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

Alexander Kuznetsov (1) , Laurent Manivel (2) , Dimitri Markushevich (3)

1 SMI | RAS - Steklov Mathematical Institute [Moscow]
2 IF - Institut Fourier
3 LPP - Laboratoire Paul Painlevé - UMR 8524

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.

Domaines: Géométrie algébrique [math.AG]

Alexander Kuznetsov, Laurent Manivel, Dimitri Markushevich. Abel-Jacobi maps for hypersurfaces and non commutative Calabi-Yau's. Communications in Contemporary Mathematics, 2010, 12 (3), pp.373-416. ⟨hal-00330557⟩