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

作者：Kuznetsov A^*; Manivel L; Markushevich D

来源：Communications in Contemporary Mathematics, 2010, 12(3): 373-416.

DOI：10.1142/S021919971000383X

• Abel-Jacobi map
• hypersurface
• Fano scheme
• Pfaffian variety
• derived category
• homological projective duality
• noncommunicative Calabi-Yau
• Hochschild homology

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.

• 出版日期2010-6