Infinitesimal categorical Torelli theorems for Fano threefolds

Bridgeland moduli spaces; Brill-Noether locus; Categorical Torelli theorems; Derived categories; Fano threefolds; Kuznetsov components; Algebra and Number Theory; Brill-No ether locus

Abstract :

[en] Let X be a smooth Fano variety and Ku(X) its Kuznetsov component. A Torelli theorem for Ku(X) states that Ku(X) is uniquely determined by a certain polarized abelian variety associated to it. An infinitesimal Torelli theorem for X states that the differential of the period map is injective. A categorical variant of the infinitesimal Torelli theorem for X states that the morphism η:H1(X,TX)→HH2(Ku(X)) is injective. In the present article, we use the machinery of Hochschild (co)homology to relate the aforementioned three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, we prove infinitesimal categorical Torelli theorems for a class of prime Fano threefolds. We then prove, infinitesimally, a restatement of the Debarre–Iliev–Manivel conjecture regarding the general fibre of the period map for ordinary Gushel–Mukai threefolds.

Disciplines :

Mathematics

Author, co-author :

JACOVSKIS, Augustinas ^✱; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) ; School of Mathematics, The University of Edinburgh, Edinburgh, United Kingdom

Lin, Xun ^✱; Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

Liu, Zhiyu ^✱; Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou, China ; College of Mathematics, Sichuan University, Chengdu, China

Zhang, Shizhuo ^✱; Max Planck Institute for Mathematics, Bonn, Germany ; Institut de Mathématiqes de Toulouse, UMR 5219, Université de Toulouse, Université Paul Sabatier, Toulouse Cedex 9, France

^✱ These authors have contributed equally to this work.

External co-authors :

yes

Language :

English

Title :

Infinitesimal categorical Torelli theorems for Fano threefolds

Publication date :

December 2023

Journal title :

Journal of Pure and Applied Algebra

ISSN :

0022-4049

eISSN :

1873-1376

Publisher :

Elsevier B.V.

Volume :

227

Issue :

Pages :

107418

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://api.elsevier.com/content/article/PII:S0022404923001019?httpAccept=text/xml

Funders :

European Research Council
ANR
National Natural Science Foundation of China

Funding text :

The first and last authors were supported by ERC Consolidator Grant WallCrossAG, no. 819864 . The last author is also supported by ANR project FanoHK, grant ANR-20-CE40-0023 . The third author is partially supported by NSFC (nos. 11890660 and 11890663 ). Part of this work was finished while the last author was visiting the Max-Planck Institute For Mathematics. He is grateful for their excellent hospitality and support.

Available on ORBilu :

since 27 November 2023

Statistics

Number of views

98 (5 by Unilu)

Number of downloads

0 (0 by Unilu)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenAlex citations

