Enumerative aspect of symplectic log Calabi-Yau divisors and almost toric fibrations

TL;DR

This paper investigates the classification and counting of symplectic log Calabi-Yau divisors in rational surfaces, establishing stability, finiteness, and rigidity results, and explores their relation to almost toric fibrations.

Contribution

It provides a general counting formula for symplectic log Calabi-Yau divisors and connects their combinatorics to almost toric fibrations, advancing understanding of symplectic geometry.

Findings

Finiteness and rigidity of symplectic log Calabi-Yau divisors established.

A counting formula for these divisors in certain regions is derived.

Relation between divisors and almost toric fibrations is analyzed.

Abstract

In this paper we are interested in the isotopy classes of symplectic log Calabi-Yau divisors in a fixed symplectic rational surface. We give several equivalent definitions and prove the stability, finiteness and rigidity results. Motivated by the problem of counting toric actions, we obtain a general counting formula of symplectic log Calabi-Yau divisors in a restrictive region of $c_1$-nef cone. A detailed count in the case of 2- and 3-point blow-ups of complex projective space for all symplectic forms is also given. In our framework the complexity of the combinatorics of analyzing Delzant polygons is reduced to the arrangement of homology classes. Then we study its relation with almost toric fibrations. We raise the problem of realizing all symplectic log Calabi-Yau divisors by some almost toric fibrations and verify it together with another conjecture of Symington in a special region.