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

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

2008-06-06

arxiv.org/abs/0806.1154v3

Mathematics

Algebraic Geometry

34 pages; exposition of Hochschild homology expanded; references added; introduction re-written; some imrecisions, typos and t

Scientific paper

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.

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