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].

Let (S, E) be a log Calabi-Yau surface pair with E a smooth divisor. We define new conjecturally integer-valued counts of A 1 -curves in (S, E). These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along E via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.

We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples and a proof for counting algebraic curves via tropical curves. To illustrate an application of tropical geometry and the Gross-Siebert program to mirror symmetry, we discuss the mirror symmetry of the projective plane.Michel van Garrel KIAS,discretizing Hitchin's Legendre duality.Enumerative aspects of the Gross-Siebert program 3 Kontsevich and Soibelman [32] demonstrated how one could reconstruct a K3 surface from an affine structure with singularities on S 2 . Using logarithmic geometry, Gross and Siebert were able to solve the reconstruction problem [20] in any dimension, obtaining a degenerating family of Calabi-Yau manifolds X → D over a holomorphic disk from the information of (B, P, ϕ) and a log structure. Furthermore, this family is parametrized by a canonical coordinate (in the usual sense in mirror symmetry). The construction features wall-crossings and scatterings, structures that encode enumerative information linking symplectic with complex geometry via tropical geometry. As will be hinted at in this exposition, Gromov-Witten theory [21] can also be incorporated in this framework. Toric conventionsWe assume familiarity with toric geometry. The interested reader is referred to the excellent exposition of Fulton [10]. As the following story is closely tied to toric geometry, it is convenient to begin by making a few conventions regarding notation.SetFor n ∈ N, set n, m to be the evaluation of n on m. Set a toric fan Σ in M R . Let Σ [n] signify the set of n dimensional cones of Σ . Let X Σ be the toric variety defined by Σ .Denote by T Σ the free abelian group generated by Σ [1] . For ρ ∈ Σ [1] , denote by v ρ the corresponding generator in T Σ . We will need the mapwhereρ is the integral vector generating ρ, that is ρ ∩ M = Z ≥0ρ .

Relative Bogomolny-Prasad-Sommerfield (BPS) state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert in [4] and conjectured by the authors to be integers. For toric del Pezzo surfaces, we provide an arithmetic proof of this conjecture, by relating these invariants to the local BPS state counts of the surfaces. The latter were shown to be integers by Peng in [15]; and more generally for toric Calabi-Yau three-folds by Konishi in [8].

Let (S, E) be a log Calabi-Yau surface pair with E a smooth divisor. We define new conjecturally integer-valued counts of A 1 -curves in (S, E). These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along E via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.