Homological projective duality for determinantal varieties

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 Yn 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.

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Babbage's conjecture, contact of surfaces, symmetric determinantal varieties and applications

Inventiones mathematicae ◽

10.1007/bf01389064 ◽

1981 ◽

Vol 63 (3) ◽

pp. 433-465 ◽

Cited By ~ 36

Author(s):

F. Catanese

Keyword(s):

Determinantal Varieties

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Introduction to Projective Duality

Encyclopaedia of Mathematical Sciences - Projective Duality and Homogeneous Spaces ◽

10.1007/3-540-26957-6_1 ◽

2005 ◽

pp. 1-16

Keyword(s):

Projective Duality

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Linear codes associated to determinantal varieties

Discrete Mathematics ◽

10.1016/j.disc.2015.03.009 ◽

2015 ◽

Vol 338 (8) ◽

pp. 1493-1500 ◽

Cited By ~ 3

Author(s):

Peter Beelen ◽

Sudhir R. Ghorpade ◽

Sartaj Ul Hasan

Keyword(s):

Linear Codes ◽

Determinantal Varieties

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Some remarks concerning Voevodsky’s nilpotence conjecture

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2015-0068 ◽

2018 ◽

Vol 2018 (738) ◽

pp. 299-312 ◽

Cited By ~ 1

Author(s):

Marcello Bernardara ◽

Matilde Marcolli ◽

Gonçalo Tabuada

Keyword(s):

Determinantal Varieties ◽

Linear Sections ◽

Intersection Of Quadrics

Abstract In this article we extend Voevodsky’s nilpotence conjecture from smooth projective schemes to the broader setting of smooth proper dg categories. Making use of this noncommutative generalization, we then address Voevodsky’s original conjecture in the following cases: quadric fibrations, intersection of quadrics, linear sections of Grassmannians, linear sections of determinantal varieties, homological projective duals, and Moishezon manifolds.

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