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199
The Intrinsic Normal Cone
- INVENT. MATH
, 1997
"... We suggest a construction of virtual fundamental classes of certain types of moduli spaces. ..."
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Cited by 347 (9 self)
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We suggest a construction of virtual fundamental classes of certain types of moduli spaces.
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
A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations
- J. DIFFERENTIAL GEOM
, 2000
"... We briefly review the formal picture in which a Calabi-Yau n-fold is the complex analogue of an oriented real n-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fol ..."
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Cited by 199 (8 self)
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We briefly review the formal picture in which a Calabi-Yau n-fold is the complex analogue of an oriented real n-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of [LT], [BF] in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in P 3, and Donaldson – and Gromov-Witten – like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold X, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general K3 fibration X, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the “Mukai-dual” 3-fold for others. As an example the invariant is shown to distinguish Gross’ diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of X.
Hodge integrals and Gromov-Witten theory
- Invent. Math
"... Let Mg,n be the nonsingular moduli stack of genus g, n-pointed, Deligne-Mumford stable curves. For each marking i, there is an associated cotangent line bundle Li → Mg,n with fiber T ∗ C,pi over the moduli point [C, p1,...,pn]. Let ψi = c1(Li) ∈ H ∗ (Mg,n, Q). The integrals of products of the ψ cla ..."
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Cited by 175 (25 self)
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Let Mg,n be the nonsingular moduli stack of genus g, n-pointed, Deligne-Mumford stable curves. For each marking i, there is an associated cotangent line bundle Li → Mg,n with fiber T ∗ C,pi over the moduli point [C, p1,...,pn]. Let ψi = c1(Li) ∈ H ∗ (Mg,n, Q). The integrals of products of the ψ classes
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
