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371
Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 343 (2 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
Localization of virtual classes
"... We prove a localization formula for the virtual fundamental class in the general context of C∗equivariant perfect obstruction theories. Let X be an algebraic scheme with a C∗action and a C∗equivariant perfect obstruction theory. The virtual fundamental class [X] vir in ..."
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Cited by 174 (26 self)
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We prove a localization formula for the virtual fundamental class in the general context of C∗equivariant perfect obstruction theories. Let X be an algebraic scheme with a C∗action and a C∗equivariant perfect obstruction theory. The virtual fundamental class [X] vir in
GromovWitten invariants in algebraic geometry, preprint
, 1996
"... GromovWitten invariants for arbitrary projective varieties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.] ..."
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Cited by 144 (2 self)
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GromovWitten invariants for arbitrary projective varieties and arbitrary genus are constructed using the techniques from [K. Behrend, B. Fantechi. The Intrinsic Normal Cone.]
Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato
, 1999
"... The deformation theory of associative algebras is a guide for developing the deformation theory of many algebraic structures. Conversely, all the ..."
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Cited by 112 (6 self)
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The deformation theory of associative algebras is a guide for developing the deformation theory of many algebraic structures. Conversely, all the
Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields
 Internat. Math. Res. Notices bf
, 1998
"... Abstract. We construct a generalization of the variations of Hodge structures on CalabiYau manifolds. It gives a Mirror partner for the theory of genus = 0 GromovWitten invariants. ..."
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Cited by 104 (4 self)
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Abstract. We construct a generalization of the variations of Hodge structures on CalabiYau manifolds. It gives a Mirror partner for the theory of genus = 0 GromovWitten invariants.
Mirror Symmetry, Mirror Map and Applications to CalabiYau Hypersurfaces
 Comm. Math. Phys
, 1995
"... Mirror Symmetry, PicardFuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of CalabiYau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa coup ..."
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Cited by 97 (23 self)
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Mirror Symmetry, PicardFuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of CalabiYau 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.
Equivariant GromovWitten invariants
 Internat. Math. Res. Notices
, 1996
"... The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the GromovWitten (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 co ..."
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Cited by 92 (10 self)
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The objective of this paper is to describe the construction and some applications of the equivariant counterpart to the GromovWitten (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 GWtheory 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 S1equivariant 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 fixedpoint 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 CalabiYau nfold and, respectively, complex and symplectic geometry on another CalabiYau nfold, called the mirror partner of the former one. The remarkable application [8]ofthe mirror conjecture to the enumeration of rational curves on CalabiYau 3folds (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 CalabiYau manifolds to more general compact symplectic manifolds if one admits non