Download PDF by Alexander Polishchuk: Abelian Varieties, Theta Functions and the Fourier Transform
By Alexander Polishchuk
This e-book is a contemporary remedy of the speculation of theta capabilities within the context of algebraic geometry. the newness of its procedure lies within the systematic use of the Fourier-Mukai rework. Alexander Polishchuk starts off through discussing the classical conception of theta capabilities from the point of view of the illustration idea of the Heisenberg team (in which the standard Fourier rework performs the famous role). He then indicates that during the algebraic method of this concept (originally as a result of Mumford) the Fourier-Mukai remodel can frequently be used to simplify the prevailing proofs or to supply thoroughly new proofs of many vital theorems. This incisive quantity is for graduate scholars and researchers with robust curiosity in algebraic geometry.
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Additional resources for Abelian Varieties, Theta Functions and the Fourier Transform
4 Indeed, splitting the series for θ11 ( τ2 + 14 , τ ) into two series, the sum over Appendix A. Theta Series and Weierstrass Sigma Function 39 even n and the sum over odd n, we get 1 3πi 3 τ + ,τ = τ+ (−1)n exp πi 4n 2 + 4n + θ11 2 4 4 4 n + exp πi(2n + 1)2 τ + 2πi n + n 1 2 2τ + 1 2 3πi (τ + 1) . 4 It remains to note that the ﬁrst sum is zero (as seen by substituting n → −n − 1). 3). 4) (1 − q n )2 n=1 where in the right-hand side we use multiplicative variables q = exp(2πiτ ), 1 u = exp(2πi z) (and where u 2 = exp(πi z)).
We will call a decomposition = 1 ⊕ 2 of the type described in the above proposition, an isotropic decomposition of . 2) for some c ∈ V which is uniquely determined modulo ⊥ . It is easy to see that the corresponding homomorphisms σα and σα are related as follows: σα (γ ) = (1, c)σα (γ )(1, c)−1 . 3) Therefore, we can deﬁne an isomorphism of the corresponding ﬁnite Heisenberg groups i c : G(E, , α) → G(E, , α ) : g → (1, c)g(1, c)−1 . 4) Now the operator U(1,c) (corresponding to the action of (1, c) on Fock representation) restricts to an isomorphism between T (H, , α) and T (H, , α ) compatible with the actions of G(E, , α) and G(E, , α ) via i c .
This is indeed true (see Exercise 4). Let us consider some examples of complex abelian varieties. Exercises 35 Examples. 1. If T = C/ is a 1-dimensional complex torus (complex elliptic curve) then for every nondegenerate Z-valued symplectic form E on the corresponding Hermitian form H on C is nondegenerate. Changing the sign of E if necessary we can achieve that H is positive. Hence, every complex elliptic curve is projective. 2. Let T = V / be a complex torus and T = V / → T be a ﬁnite unramiﬁed covering of T corresponding to a sublattice ⊂ of ﬁnite index.
Abelian Varieties, Theta Functions and the Fourier Transform by Alexander Polishchuk