R. Simha, this approach appears already in the work of Matsushima (see [Ma]).