Low degree GW invariants of surfaces II

Kiem, Young-Hoon; Li, Jun

doi:10.1007/s11425-011-4258-x

Cited by 17 publications

(15 citation statements)

References 23 publications

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“…In the case of no insertions Lee-Parker [LP] proved that (6) is equal to the degree d unramified spin Hurwitz number of C with theta characteristic K S | C . This result was proved algebro-geometrically by KL2]. The spin Hurwitz numbers were recently computed explicitly using TQFT by Gunningham [Gun].…”

Section: Many Of These Invariants Vanishmentioning

confidence: 92%

“…More generally one can localise the calculation of P χ,β X, τ α 1 (σ 1 ) · · · τ αm (σ m ) to (thickenings of) a canonical divisor C. In the context of Seiberg-Witten and Gromov-Witten theory on S this goes back to ideas of Witten, Taubes and Lee-Parker [LP], formalised in algebraic geometry as Kiem-Li's cosection localisation [KL1,KL2,KL3].…”

Section: Many Of These Invariants Vanishmentioning

confidence: 99%

“…This was originally proved by Lee-Parker [LP] using analytical techniques rather than the MNOP conjecture. See [MP] and [KL1,KL2] for algebrogeometric proofs.…”

Section: Links To Gromov-witten Theory Of Smentioning

confidence: 99%

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Stable pairs with descendents on local surfaces I: the vertical component

Kool

Thomas

2017

Pure and Applied Mathematics Quarterly

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We study the full stable pair theory -with descendents -of the Calabi-Yau 3-fold X = K S , where S is a surface with a smooth canonical divisor C.By both C * -localisation and cosection localisation we reduce to stable pairs supported on thickenings of C indexed by partitions. We show that only strict partitions contribute, and give a complete calculation for length-1 partitions. The result is a surprisingly simple closed product formula for these "vertical" thickenings.This gives all contributions for the curve classes [C] and 2[C] (and those which are not an integer multiple of the canonical class). Here the result verifies, via the descendent-MNOP correspondence, a conjecture of Maulik-Pandharipande, as well as various results about the Gromov-Witten theory of S and spin Hurwitz numbers. 1 We emphasise that in this paper we are concerned with the full stable pair and Gromov-Witten invariants of X, not their reduced cousins computed in [KT1, KT2].2 In fact all we require, by the deformation invariance of stable pair and Gromov-Witten invariants, is that some deformation of S should have such a divisor.

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Section: Many Of These Invariants Vanishmentioning

confidence: 92%

Section: Many Of These Invariants Vanishmentioning

confidence: 99%

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Stable pairs with descendents on local surfaces I: the vertical component

Kool

Thomas

2017

Pure and Applied Mathematics Quarterly

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“…This is proved in [5], and will appear in the proof of the degeneration formula of GW invariants for spin surfaces [7].…”

Section: Proposition 41 ([5]) the Localized Gw Invariants Of The Spmentioning

confidence: 99%

“…Since the proof of a degeneration formula requires an extensive study of obstruction theory of the family, we will prove it in a separate paper [7]. In this section, we prove a special case of (1.3) that will be part of the proof of (1.3) using degeneration.…”

Section: Low-degree Gw Invariantsmentioning

confidence: 99%

Low Degree GW Invariants of Spin Surfaces

Kiem¹,

Li²

2011

Pure and Applied Mathematics Quarterly

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A spin surface S refers to the total space of a line bundle L over a smooth projective curve D satisfying L 2 = K D . A spin surface is canonically equipped with a holomorphic 2-form θ, which gives rise to a cosection σ of the obstruction sheaf of the moduli stack M g,n (S, d) of stable maps, thus by [6] the localized GW invariants of S. In this paper, we first relate the localized GW invariants of S with the twisted GW invariants of D when certain locally freeness assumption holds. We then analyze in detail a situation in which the aforementioned locally freeness does not hold. Both studies are part of our proof of the Maulik-Pandharipande formulas for low degree GW invariants of surfaces with smooth canonical divisors.

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