MIRROR SYMMETRY FOR LATTICE POLARIZED K3 SURFACES
I. V. Dolgachev
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
MS3 For some open subset U of oo, for any V E U n 5 ~-, the Laurent expansion of the canonical symmetric
S3(HI(V, ev)) --4
(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
H2(V'), V I E :7
MS4 The period map induces a holomorphic multivalued mapping from the subset Uft5 v to the tube domain
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  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 . 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  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 , 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