Diffeomorphism classes of the doubling Calabi-Yau threefolds with Picard number two

TL;DR

This paper classifies the diffeomorphism types of certain Calabi-Yau threefolds constructed via a gluing method from Fano threefolds, showing that different initial Fano pairs lead to non-diffeomorphic Calabi-Yau threefolds.

Contribution

It establishes that doubling Calabi-Yau threefolds with Picard number two are not diffeomorphic when derived from different Fano threefold families, using differential topology and the $$-invariant.

Findings

Doubling Calabi-Yau threefolds with Picard number two are non-diffeomorphic for different Fano pairs.

Classification relies on the $$-invariant and 6-manifold topology.

The method distinguishes Calabi-Yau threefolds based on initial Fano data.

Abstract

Previously we constructed Calabi-Yau threefolds by a differential-geometric gluing method using Fano threefolds with their smooth anticanonical $K3$ divisors (New York J. Math. 20: 1-33, 2014). In this paper, we further consider the diffeomorphism classes of the resulting Calabi-Yau threefolds (which are called the doubling Calabi-Yau threefolds) starting from different pairs of Fano threefolds with Picard number one. Using the classifications of simply-connected $6$-manifolds in differential topology and the $\lambda$-invariant introduced by Lee (J. Math. Pures Appl. 141: 195-219, 2020), we prove that any two of the doubling Calabi-Yau threefolds with Picard number two are not diffeomorphic to each other when the underlying Fano threefolds are distinct families.