Differential operators on Hitchin variety

TL;DR

This paper introduces Hitchin varieties over complex numbers, studies differential operators on line bundles over them, and shows that certain spaces of differential operators are one-dimensional for projective cases, implying constraints on holomorphic connections.

Contribution

It defines Hitchin varieties over , analyzes differential operators on line bundles over these varieties, and establishes their dimensional properties in the projective case.

Findings

Spaces of global differential operators are one-dimensional on projective Hitchin varieties.

The space of holomorphic connections on a line bundle has no non-constant regular functions.

Provides foundational properties of differential operators on Hitchin varieties.

Abstract

We introduce the notion of Hitchin variety over $\C$. Let $L$ be a holomorphic line bundle over a Hitchin variety $X$. We investigate the space of all global sections of sheaf of differential operators $\cat{D}^k (L)$ and symmetric powers of sheaf of first order differential operators $\cat{S}^k(\cat{D}^1 (L))$ over $X$ and show that for a projective Hithcin variety both the spaces are one dimensional. As an application, we show that the space $\cat{C}(L)$ of holomorphic connections on $L$ does not admit any non-constant regular function.