mirror - M I R R O R S Y M M E T R Y F O R L A T T I C E P...

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MIRROR SYMMETRY FOR LATTICE POLARIZED K3 SURFACES I. V. Dolgachev UDC 512.774.5 Introduction There has been a recent explosion in the number of mathematical publications due to the discovery of a certain duality between some families of Calabi-Yau threefolds made by a group of theoretical physicists (see [11, 26] for references). Roughly speaking, this duality, called mirror symmetry, pairs two families ~-and .T'* of Calabi-Yau threefolds in such a way that the following properties are satisfied: MS1 The choice of the mirror family ~* involves the choice of a boundary point c~ of a compactification ~" of the moduli space for ~" at which the monodromy is "maximally unipotent." MS2 For each V E ~" and W E .T* the H0dge numbers satisfy hl,l(V) = h2, (V) = MS3 For some open subset U of oo, for any V E U n 5 ~-, the Laurent expansion of the canonical symmetric trilinear form S3(HI(V, ev)) --4 H~ | (the Griffiths-Yukawa cubic) at oo can be identified, after some special choice of local parameters and a basis of H~ with the quantum intersection form on H2(V'), V I E :7 c*. MS4 The period map induces a holomorphic multivalued mapping from the subset Uft5 v to the tube domain H2(V ', R) + ilCv, where/Cv, is the Ks cone of V ~ E 5 v* (the mirror mapping). Although known to some experts but never stated explicitly, it is a fact that mirror symmetry is a very beautiful and nontrivial (in many respects still hypothetical) generalization to the next dimension of the du- ality for K3 surfaces discovered almost 20 years ago by H. Pinkham [34] and independently by the author and V. Nikulin [8, 9, 31]. This duality was used to explain Arnold's Strange Duality for exceptional unimodal critical points [1]. There are repeated hints on the relationship between the latter duality and mirror symme- try in the physics ([2, 14, 23]) and mathematics literature ([6, 19, 35, 41]). Some of the results of this paper were independently obtained in [3, 19, 21, 27, 35] and some must be known to V. Batyrev and V. Nikulin. The paper [40] of Todorov is probably most relevant. Nevertheless I believe that it is worthwhile to give a detailed account of how the ideas of Arnold's strange duality allow one to state (and prove) precise analogs of properties MS1-MS4 for K3 surfaces. Note that property MS2 says that the local moduli number of V E ~" is equal to the second Betti number of V ~ E 5 v*. In the case of K3 surfaces, the first number is always equal to 20, and the second number is equal to 22. The key observation is that in the three-dimensional case the second Betti number is equal to the rank of the Picard group of algebraic cycles. This suggests that one create different moduli families of K3 surfaces with a condition on the Picard group. The simplest realization of this idea is based on the notion of a polarized K3 surface. This is a pair (X, h), where X is a K3 surface and h E Pic(X) is an ample (or pseudo-ample) divisor class. A generalization of this notion, due to V. Nikulin [30], is the notion of a lattice polarized K3 surface. We fix a lattice M (a free abelian group equipped with an integral quadratic form) and consider a pair (X,j),
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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mirror - M I R R O R S Y M M E T R Y F O R L A T T I C E P...

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