On Calabi-Yau Complete Intersections in Toric Varieties

TL;DR

The paper investigates mirror symmetry for Calabi–Yau complete intersections inside Gorenstein toric Fano varieties with mild singularities by leveraging nef-partitions and MPCP-desingularizations. It proves dualities for string-theoretic Hodge numbers in the (0,q) and (1,q) regimes, including an Euler-characteristic framework for Ω^1 and explicit formulas in the ample/nef-big and projective-space settings. The authors develop a semi-simplicity principle for nef-partitions, derive combinatorial expressions linking Hodge data to lattice-point counts, and verify concrete instances in projective spaces, thereby strengthening the geometric understanding of mirror symmetry in singular toric CYs. These results provide practical tools for computing Hodge numbers via toric combinatorics and support the broader applicability of mirror symmetry beyond smooth CY varieties.

Abstract

We investigate Hodge-theoretic properties of Calabi-Yau complete intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano varieties having at most Gorenstein singularities. Our main purpose is to show that the combinatorial duality proposed by second author agrees with the duality for Hodge numbers predicted by mirror symmetry. It is expected that the complete verification of mirror symmetry predictions for singular Calabi-Yau varieties $V$ of arbitrary dimension demands considerations of so called {\em string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$. We restrict ourselves to the string-theoretic Hodge numbers $h^{0,q}_{\rm st}(V)$ and $h^{1,q}_{\rm st}(V)$ $(0 \leq q \leq d-r) which coincide with the usual Hodge numbers $h^{0,q}(\widehat{V})$ and $h^{1,q}(\widehat{V})$ of a $MPCP$-desingularization $\widehat{V}$ of $V$.

Theorems & Definitions (67)

• Proposition 1.1
• Theorem 2.1
• Proposition 2.2
• Definition 2.3
• Proposition 2.4
• Theorem 2.5
• Theorem 2.6
• Definition 3.1
• Remark 3.2
• Theorem 3.3