Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee

Kim, Bumsig; Kresch, Andrew; Pantev, Tony

doi:10.1016/s0022-4049(02)00293-1

Cited by 104 publications

(133 citation statements)

References 11 publications

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“…This is a consequence of the fact that [12, Proposition 6.2.2] holds equally in the orbifold case (the same argument, based on [22], works, see also [16, Proposition 5.1]).…”

Section: 7mentioning

confidence: 81%

Orbifold quasimap theory

2015

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“…This is a consequence of the fact that [12, Proposition 6.2.2] holds equally in the orbifold case (the same argument, based on [22], works, see also [16, Proposition 5.1]).…”

Section: 7mentioning

confidence: 81%

Orbifold quasimap theory

2015

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“…] (see for instance [30]). The long subscript here is to remind us that the corresponding homomorphism between Novikov rings should be applied to the RHS.…”

Section: Mirror Formulasmentioning

confidence: 99%

“…(iii) The composition rule: recall that the double coverZ g,n+1,d of the nodal locus coincides with the total range of the gluing maps (30) and…”

Section: Appendix 1 the Proof Of Theoremmentioning

confidence: 99%

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Quantum Riemann–Roch, Lefschetz and Serre

2007

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Given a holomorphic vector bundle E over a compact Kähler manifold X, one defines twisted Gromov-Witten invariants of X to be intersection numbers in moduli spaces of stable maps f : Σ → X with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle H 0 (Σ, f * E) H 1 (Σ, f * E). Using the formalism of quantized quadratic Hamiltonians [25], we express the descendant potential for the twisted theory in terms of that for X. This result (Theorem 1) is a consequence of Mumford's Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps. It determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants.When E is concave and the C × -equivariant inverse Euler class is chosen as the characteristic class, the twisted invariants of X give Gromov-Witten invariants of the total space of E. "Nonlinear Serre duality" [21], [23] expresses Gromov-Witten invariants of E in terms of those of the super-manifold ΠE: it relates Gromov-Witten invariants of X twisted by the inverse Euler class and E to Gromov-Witten invariants of X twisted by the Euler class and E * . We derive from Theorem 1 nonlinear Serre duality in a very general form (Corollary 2).When the bundle E is convex and a submanifold Y ⊂ X is defined by a global section of E, the genus-zero Gromov-Witten invariants of ΠE coincide with those of Y . We establish a "quantum Lefschetz hyperplane section principle" (Theorem 2) expressing genus-zero Gromov-Witten invariants of a complete intersection Y in terms of those of X. This extends earlier results [4], [9], [18], [29], [33] and yields most of the known mirror formulas for toric complete intersections.

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“…The tools used in this paper are: the degeneration formula (see Ionel and Parker [15], A-M Li and Ruan [21], J Li [22] and Maulik and Pandharipande [26]), compatibility of perfect obstruction theories (see Definition 3.3 and Behrend and Fantechi [2], Kim, Kresch and Pantev [17] and J Li and Tian [23]) and deformation invariance of virtual classes. Since there is no assumption on the manifold X , the moduli of stable maps of X can be highly singular.…”

Section: Then Hmentioning

confidence: 99%

Gromov–Witten invariants of blow-ups along submanifolds with convex normal bundles

Lai

2009

Geom. Topol.

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When the normal bundle N Z =X is convex with a minor assumption, we prove that genus 0 GW-invariants of the blow-up Bl Z X of X along a submanifold Z , with cohomology insertions from X , are identical to GW-invariants of X . Under the same hypothesis, a vanishing theorem is also proved. An example to which these two theorems apply is when N Z =X is generated by its global sections. These two main theorems do not hold for arbitrary blow-ups, and counterexamples are included.

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Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee

Abstract: A functoriality property of the virtual fundamental class on the moduli of stable maps is proven. The property is used to supply a proof of a conjecture of Cox, Katz and Lee.

Cited by 104 publications

References 11 publications

Orbifold quasimap theory

Orbifold quasimap theory

Quantum Riemann–Roch, Lefschetz and Serre

Gromov–Witten invariants of blow-ups along submanifolds with convex normal bundles

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