Complex Tori pp 63-90 | Cite as

# Embeddings into Projective Space

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## 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 3*H*. 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* = *T*_{0}*X* 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*) = *h*^{0}(*M*). As we learned from R. R. Simha, this approach appears already in the work of Matsushima (see [Ma]).

## Keywords

Vector Space Line Bundle Projective Space Elliptic Curf Theta Function## Preview

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