Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles

TL;DR

This paper studies the Hodge structures arising from hypergeometric differential operators on the punctured projective line, computes Hodge invariants, and interprets results via non-abelian Hodge theory and parabolic Higgs bundles.

Contribution

It provides explicit calculations of Hodge invariants for hypergeometric variations and connects these to parabolic Higgs bundles, confirming a conjecture of Corti and Golyshev.

Findings

Computed Hodge numbers for hypergeometric variations.

Established a link between hypergeometric differential operators and parabolic Higgs bundles.

Proposed a conjecture relating Hodge invariants to hypergeometric data.

Abstract

Consider the holomorphic bundle with connection on $\mathbb P^1-\{0,1,\infty\}$ corresponding to the regular hypergeometric differential operator \[ \prod_{j=1}^h(D-\alpha_j)-z\prod_{j=1}^h(D-\beta_j), \qquad D=z\frac{d}{dz}. \] If the numbers $\alpha_i$ and $\beta_j$ are real and for all $i$ and $j$ the number $\alpha_i-\beta_j$ is not integer, then the bundle with connection is known to underlie a complex polarizable variation of Hodge structures. We calculate some Hodge invariants for this variation, in particular, the Hodge numbers. From this we derive a conjecture of Corti and Golyshev. We also use non-abelian Hodge theory to interpret our theorem as a statement about parabolic Higgs bundles.