ABEL-JACOBI MAPS FOR HYPERSURFACES AND NONCOMMUTATIVE CALABI-YAU'S

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 (2n-4)-form on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y-n 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 Yn we show that the Fano scheme is birational to a certain moduli space of sheaves of a (2n-4)-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 noncommutative.