Abstract
We show that, after the change of variables q=e^{iu}, refined floor diagrams for {mathbb {P}}^2 and Hirzebruch surfaces compute generating series of higher genus relative Gromov–Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov–Witten theory and an explicit result in relative Gromov–Witten theory of {mathbb {P}}^1. Combining this result with the similar looking refined tropical correspondence theorem for log Gromov–Witten invariants, we obtain a non-trivial relation between relative and log Gromov–Witten invariants for {mathbb {P}}^2 and Hirzebruch surfaces. We also prove that the Block–Göttsche invariants of {mathbb {F}}_0 and {mathbb {F}}_2 are related by the Abramovich–Bertram formula.
Highlights
Floor diagrams, introduced by Brugallé and Mikhalkin [11,12], are combinatorial objects that are used to provide a solution to enumerative problems concerning real and complex curves in h-transverse toric surfaces
For h-transverse toric surfaces, one can consider a particular choice of tropical incidence conditions, known as “vertically stretched”, for which the combinatorics of the tropical curves can be encoded by floor diagrams
Relative Gromov–Witten theory [22] allows the definition of counts of curves in P2 and Hirzebruch surfaces with tangency conditions along smooth divisors, and floor diagrams naturally appear [2,9] as describing the combinatorics of successive applications of the degeneration formula in relative Gromov–Witten theory [23]
Summary
Floor diagrams, introduced by Brugallé and Mikhalkin [11,12], are combinatorial objects that are used to provide a solution to enumerative problems concerning real and complex curves in h-transverse toric surfaces. Page 3 of 42 43 refined counts of tropical curves in R2 [5] and generating series of higher genus log Gromov–Witten invariants of toric surfaces with insertion of a lambda class after the change of variables q = eiu. Theorem 1.1 For P2 and Hirzebruch surfaces, q-refined counts of floor diagrams are, after the change of variables q = eiu, generating series of higher genus relative Gromov–Witten invariants with insertion of a lambda class. The conjectural correspondence between Gromov–Witten invariants and Pandharipande–Thomas stable pair invariants for threefolds [25,26,31] is formulated in terms of a change of variables q = eiu As this correspondence is known for the equivariant relative theories of toric threefolds [27,28], we can rephrase Theorem 1.1 as follows (see Theorem 6.1 for the precise statement).
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