Almost toric presentations of symplectic log Calabi-Yau pairs

Abstract

It is known that the union of fibers over elliptic singularities of an almost toric fibered (ATF) closed symplectic four-manifold forms a symplectic log Calabi-Yau (LCY) divisor. In this paper, we show the converse: any symplectic LCY divisor can be realized as the boundary divisor of an ATF. For divisors in elliptic ruled surfaces, this realization occurs over the M\"obius strip; for divisors in rational surfaces, the realization occurs over the disk and becomes canonical once we choose an additional datum, called the framing, on the space of LCYs in rational surfaces. The construction for rational surfaces is achieved by considering the symplectic analogue of the toric model used in algebraic geometry, which motivates the introduction of a new combinatorial object that we call the bitten Delzant polygon.