Low Degree GW Invariants of Spin Surfaces

Abstract

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.

Low degree GW invariants of surfaces II

“…In the end, by applying this degeneration formula, and a calculation of low degree GW invariants of surfaces [9], we prove the following low degree formulas originally conjectured by Maulik and Pandharipande [21]. …”

“…The part of [7] on explicit calculation of low degree GW-invariants of surfaces, and the formula relating localized GW-invariants of spin surfaces with the twisted GW-invariants of surfaces form the preprint [9].…”

“…1.3] (see also [7]). Before we prove (7.4) using the degeneration formula proved above, we recall the following two identities proved in [9].…”

Spin Hurwitz numbers and the Gromov-Witten invariants of Kähler surfaces

“…The results of [8,10] show that the GW invariant of X is a sum over the components of (D, N ) of certain local GW invariants GW loc g,n . As usual, one can work either with the local GW invariants that count maps from connected domains of genus g or with the local "Gromov-Taubes" invariants GT loc g,n that count maps from possibly disconnected domains of Euler characteristic χ.…”

“…For h = 0, 1, these invariants were calculated for all degrees d in [8,10]. As an immediate application of Theorem 1.1, one can express the local invariants (1.11) with h ≥ 2 in terms of h = 1 spin Hurwitz numbers calculated in [3]: The proof of Theorem 1.1 involves five main steps, described below.…”