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 Mosc. Math. J., 2016, Volume 16, Number 2, Pages 205–235 (Mi mmj598)

From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives

a Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France
b Department of Mathematics, MIT, Cambridge, MA 02139, USA
c Centro de Matemática e Aplicações (CMA), FCT, UNL, Portugal
d Departamento de Matemática, FCT, UNL, Portugal

Abstract: Let $X$ and $Y$ be complex smooth projective varieties, and $\mathcal D^b(X)$ and $\mathcal D^b(Y)$ the associated bounded derived categories of coherent sheaves. Assume the existence of a triangulated category $\mathcal T$ which is admissible both in $\mathcal D^b(X)$ as in $\mathcal D^b(Y)$. Making use of the recent theory of Jacobians of noncommutative motives, we construct out of this categorical data a morphism $\tau$ 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 $\mathcal T^\perp$ of $\mathcal T\subset\mathcal D^b(X)$ has a trivial Jacobian (e.g., when $\mathcal T^\perp$ is generated by exceptional objects), the morphism $\tau$ is split injective. When this also holds for the orthogonal complement $\mathcal T^\perp$ of $\mathcal T\subset\mathcal D^b(Y)$, $\tau$ becomes an isomorphism. Furthermore, in the case where $X$ and $Y$ have a unique principally polarized intermediate Jacobian, we prove that $\tau$ 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.

Key words and phrases: intermediate Jacobians, polarizations, noncommutative motives, semi-orthogonal decompositions, Torelli theorem, Fano threefolds, quadric fibrations, blow-ups.

 Funding Agency Grant Number National Science Foundation CAREER #1350472 Fundação para a Ciência e a Tecnologia UID/MAT/00297/2013 G. Tabuada was supported in part by the National Science Foundation CAREER Award #1350472 and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project grant UID/MAT/00297/2013 (Centro de Matemática e Aplicações).

Full text: http://www.mathjournals.org/.../2016-016-002-001.html
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Document Type: Article
MSC: 14A22, 14C34, 14E08, 14J30, 14J45, 14K30, 18E30
Received: July 5, 2014; in revised form May 26, 2015
Language: English

Citation: Marcello Bernardara, Gonçalo Tabuada, “From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives”, Mosc. Math. J., 16:2 (2016), 205–235

Citation in format AMSBIB
\Bibitem{BerTab16} \by Marcello~Bernardara, Gon{\c c}alo~Tabuada \paper From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives \jour Mosc. Math.~J. \yr 2016 \vol 16 \issue 2 \pages 205--235 \mathnet{http://mi.mathnet.ru/mmj598} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3480702} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000391209600001} 

• http://mi.mathnet.ru/eng/mmj598
• http://mi.mathnet.ru/eng/mmj/v16/i2/p205

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This publication is cited in the following articles:
1. A. Auel, M. Bernardara, “Cycles, derived categories, and rationality”, Surveys on Recent Developments in Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, 95, eds. I. Coskun, T. de Fernex, A. Gibney, Amer. Math. Soc., 2017, 199–266