From Semi-Orthogonal Decompositions to Polarized Intermediate Jacobians Via Jacobians of Noncommutative Motives
Name
Tabuada_From semi.pdf
Size
466.41 KB
Format
Adobe PDF
Checksum (MD5)
e159537bfced7b70dc2c39c39ae34b0a
Author(s)
Bernardara, Marcello
Trigo Neri Tabuada, Goncalo Jorge
Date Issued
March 2016
Journal
Moscow Mathematical Journal
Publisher
Independent University of Moscow
Citation
Bernardara, Marcello and Gonçalo Tabuada "From Semi-Orthogonal Decompositions to Polarized Intermediate Jacobians via Jacobians of Noncommutative Motives." Moscow Mathematical Journal 16.2 (2016): 205-243.
Version
Original manuscript
Abstract
Let X and Y be complex smooth projective varieties, and D[superscript b](X) and D[superscript b](Y) the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category T which is admissible both in D[superscript b](X) as in D[superscript b](Y). Making use of the recent theory of Jacobians of noncommutative motives, we construct out of this categorical data a morphism τ of abelian varieties (up to isogeny) from the product of the intermediate algebraic Jacobians of X to the product of the intermediate algebraic Jacobians of Y. Our construction is conditional on a conjecture of Kuznetsov concerning functors of Fourier–Mukai type and on a conjecture concerning intersection bilinear pairings (which follows from Grothendieck’s standard conjecture of Lefschetz type). We describe several examples where these conjectures hold and also some conditional examples. When the orthogonal complement T⊥ of T⊂D[superscript b](X) has a trivial Jacobian (e.g., when T[superscript ⊥] is generated by exceptional objects), the morphism τ is split injective. When this also holds for the orthogonal complement T[superscript ⊥] of T⊂D[superscript b](Y), τ becomes an isomorphism. Furthermore, in the case where X and Y have a unique principally polarized intermediate Jacobian, we prove that τ preserves the principal polarization. As an application, we obtain categorical Torelli theorems, an incompatibility between two conjectures of Kuznetsov (one concerning functors of Fourier–Mukai type and another one concerning Fano threefolds), and also several new results on quadric fibrations and intersections of quadrics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
http://www.mathjournals.org/mmj/2016-016-002/2016-016-002-001.html
