Dans cet article nous étudions et construisons des relations entre les sous-structures de Hodge de type Calabi-Yau sur des variétés de Fano qui sont des sous-variétés de grassmanniennes. En particulier, nous construisons un isomorphisme entre les sous-structures de Hodge de type Calabi-Yau des sections hyperplanes de et celles d’autres variétés provenant de grassmanniennes symplectiques et de congruences de droites ou de plans. Nous détaillons le cas des sections hyperplanes de , qui sont des variétés de Fano de type K3 dont la structure K3 est isomorphe à celle d’autres variétés de Fano comme la variété de Peskine. Ces isomorphismes sont obtenus via des correspondances géométriques entre différentes grassmanniennes, notamment des projections et des sauts via des variétés de drapeaux. Nous montrons aussi que ces correspondances permettent de construire une résolution catégorielle crépante de toute cubique de Coble. De plus, on montre une généralisation de la formule d’Orlov sur les décompositions semi-orthogonales des éclatements, qui permet de donner des versions (conjecturales) des résultats ci-dessus.
In this paper, we study and relate Calabi-Yau subHodge structures of Fano subvarieties of different Grassmannians. In particular, we construct isomorphisms between Calabi-Yau subHodge structures of hyperplane sections of and those of other varieties arising from symplectic Grassmannians and congruences of lines or planes. We describe in details the case of the hyperplane sections of , which are Fano varieties of K3 type whose K3 Hodge structures are isomorphic with those of other Fano varieties such as the Peskine variety. These isomorphisms are obtained via the study of geometrical correspondences between different Grassmannians, such as projections and jumps via two-step flags. We also show how these correspondences allow to construct crepant categorical resolutions of the Coble cubics. Finally, we prove a generalization of Orlov’s formula on semiorthogonal decompositions for blow-ups, which provides conjectural categorical counterparts of our Hodge-theoretical results.
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@article{JEP_2021__8__733_0,
author = {Marcello Bernardara and Enrico Fatighenti and Laurent Manivel},
title = {Nested varieties of {K3} type},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {733--778},
publisher = {\'Ecole polytechnique},
volume = {8},
year = {2021},
doi = {10.5802/jep.156},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.156/}
}
TY - JOUR AU - Marcello Bernardara AU - Enrico Fatighenti AU - Laurent Manivel TI - Nested varieties of K3 type JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 733 EP - 778 VL - 8 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.156/ DO - 10.5802/jep.156 LA - en ID - JEP_2021__8__733_0 ER -
%0 Journal Article %A Marcello Bernardara %A Enrico Fatighenti %A Laurent Manivel %T Nested varieties of K3 type %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 733-778 %V 8 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.156/ %R 10.5802/jep.156 %G en %F JEP_2021__8__733_0
Marcello Bernardara; Enrico Fatighenti; Laurent Manivel. Nested varieties of K3 type. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 733-778. doi : 10.5802/jep.156. https://jep.centre-mersenne.org/articles/10.5802/jep.156/
[1] - “Hodge theory and derived categories of cubic fourfolds”, Duke Math. J. 163 (2014) no. 10, p. 1885-1927 | DOI | MR | Zbl
[2] - “A note on nonvanishing and applications”, Duke Math. J. 72 (1993) no. 3, p. 739-755 | DOI | MR | Zbl
[3] - “Stability conditions on Kuznetsov components”, 2017, Appendix joint with X. Zhao | arXiv
[4] - “The Coble hypersurfaces”, Comptes Rendus Mathématique 337 (2003) no. 3, p. 189-194 | DOI | MR | Zbl
