Abel-Jacobi maps for hypersurfaces and non commutative Calabi-Yau's

TL;DR

This paper generalizes the symplectic structure on Fano schemes of lines on hypersurfaces, constructing a closed form via multiple methods, and relates these schemes to Calabi-Yau varieties, including non-commutative cases.

Contribution

It introduces a new closed p-form on Fano schemes of lines for higher-dimensional hypersurfaces and links these schemes to Calabi-Yau varieties through homological duality.

Findings

Constructed a closed p-form on Fano schemes for hypersurfaces of degree n.

Established birational equivalence between Fano schemes and moduli spaces of sheaves on Calabi-Yau varieties.

Extended the relation to non-Pfaffian hypersurfaces with non-commutative Calabi-Yau duals.

Abstract

It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed p-form with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y of degree n. We provide several definitions of this form - via the Abel-Jacobi map, via Hochschild homology, and via the linkage class, and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show that the Fano scheme is birational to a certain moduli space of sheaves on a p-dimensional Calabi--Yau variety X arising naturally in the context of homological projective duality, and that the constructed form is induced by the holomorphic volume form on X. This remains true for a general non Pfaffian hypersurface but the dual Calabi-Yau becomes non commutative.