Deformation of product of complex Fano manifolds

TL;DR

This paper proves that in a family of complex Fano manifolds, if one fiber is a product of lower-dimensional manifolds, then all fibers are also products, leading to uniformity in Hermitian symmetric spaces.

Contribution

It establishes the deformation invariance of product structures in families of complex Fano manifolds, extending previous results on Hermitian symmetric spaces.

Findings

Product structure is preserved in deformations of Fano manifolds.

All fibers in a family with a product fiber are themselves products.

Hermitian symmetric spaces of compact type are deformation rigid.

Abstract

Let X be a connected family of complex Fano manifolds. We show that if some fiber is the product of two manifolds of lower dimensions, then so is every fiber. Combining with previous work of Hwang and Mok, this implies immediately that if a fiber is (possibly reducible) Hermitian symmetric space of compact type, then all fibers are isomorphic to the same variety.