By Alexander Polishchuk

The purpose of this ebook is to provide a contemporary therapy of the idea of theta features within the context of algebraic geometry. the newness of its strategy lies within the systematic use of the Fourier-Mukai rework. the writer starts off by means of discussing the classical concept of theta services from the perspective of the illustration conception of the Heisenberg workforce (in which the standard Fourier remodel performs the in demand role). He then indicates that during the algebraic method of this thought, the Fourier–Mukai remodel can frequently be used to simplify the present proofs or to supply thoroughly new proofs of many very important theorems. Graduate scholars and researchers with powerful curiosity in algebraic geometry will locate a lot of curiosity during this quantity.

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**Additional info for Abelian varieties and the Fourier transform**

**Example text**

3]). m; n/ and rank m C n, with discriminant lattice AL . L/ denote the minimal number of generators of AL . L/ Ä m C n 2, then any other lattice with the same rank, signature and discriminant lattice is isomorphic to L. Overlattices Now assume that L and M are two even lattices of the same rank, such that L embeds inside of M. Then we say that M is an overlattice of L. If we begin with a lattice M, then it is easy to compute all possible sublattices of maximal rank of L, but the problem of computing all possible overlattices of L is more subtle.

One may check that for a general choice of a; b; c, the discriminant vanishes simply at four other points in P1 , giving four further singular fibres of type I1 . 1/ D 24, as expected. For a singular fibre of type II , the root lattice Rp is isomorphic to . E8 /. Therefore, the lattice L for this K3 surface is isomorphic to H ˚ . E8 / ˚ . E8 / (recall that singular fibres of type I1 are irreducible, so do not contribute to L ). S; /tors is trivial, so Corollary 2 shows that S is in fact H ˚ . E8 / ˚ .

J. 116, 1–15 (1989) 35. : Abelian varieties attached to polarized K3 -surfaces. Math. Ann. 169, 239–242 (1967) 36. : Degenerations of K3 surfaces and Enriques surfaces. Math. USSR Izv. 11(5), 957–989 (1977) 37. : On modifications of degenerations of surfaces with Ä D 0. Math. USSR Izv. 17(2), 339–342 (1981) 38. : On minimally elliptic singularities. Am. J. Math. 99(6), 1257–1295 (1977) 42 A. Harder and A. Thompson 39. : The KSBA compactification for the moduli space of degree two K3 pairs (2012, preprint).

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