On the Complex Affine Structures of SYZ Fibration of Del Pezzo Surfaces

Abstract

Given any smooth cubic curve E ⊆ P 2 , we show that the complex affine structure of the special Lagrangian fibration of P 2 \ E constructed by Collins-Jacob-Lin [12] coincides with the affine structure used in Carl-Pomperla-Siebert [15] for constructing mirror. Moreover, we use the Floer-theoretical gluing method to construct a mirror using immersed Lagrangians, which is shown to agree with the mirror constructed by Carl-Pomperla-Siebert.

Some Examples of Family Floer Mirrors

“…We first prove that the complex affine structures of the bases of special Lagrangian fibrations coincide with the affine manifolds with singularities constructed in Gross-Hacking-Keel [19] from some log Calabi-Yau surfaces. See the similar results in [29] for the case of P 2 , general del Pezzo surfaces relative smooth anti-canoncial divisors [28] and rational elliptic surfaces [9] and the case of Fermat hypersurfaces [31]. When the 2-dimensional Calabi-Yau admits a special Lagrangian fibration, it is well-known that the special Lagrangian torus fibres bounding holomorphic discs supports along affine lines with respect to the complex affine coordinates.…”

Enumerative Geometry of Del Pezzo Surfaces

“…A priori, the integral affine manifold used in the Gross-Siebert program is unclear to be the same as the one from the base of special Lagrangian fibration. In the case of Y = P 2 , Lau-Lee-Lin [57] prove that the affine structure used in Carl-Pomperla-Siebert coincides with the complex affine structure of the special Lagrangian fibration constructed by Collins-Jacob-Lin. With the explicit affine structure, we provide some explicit calculation of the enumerative invariants and explains the relation with the previous works of Bouseau [5], Vianna [76], Gabele [30].…”

“…Theorem 7.4. [57] [65] Let Y = P 2 . The complex affine structure on B induced from the special Lagrangian fibration on X coincides with the the affine structure from Carl-Pomperla-Siebert [16].…”