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

Abstract

Consider the holomorphic bundle with connection on ${\mathbb{P}}^1-\{0,1,\infty \}$ corresponding to the regular hypergeometric differential operator \[ \prod_{j=1}^h(D-α_j)-z\prod_{j=1}^h(D-β_j), \qquad D=z\frac{d}{dz}. \] 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.

Minor corrections throughout the text, including the statement of Theorem 4. Exposition improved

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• Algebraic aspects of hypergeometric differential equations
• Lower bounds for Lyapunov exponents of flat bundles on curves
• Irregular Hodge filtration of some confluent hypergeometric systems
• Examples of hypergeometric twistor $\mathcal{D}$-modules
• Stokes matrices for confluent hypergeometric equations
• Middle multiplicative convolution and hypergeometric equations
• Mixed Hodge numbers and factorial ratios
• A functional logarithmic formula for hypergeometric functions 3F2
• Uniformization of some weight 3 variations of Hodge structure, Anosov representations, and Lyapunov exponents
• Irregular Hodge numbers for rigid $G_2$-connections
• Irregular Hodge filtration of hypergeometric differential equations