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161
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
Compactifying the space of stable maps
 electronic), 2002. OLSSON AND STARR
"... Abstract. In this paper we study a notion of twisted stable map, from a curve to a tame Deligne–Mumford stack, which generalizes the wellknown notion of stable map to a projective variety. Contents ..."
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Cited by 182 (23 self)
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Abstract. In this paper we study a notion of twisted stable map, from a curve to a tame Deligne–Mumford stack, which generalizes the wellknown notion of stable map to a projective variety. Contents
Gromov–Witten theory of Deligne–Mumford stacks
, 2006
"... 2. Chow rings, cohomology and homology of stacks 5 3. The cyclotomic inertia stack and its rigidification 10 4. Twisted curves and their maps 18 ..."
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Cited by 129 (10 self)
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2. Chow rings, cohomology and homology of stacks 5 3. The cyclotomic inertia stack and its rigidification 10 4. Twisted curves and their maps 18
Gröbner geometry of Schubert polynomials
 Ann. Math
"... Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix S ..."
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Cited by 99 (15 self)
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Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix Schubert varieties” are Gröbner bases for any antidiagonal term order. The Schubert polynomials are therefore positive sums of monomials, each monomial representing the torusequivariant cohomology class of a component (a schemetheoretically reduced coordinate subspace) in the limit of the resulting Gröbner degeneration. Interpreting the Hilbert series of the flat limit in equivariant Ktheory, another corollary of the proof is that Grothendieck polynomials represent the classes of Schubert varieties in Ktheory of the flag manifold. An inductive procedure for listing the limit coordinate subspaces is provided by the proof of the Gröbner basis property, bypassing what has come to be known as Kohnert’s conjecture [Mac91]. The coordinate subspaces, which are
Cycle groups for Artin stacks
 Invent. Math
, 1999
"... 2. Definition and basic properties 2 3. Elementary intersection theory 16 4. Extended excision axiom 22 ..."
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Cited by 93 (3 self)
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2. Definition and basic properties 2 3. Elementary intersection theory 16 4. Extended excision axiom 22
Instanton counting on blowup, I. 4Dimensional pure gauge theory
 INVENT. MATH
, 2005
"... We give a mathematically rigorous proof of Nekrasov’s conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on R 4 gives a deformation of the SeibergWitten prepotential for N = 2 SUSY YangMills theory. Through a study of moduli spaces on the blowup of R 4, ..."
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Cited by 83 (5 self)
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We give a mathematically rigorous proof of Nekrasov’s conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on R 4 gives a deformation of the SeibergWitten prepotential for N = 2 SUSY YangMills theory. Through a study of moduli spaces on the blowup of R 4, we derive a differential equation for the Nekrasov’s partition function. It is a deformation of the equation for the SeibergWitten prepotential, found by Losev et al., and further studied by Gorsky et al.
Positivity in equivariant Schubert calculus
 Duke Math. J
"... Let X = G/B be the flag variety of a complex semisimple group G with B ⊃ T a Borel subgroup and maximal torus, respectively. The homology H∗(X) has as a basis the fundamental classes [Xw] of Schubert varieties Xw ⊂ X; if {xw} ⊂ H ∗ (X) is the ..."
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Cited by 54 (2 self)
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Let X = G/B be the flag variety of a complex semisimple group G with B ⊃ T a Borel subgroup and maximal torus, respectively. The homology H∗(X) has as a basis the fundamental classes [Xw] of Schubert varieties Xw ⊂ X; if {xw} ⊂ H ∗ (X) is the
Moduli of Twisted Sheaves
, 2004
"... Abstract. We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of su ..."
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Cited by 48 (9 self)
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Abstract. We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asympotically geometrically irreducible, normal, generically smooth, and l.c.i. over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, twisted Bogomolov inequalities,