Toric Representation and Positive Cone of Picard Group and Deformation Space in Mirror Symmetry of Calabi-Yau Hypersurfaces in Toric Varieties

TL;DR

This paper develops combinatorial methods to represent the Picard group and deformation space of Calabi-Yau hypersurfaces in toric varieties, establishing mirror symmetry correspondences for arbitrary dimensions.

Contribution

It introduces a combinatorial framework for Picard and deformation spaces in toric Calabi-Yau hypersurfaces and clarifies mirror symmetry relations between cones and flops.

Findings

Established mirror cohomology correspondence for arbitrary dimension CY spaces.

Identified interchangeability of Kahler and degeneration cones under mirror symmetry.

Linked different degeneration cones to flops in mirror CY 3-folds.

Abstract

We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry. In particular, different degeneration cones of a CY 3-fold are corresponding to flops of its mirror 3-fold.