We consider families F(∆) consisting of complex (n − 1)-dimensional projective algebraic compactifications of ∆-regular affine hypersurfaces Zf defined by Laurent polynomials f with a fixed n-dimensional Newton polyhedron ∆ in n-dimensional algebraic torus T = (C ∗ ) n. If the family F(∆) defined by a Newton polyhedron ∆ consists of (n − 1)-dimensional Calabi-Yau varieties then the dual, or polar, polyhedron ∆ ∗ in the dual space defines another family F( ∆ ∗ ) of Calabi-Yau varieties, so that we obtain the remarkable duality between two different families of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of Mirror Symmetry discovered by physicists for Calabi-Yau 3-folds. Our method allows to construct many new examples of Calabi-Yau 3-folds and new candidates for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families F(∆) and F( ∆ ∗). 1

The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the Gromov-Witten (GW) theory, i.e., intersection theory on spaces of (pseudo-) holomorphic curves in (almost) Kähler manifolds. Given a Killing action of a compact Lie group G on a compact Kähler manifold X, the equivariant GW-theory provides, as we will show in Section 3, the equivariant cohomology space H ∗ G (X) with a Frobenius structure (see [11]). We will discuss applications of the equivariant theory to the computation ([15], [18]) of quantum cohomology algebras of flag manifolds (Section 5), to the simultaneous diagonalization of the quantum cupproduct operators (Sections 7, 8), to the S1-equivariant Floer homology theory on the loop space LX (see Section 6 and [14], [13]), and to a “quantum ” version of the Serre duality theorem (Section 12). In Sections 9–11 we combine the general theory developed in Sections 1–6 with the fixed-point localization technique [21], in order to prove the mirror conjecture (in the form suggested in [14]) for projective complete intersections. By the mirror conjecture, one usually means some intriguing relations (discovered by physicists) between symplectic and complex geometry on a compact Kähler Calabi-Yau n-fold and, respectively, complex and symplectic geometry on another Calabi-Yau n-fold, called the mirror partner of the former one. The remarkable application [8]ofthe mirror conjecture to the enumeration of rational curves on Calabi-Yau 3-folds (1991, see the theorem below) raised a number of new mathematical problems—challenging tests of maturity for modern methods of symplectic topology. On the other hand, in 1993 I suggested that the relation between symplectic and complex geometry predicted by the mirror conjecture can be extended from the class of Calabi-Yau manifolds to more general compact symplectic manifolds if one admits non-

Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

Abstract. We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich’s stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced by any concavex bundles – including any direct sum of line bundles – on Pn. This includes proving the formula of Candelas-de la Ossa-Green-Parkes hence completing the program of Candelas et al, Kontesevich, Manin, and Givental, to compute rigorously the instanton prepotential function for the quintic in P4. We derive, among many other examples, the multiple cover formula for Gromov-Witten invariants of P1, computed earlier by Morrison-Aspinwall and by Manin in different approaches. We also prove a formula for enumerating Euler classes which arise in the so-called local mirror symmetry for some noncompact Calabi-Yau manifolds. At the end we interprete an infinite dimensional transformation group, called the mirror group, acting on Euler data, as a certain duality group of the linear sigma

We introduce the notion of stringy E-function for an arbitrary normal irreducible algebraic variety X with at worst log-terminal singularities. We prove some basic properties of stringy E-functions and compute them explicitly for arbitrary Q-Gorenstein toric varieties. Using stringy E-functions, we propose a general method to define stringy Hodge numbers for projective algebraic varieties with at worst Gorenstein canonical singularities. This allows us to formulate the topological mirror duality test for arbitrary Calabi-Yau varieties with canonical singularities. In Appendix we explain non-Archimedian integrals over spaces of arcs. We need these integrals for the proof of the main technical statement used in the definition of stringy Hodge numbers. 1

We propose a combinatorical duality for lattice polyhedra which conjecturally gives rise to the pairs of mirror symmetric families of Calabi-Yau complete intersections in toric Fano varieties with Gorenstein singularities. Our construction is a generalization of the polar duality proposed by Batyrev for the case of hypersurfaces. 1

We study aspects related to Kontsevich’s homological mirror symmetry conjecture [42] in the case of Calabi–Yau complete intersections in toric varieties. In a 1996 lecture, Kontsevich [43] indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in the homological mirror symmetry conjecture should also be identified. Our results provide an explicit geometric construction of the correspondence between the automorphisms of the two types of categories. We compare the monodromy calculations for the Picard–Fuchs system associated with the periods of a Calabi–Yau manifold M with the algebro-geometric computations of the cohomology action of Fourier– Mukai functors on the bounded derived category of coherent sheaves on the mirror Calabi–Yau manifold W. We obtain the complete dictionary between the two sides for the one complex parameter case of Calabi–Yau complete intersections in weighted projective spaces, as well as for some two parameter cases. We also find the complex of sheaves on W × W that corresponds to a loop in the moduli space of complex structures on M induced by a phase transition of W.

Abstract: We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror map in each case. For algebraic K3 surfaces, we solve the equation in terms of the J-function. By deriving explicit modular relations we prove that some K3 mirror maps are algebraic over the genus zero function field Q(J). This leads to a uniform proof that those mirror maps have integral Fourier coefficients. Regarding the maps as Riemann mappings, we prove that they are genus zero functions. By virtue of the Conway-Norton conjecture (proved by Borcherds using Frenkel-Lepowsky-Meurman’s Moonshine module), we find that these maps are actually the reciprocals of the Thompson series for certain conjugacy classes in the Griess-Fischer group. This also gives, as an immediate consequence, a second proof that those mirror maps are integral. We thus conjecture a surprising connection between K3 mirror maps and the Thompson series. For threefolds, we construct a formal nonlinear ODE for the quantum coupling reduced mod p. Under the mirror hypothesis and an integrality assumption, we derive mod p congruences for the Fourier coefficients. For the quintics, we deduce (at least for 5 ̸ |d) that the degree d instanton numbers nd are divisible by 53 – a fact first conjectured by Clemens.

We present a detailed study of the generalized hypergeometric system introduced by Gel’fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of Calabi-Yau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Gröbner basis for the so-called toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the special point called the large radius limit, we find a close relationship between the principal parts of the operators in the GKZ system and the intersection ring of a toric variety. As applications, we analyze general three dimensional hypersurfaces of Fermat and non-Fermat types with Hodge numbers up to h1,1 = 3. We also find and analyze several non Landau-Ginzburg models which are related to singular models.

In this paper we show that conifold transitions between Calabi-Yau 3-folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection Calabi-Yau 3-folds in Grassmannians. Using a natural degeneration of Grassmannians G(k,n) to some Gorenstein toric Fano varieties P(k,n) with conifolds singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for Calabi-Yau complete intersections X ⊂ G(k,n) of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational