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 CalabiYau 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 CalabiYau 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 GriffithsYukawa 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 MS1MS4 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 threedimensional 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 pseudoample)
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),