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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 523 (3 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 258 (36 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
, 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 199 (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.]
Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties
, 1995
"... We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety V or a Calabi–Yau hypersurface M ⊂ V. In the linear model the instanton moduli spaces are relatively simple objects and ..."
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Cited by 162 (14 self)
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We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety V or a Calabi–Yau hypersurface M ⊂ V. In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth V, our results reproduce and clarify an algebraic solution of the V model due to Batyrev. In addition, we find an algebraic relation determining the solution for M in terms of that for V. Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the M model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured “monomialdivisor mirror map” of Aspinwall, Greene, and Morrison.
Mirror symmetry, mirror map and applications to complete . . .
 EXPERIMENTAL NUCLEAR PHYSICS B
, 1995
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