No abstract

In [15] we established a series of correspondences relating five enumerative theories of log Calabi-Yau surfaces, i.e. pairs (Y, D) with Y a smooth projective complex surface and D = D1 + • • • + D l an anticanonical divisor on Y with each Di smooth and nef. In this paper we explore the generalisation to Y being a smooth Deligne-Mumford stack with projective coarse moduli space of dimension 2, and Di nef Q-Cartier divisors. We consider in particular three infinite families of orbifold log Calabi-Yau surfaces, and for each of them we provide closed form solutions of the maximal contact log Gromov-Witten theory of the pair (Y, D), the local Gromov-Witten theory of the total space of i OY (−Di), and the open Gromov-Witten of toric orbi-branes in a Calabi-Yau 3-orbifold associated to (Y, D). We also consider new examples of BPS integral structures underlying these invariants, and relate them to the Donaldson-Thomas theory of a symmetric quiver specified by (Y, D), and to a class of open/closed BPS invariants.

Let 𝑋 be a smooth projective complex variety and let 𝐷 = 𝐷 1 + ⋯ + 𝐷 𝑙 be a reduced normal crossing divisor on 𝑋 with each component 𝐷 𝑗 smooth, irreducible and numerically effective. The log-local principle put forward in van Garrel et al. (Adv. Math. 350 (2019) 860-876) conjectures that the genus 0 log Gromov-Witten theory of maximal tangency of (𝑋, 𝐷) is equivalent to the genus 0 local Gromov-Witten theory of 𝑋 twisted by ⨁ 𝑙 𝑗=1 (−𝐷 𝑗 ). We prove that an extension of the loglocal principle holds for 𝑋 a (not necessarily smooth) ℚ-factorial projective toric variety, 𝐷 the toric boundary, and descendant point insertions. M S C ( 2 0 2 0 )14N35 (primary), 14M25, 14J33, 14T90 (secondary) INTRODUCTIONLet 𝑋 be a smooth projective complex variety of dimension 𝑛 and let 𝐷 = 𝐷 1 + ⋯ + 𝐷 𝑙 be an effective reduced normal crossing divisor with each component 𝐷 𝑗 smooth, irreducible and numerically effective. We can then consider two, a priori very different, geometries associated to the pair (𝑋, 𝐷):-the 𝑛-dimensional log geometry of the pair (𝑋, 𝐷), -the (𝑛 + 𝑙)-dimensional local geometry of the total space Tot( ⨁ 𝑙 𝑗=1  𝑋 (−𝐷 𝑗 )). The genus 0 log Gromov-Witten invariants of (𝑋, 𝐷) virtually count rational curves 𝑓 ∶ ℙ 1 → 𝑋

We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface S. We calculate the Poincaré polynomials of the moduli spaces for the curve classes β having arithmetic genus at most 2. We formulate a conjecture that these Poincaré polynomials are divisible by the Poincaré polynomials of ((−KS).β−1)-dimensional projective space. This conjecture motivates upcoming work on log BPS numbers [8].