On the Topology of Fano Smoothings

TL;DR

This paper investigates the topology of Fano smoothings related to mirror symmetry, using combinatorial data to confirm conjectures and demonstrate compatibility with theoretical expectations.

Contribution

It determines the topology of Fano smoothings from combinatorial data, supporting conjectures by Corti--Hacking--Petracci and aligning with mirror symmetry principles.

Findings

Topology of general fiber derived from combinatorial data.

Supports Corti--Hacking--Petracci conjectures.

Verifies compatibility with mirror symmetry expectations.

Abstract

Suppose that X is a Fano manifold that corresponds under Mirror Symmetry to a Laurent polynomial f, and that P is the Newton polytope of f. In this setting it is expected that there is a family of algebraic varieties over the unit disc with general fiber X and special fiber the toric variety defined by the spanning fan of P. Building on recent work and conjectures by Corti--Hacking--Petracci, who construct such families of varieties, we determine the topology of the general fiber from combinatorial data on P. This provides evidence for the Corti--Hacking--Petracci conjectures, and verifies that their construction is compatible with expectations from Mirror Symmetry.