WoS citations^™

Bibliography

Armenta, Marco Antonio, Keller, Bernhard, Derived invariance of the Tamarkin–Tsygan calculus of an algebra. C. R. Math. 357:3 (2019), 236–240.
Belmans, Pieter, Fanography. https://www.fanography.info//, 2021. (Accessed 16 July 2022)
Brambilla, Maria Chiara, Faenzi, Daniele, Rank 2 stable sheaves with odd determinant on Fano threefolds of genus 9. arXiv preprint arXiv:0905.1803, 2009.
Belmans, Pieter, Fatighenti, Enrico, Tanturri, Fabio, Polyvector fields for Fano 3-folds. arXiv preprint arXiv:2104.07626, 2021.
Blanc, Anthony, Topological k-theory of complex noncommutative spaces. Compos. Math. 152:3 (2016), 489–555.
Bayer MartíLahoz, Arend, Macrì, Emanuele, Stellari, Paolo, Stability conditions on Kuznetsov components. arXiv preprint arXiv:1703.10839, 2017.
Bernardara, Marcello, Macrì, Emanuele, Mehrotra, Sukhendu, Stellari, Paolo, A categorical invariant for cubic threefolds. Adv. Math. 229:2 (2012), 770–803.
Bondal, Alexei, Orlov, Dmitri, Reconstruction of a variety from the derived category and groups of autoequivalences. Compos. Math. 125:3 (2001), 327–344.
Bernardara, Marcello, Tabuada, Gonçalo, From semi-orthogonal decompositions to polarized intermediate Jacobians via Jacobians of noncommutative motives. Mosc. Math. J. 94:2 (2016), 205–235.
Căldăraru, Andrei, The Mukai pairing—II: the Hochschild–Kostant–Rosenberg isomorphism. Adv. Math. 194:1 (2005), 34–66.
Carlson, James, Green, Mark, Griffiths, Phillip, Harris, Joe, Infinitesimal variations of Hodge structure (I). Compos. Math. 50:2–3 (1983), 109–205.
Calaque, Damien, Rossi, Carlo A., Van den Bergh, Michel, Căldăraru's conjecture and Tsygan's formality. Ann. Math., 2012, 865–923.
Debarre, Olivier, Iliev, Atanas, Manivel, Laurent, On the period map for prime Fano threefolds of degree 10. J. Algebraic Geom. 21:1 (2012), 21–59.
Donagi, Ron, Generic Torelli for projective hypersurfaces. Compos. Math. 50:2–3 (1983), 325–353.
Flenner, Hubert, The infinitesimal Torelli problem for zero sets of sections of vector bundles. Math. Z. 193:2 (1986), 307–322.
Fatighenti, Enrico, Rizzi, Luca, Zucconi, Francesco, Weighted Fano varieties and infinitesimal Torelli problem. J. Geom. Phys. 139 (2019), 1–16.
Huybrechts, Daniel, Fourier–Mukai Transforms in Algebraic Geometry. 2006, Oxford University Press.
Iliev, Atanas, Markushevich, Dimitri, Parametrization of Sing(Θ) for a Fano 3-fold of genus 7 by moduli of vector bundles. Asian J. Math. 11:3 (2007), 427–458.
Jacovskis, Augustinas, Lin, Xun, Liu, Zhiyu, Zhang, Shizhuo, Categorical Torelli theorems for Gushel–Mukai threefolds. arXiv preprint arXiv:2108.02946, 2021.
Keller, Bernhard, Invariance and localization for cyclic homology of DG algebras. J. Pure Appl. Algebra 123:1 (1998), 223–273.
Keller, Bernhard, On differential graded categories. Proceedings of the International Congress of Mathematicians, Madrid, August 22–30, 2006, 2007, 151–190.
Konno, Kazuhiro, On deformations and the local Torelli problem of cyclic branched coverings. Math. Ann. 271:4 (1985), 601–617.
Konno, Kazuhiro, Infinitesimal Torelli theorem for complete intersections in certain homogeneous Kähler manifolds. Tohoku Math. J. (2) 38:4 (1986), 609–624.
Kuznetsov, Alexander, Perry, Alexander, Derived categories of Gushel–Mukai varieties. Compos. Math. 154:7 (2018), 1362–1406.
Kuznetsov, Alexander, Derived categories of cubic and V14 threefolds. Proc. Steklov Inst. Math. 246 (2004), 183–207.
Kuznetsov, Alexander, Derived categories of Fano threefolds. Proc. Steklov Inst. Math. 264:1 (2009), 110–122.
Kuznetsov, Alexander, Hochschild homology and semiorthogonal decompositions. arXiv preprint arXiv:0904.4330, 2009.
Kuznetsov, Alexander, Derived categories view on rationality problems. Rationality Problems in Algebraic Geometry, 2016, 67–104.
Kuznetsov, Alexander, Semiorthogonal decompositions in families. arXiv preprint arXiv:2111.00527, 2021.
Licht, Philipp, Infinitesimal Torelli for weighted complete intersections and certain Fano threefolds. arXiv preprint arXiv:2203.11127, 2022.
Lunts, Valery A., Orlov, Dmitri O., Uniqueness of enhancement for triangulated categories. J. Am. Math. Soc. 23:3 (Sep 2010), 853–907.
Ottaviani, Giorgio, Spinor bundles on quadrics. Trans. Am. Math. Soc. 307:1 (1988), 301–316.
Perry, Alexander, The integral Hodge conjecture for two-dimensional Calabi–Yau categories. Compos. Math. 158:2 (2022), 287–333.
Peters, C., The local Torelli theorem I. Complete intersections. Math. Ann. 223:2 (1976), 191–192.
Pertusi, Laura, Yang, Song, Some remarks on Fano three-folds of index two and stability conditions. Int. Math. Res. Not., 2021.
Saito, Masa-Hiko, Weak global Torelli theorem for certain weighted projective hypersurfaces. Duke Math. J. 53:1 (1986), 67–111.
Tabuada, Gonçalo, Invariants additifs de dg-catégories. Int. Math. Res. Not. 2005:53 (01 2005), 3309–3339.
Terasoma, Tomohide, Infinitestimal variation of Hodge structures and the weak global Torelli theorem for complete intersections. Ann. Math. 132:2 (1990), 213–235.
Usui, Sampei, Local Torelli theorem for non-singular complete intersections. Jpn. J. Math. New Ser. 2:2 (1976), 411–418.
Usui, Sampei, Local Torelli theorem for some non-singular weighted complete intersections. Proceedings of the International Symposium on Algebraic Geometry, Kyoto, 1977, 1978, Kinokuniya Book-Store Co., 723–734.