Finiteness of Subfamilies of Calabi-Yau n-Folds over Curves with Maximal Length of Yukawa-Coupling

TL;DR

This paper proves that there are only finitely many families of Calabi-Yau n-folds over a fixed curve with non-zero Yukawa-coupling, using algebraic methods involving Hodge structures and Higgs bundles.

Contribution

It provides a simplified, algebraic proof of the finiteness of such Calabi-Yau families and extends the result to more general cases.

Findings

Finiteness of Calabi-Yau n-fold families over curves with fixed degeneration locus

Application of variation of Hodge structure and Higgs bundle stability

Generalization of finiteness result to broader settings

Abstract

We give a simplified and more algebraic proof of the finiteness of the families of Calabi-Yau n-folds with non-vanishing of Yukawa-coupling over a fixed base curve and with fixed degeneration locus. We also give a generalization of this result. Our method is variation of Hodge structure and poly-stability of Higgs bundles.