Hodge Laplacian and geometry of Kuranishi family of Fano manifolds

TL;DR

This paper derives eigenvalue estimates for the Hodge Laplacian on Fano manifolds and explores the geometry of their Kuranishi families, revealing stability of Kähler forms and providing explicit Ricci potential formulas.

Contribution

It introduces new eigenvalue bounds for the Hodge Laplacian and applies them to analyze the deformation geometry of Fano manifolds, connecting to the Donaldson-Fujiki framework.

Findings

Eigenvalue estimates for the Hodge Laplacian on Fano manifolds.

Kähler form stability across the Kuranishi family.

Explicit Ricci potential formula for deformed Fano manifolds.

Abstract

We first obtain eigenvalue estimates for the Hodge Laplacian on Fano manifolds, which follow from the Bochner-Kodaira formula. Then we apply it to study the geometry of the Kuranishi family of deformations of Fano manifolds. We show that the original K\"ahler form remains to be a K\"ahler form for other members of the Kuranishi family, and give an explicit formula of the Ricci potential. We also show that our set-up gives another account for the Donaldson-Fujiki picture.