Hodge Theory of Hypersurfaces in Toric Varieties and Recent Developments in Quantum Physics

TL;DR

This paper explores the Hodge theory of hypersurfaces in toric varieties, focusing on mirror symmetry, mixed Hodge structures, and applications to Calabi-Yau hypersurfaces, integrating combinatorial and algebraic approaches.

Contribution

It introduces a combinatorial construction of mirrors for Calabi-Yau hypersurfaces and analyzes the variation of mixed Hodge structures related to hypergeometric functions.

Findings

Constructed mirror pairs of Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties.

Connected variation of mixed Hodge structures to hypergeometric functions.

Applied results to mirror symmetry in toric geometry.

Abstract

This is the author's Habilitation which took place at University of Essen on July 11, 1993. The manuscript contains two parts. The first one is devoted to the author's combinatorial construction of mirrors of Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties. The second one contains author's results on the variation of mixed Hodge structures of affine hypersurfaces in algebraic tori and their connection to Gelfand-Kapranov-Zelevinsky theory of generalized hypergeometric functions and their applications to the mirror symmetry for Calabi-Yau hypersurfaces in toric varieties.