By Alexander Polishchuk

This e-book is a contemporary therapy of the idea of theta services within the context of algebraic geometry. the newness of its technique lies within the systematic use of the Fourier-Mukai rework. Alexander Polishchuk begins via discussing the classical idea of theta features from the point of view of the illustration thought of the Heisenberg team (in which the standard Fourier remodel performs the favourite role). He then exhibits that during the algebraic method of this concept (originally as a result of Mumford) the Fourier-Mukai rework can frequently be used to simplify the present proofs or to supply thoroughly new proofs of many very important theorems. This incisive quantity is for graduate scholars and researchers with powerful curiosity in algebraic geometry.

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**Sample text**

I) The theta series π α S(v, v) θ H, ,L (v) = exp 2 × α(γ ) · exp π(H − S)(v, γ ) − γ ∈ / ∩L π (H − S)(γ , γ ) . 1) is well deﬁned and is absolutely convergent (uniformly on compacts in V ). α (ii) θ H, ,L is a nonzero element of the space T (H, , α) of canonical theta functions. Proof. 1) depends only on γ mod ∩ L. Indeed, if we change γ to γ + γ1 where γ1 ∈ ∩ L, then this expression gets multiplied by π α(γ )−1 α(γ + γ1 ) exp − (H − S)(γ1 , γ ) . 2 But we have (H − S)(γ1 , γ ) = H (γ1 , γ ) − S(γ , γ1 ) = H (γ1 , γ ) − H (γ , γ1 ) = 2πi E(γ1 , γ ).

It follows that π H (v, δ) − l(v) = 2πi E(v, δ) + 2πim(v). But the LHS is C-linear and the RHS takes values in i R. It follows that both sides are zero which implies our claim. 1) as follows: π θ(v + δ) = A exp π H (v, δ) + H (δ, δ) θ(v). 2) 2 = + Zδ. We have seen that ⊂ ⊥ and the assumption δ ∈ Set implies that is stricly bigger than . 2) and from the condition θ ∈ T (H, , α) that in fact θ belongs to the space T (H, , α ) for some α extending α (see Exercise 6). 1. Hence, for a generic θ this situation does not occur.

3 of [18]. 2. For given data (H, , α), a Lagrangian subspace L compatible with ( , α) does not always exist. 2. We will also show there that for every given (H, ) as above, there exists α and a Lagrangian subspace L compatible with ( , α). 4. Lefschetz Theorem The standard application of the theory of theta functions is the following theorem of Lefschetz. 8. Let L be a holomorphic line bundle on the complex torus T = V / with c1 (L) = E; then for n ≥ 3 global holomorphic sections of L n deﬁne an embedding of T as a complex submanifold into P N .