Sum formulas for local Gromov-Witten invariants of spin curves

Abstract

Holomorphic 2-forms on Kähler surfaces lead to "Local Gromov-Witten invariants" of spin curves. This paper shows how to derive sum formulas for such local GW invariants from the sum formula for GW invariants of certain ruled surfaces. These sum formulas also verify the Maulik-Pandharipande formulas that were recently proved by Kiem and Li.Let X be a Kähler surface with a holomorphic 2-form α. The real part of α, also denoted by α, then induces an almost complex structure on X :Here J is the Kähler structure on X and K α is the endomorphism of T X defined by the formula u, K α v = α(u, v) where , is the Kähler metric on X. The almost complex structure J α satisfies a remarkable Image Localization Property :• if f is a J α -holomorphic map into X that represents a non-zero (1,1) class then the image of f lies in the zero set D of the holomorphic 2-form αFor simplicity, assume X is a (minimal) surface of general type and D is smooth. The normal bundle N to D is then a theta characteristic on D, i.e., N is a square root of the canonical bundle of D. The pair (D, N ) is called a spin curve of genus h where h is the genus of D. The total space N D of N has a tautological holomorphic 2-form α that induces, by the same manner as in (0.1), an almost complex structure J α on N D satisfying the image localization property, namely M g,n (N D , d[D], J α ) = M g,n (D, d).Consequently, the moduli space of J α -holomorphic maps is compact, so it represents a (virtual) fundamental class that defines local GW invariants of the spin curve (D, N ). These local GW invariants depend only on the genus h and the parity p ≡ h 0 (N ) (mod 2) and GW invariants of Kähler surfaces with p g > 0 are the sum of local GW invariants associated to spin curves [LP1]. GW invariants count maps from connected domains, while Gromov-Taubes invariants count maps from not necessarily connected domains. These GT invariants can be obtained from GW * χ,n (U, dS) every genus zero (irreducible) component is ghost component. It thus follows from the stability and the relation between ψ i class and st * φ i class (cf.[KM] page 388) that

Low degree GW invariants of surfaces II

“…Shortly after, in [14] Lee proved the Maulik-Pandharipande's formulas of low degree GW-invariants by developing a degeneration formula of localized GW-invariants of surfaces in symplectic geometry.…”

An obstruction bundle relating Gromov-Witten invariants of curves and Kähler surfaces

“…Recently, Kiem and Li [KL] defined the local invariants by algebraic geometry methods and proved the formulas for degree 1 and 2 local invariants conjectured by Maulik and Pandharipande [MP]. The first author [Lee2] also reproved those formulas by adapting the symplectic sum formula of [IP2] to local GW invariants.…”

A square root of Hurwitz numbers

“…Using this isomorphism, one can also extend the Gunningham formula [3,14] for spin Hurwitz numbers of the spin curve of genus h and parity p to the following multilinear form on OP Q : Then the Maulik-Pandharipande formulae (( 8) and ( 9) of [18]), which were proved in [11,13,12], show that for degree d = 1, 2, the descendent GW invariants of the spin curve of genus h and parity p are given by τ k1 (ω) • • • τ kn (ω)…”

“…Then the Maulik-Pandharipande formulae ( (8) and (9) of [18]), which were proved in [11,13,12], show that for degree d = 1, 2, the descendent GW invariants of the spin curve of genus h and parity p are given by τ k1 (ω) · · · τ kn (ω)…”