Log BPS numbers of log Calabi-Yau surfaces

Abstract

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.

Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs

“…Proof. This was shown in [Bou4], Lemma 1.2, using ideas from [CGKT2]. The freedom of choice of O and the fact that the monodromy of the family of smooth cubics in P 2 maps surjectively to SL(2, Z) acting on T k ≃ Z 3k × Z 3k implies that two points P, P ′ with k(P ) = k(P ′ ) are related to each other via a monodromy transformation.…”

Enumerative Geometry of Del Pezzo Surfaces

“…Assume that {m k } k≥0 is another A ∞ structure due to different choices in the construction and {f k } k≥0 is the induced A ∞ -homomorphism. Let W (u) = m (e b ) and one has W (u) = KW (u), where K is a product of transformations of the form (12) and K γ appears in the product only if L u bounds a holomorphic disc of relative γ. Notice that the coefficient of T ω(β) z ∂β in W (u), W (u) is the same unless L u bound holomorphic discs β , γ i , where β is of Maslov index two and γ i of Maslov index zero such that β = β + i γ i .…”

“…It is actually an isomorphism[71] 2. See also the corresponding integrality of log Gromov-Witten invariants for log Calabi-Yau surfaces[12] …”

Scattering diagrams, sheaves, and curves

“…was initiated by N. Takahashi [37,38] around 1999 and some form of Theorem 1.1.5 was then conjectured. A more recent study of this question has been done by Choi-van Garrel-Katz-Takahashi [14][15][16]. In particular, the statement of Theorem 1.1.5 can be found as [15,Conjecture 1.3].…”

“…A more recent study of this question has been done by Choi-van Garrel-Katz-Takahashi [14][15][16]. In particular, the statement of Theorem 1.1.5 can be found as [15,Conjecture 1.3]. The natural analogue of Theorem 1.1.5 should hold for any pair (S, D) with S a del Pezzo surface and D a smooth anticanonical divisor.…”