[5] - “La variété des droites d’une hypersurface cubique de dimension ”, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) no. 14, p. 703-706 | Zbl
[6] - “Bisymplectic Grassmannians of planes”, 2018 | arXiv
[7] - “Semiorthogonal decomposition for algebraic varieties”, 1995 | arXiv
[8] - “An introduction to motivic integration”, in Strings and geometry, Clay Math. Proc., vol. 3, American Mathematical Society, Providence, RI, 2004, p. 203-225 | MR | Zbl
[9] - “Fano congruences of index 3 and alternating 3-forms”, Ann. Inst. Fourier (Grenoble) 67 (2017) no. 5, p. 2099-2165 | DOI | Numdam | MR | Zbl
[10] - “Hyper-Kähler fourfolds and Grassmann geometry”, J. reine angew. Math. 649 (2010), p. 63-87 | DOI | Zbl
[11] - “Some special Cremona transformations”, Amer. J. Math. 111 (1989) no. 5, p. 783-800 | DOI | MR | Zbl
[12] - “A note on a Griffiths-type ring for complete intersections in Grassmannians”, 2018 | arXiv
[13] - “Fano varieties of K3 type and IHS manifolds”, Internat. Math. Res. Notices (2021) no. 4, p. 3097-3142 | DOI | MR
[14] - “On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type”, Pure Appl. Math. Q 10 (2014) no. 1, p. 1-55 | DOI | MR | Zbl
[15] - “Moduli of abelian varieties, Vinberg -groups, and free resolutions”, in Commutative algebra, Springer, New York, 2013, p. 419-469 | DOI | Zbl
[16] - “Special cubic fourfolds”, Compositio Math. 120 (2000) no. 1, p. 1-23 | DOI | MR | Zbl
[17] - Fourier-Mukai transforms in algebraic geometry, Oxford Math. Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006 | DOI | Zbl
[18] - “Hochschild cohomology versus the Jacobian ring and the Torelli theorem for cubic fourfolds”, Algebraic Geom. 6 (2019) no. 1, p. 76-99 | DOI | MR | Zbl
[19] - “Fano manifolds of Calabi-Yau Hodge type”, J. Pure Appl. Algebra 219 (2015) no. 6, p. 2225-2244 | DOI | MR | Zbl
[20] - “Derived category of projectivization and flops”, 2018 | arXiv
[21] - “Lefschetz decompositions and categorical resolutions of singularities”, Selecta Math. (N.S.) 13 (2008) no. 4, p. 661-696 | DOI | MR | Zbl
[22] - “Hochschild homology and semiorthogonal decompositions”, 2009 | arXiv
[23] - “Derived categories of cubic fourfolds”, in Cohomological and geometric approaches to rationality problems, Progress in Math., vol. 282, Birkhäuser Boston, Boston, MA, 2010, p. 219-243 | DOI | MR | Zbl
[24] - “Calabi-Yau and fractional Calabi-Yau categories”, J. reine angew. Math. 753 (2019), p. 239-267 | DOI | MR | Zbl
[25] - “Non-commutative crepant resolutions: scenes from categorical geometry”, in Progress in commutative algebra 1, de Gruyter, Berlin, 2012, p. 293-361 | MR | Zbl
[26] - “Twisted cubics on cubic fourfolds and stability conditions”, 2018 | arXiv
[27] - “Categorical resolution of singularities”, J. Algebra 323 (2010) no. 10, p. 2977-3003 | DOI | MR | Zbl
[28] - Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, vol. 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001 | MR | Zbl
[29] - “Projective bundles, monoidal transformations, and derived categories of coherent sheaves”, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992) no. 4, p. 852-862 | DOI
[30] - “Hori-mological projective duality”, Duke Math. J. 168 (2019) no. 11, p. 2127-2205 | DOI | MR | Zbl
[31] - “Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces”, Math. Ann. 276 (1986) no. 1, p. 159-176 | DOI | MR | Zbl
[32] - “Submanifolds of Abelian varieties”, Math. Ann. 233 (1978) no. 3, p. 229-256 | DOI | MR | Zbl
[33] - Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés, vol. 10, Société Mathématique de France, Paris, 2002 | DOI | Zbl
[34] - Cohomology of vector bundles and syzygies, Cambridge Tracts in Math., vol. 149, Cambridge University Press, Cambridge, 2003 | DOI | MR | Zbl
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