A category of kernels for equivariant factorizations and its implications for Hodge theory

Ballard, Matthew; Favero, David; Katzarkov, Ludmil

Publications Mathématiques de l'IHÉS, Tome 120 (2014), pp. 1-111.

We provide a factorization model for the continuous internal Hom, in the homotopy category of k-linear dg-categories, between dg-categories of equivariant factorizations. This motivates a notion, similar to that of Kuznetsov, which we call the extended Hochschild cohomology algebra of the category of equivariant factorizations. In some cases of geometric interest, extended Hochschild cohomology contains Hochschild cohomology as a subalgebra and Hochschild homology as a homogeneous component. We use our factorization model for the internal Hom to calculate the extended Hochschild cohomology for equivariant factorizations on affine space.

Combining the computation of extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov recovers and extends Griffiths’ classical description of the primitive cohomology of a smooth, complex projective hypersurface in terms of homogeneous pieces of the Jacobian algebra. In the process, the primitive cohomology is identified with the fixed subspace of the cohomological endomorphism associated to an interesting endofunctor of the bounded derived category of coherent sheaves on the hypersurface. We also demonstrate how to understand the whole Jacobian algebra as morphisms between kernels of endofunctors of the derived category.

Finally, we present a bootstrap method for producing algebraic cycles in categories of equivariant factorizations. As proof of concept, we show how this reproves the Hodge conjecture for all self-products of a particular K3 surface closely related to the Fermat cubic fourfold.

Mots clés : Algebraic Group, equivariant factorization, Triangulate Category, Homotopy Category, Hochschild Cohomology

[Ana73] Anantharaman, S. Schémas en groupes, espaces homogẽnes et espaces algébriques sur une base de dimension 1. Sur les groupes algébriques (1973), pp. 5-79 | Numdam | MR | Zbl

[Aok83] Aoki, N. On some arithmetic problems related to the Hodge cycles on the Fermat varieties, Math. Ann., Volume 266 (1983), pp. 23-54 | DOI | MR | Zbl

[BFK11] Ballard, M.; Favero, D.; Katzarkov, L. Orlov spectra: bounds and gaps, Invent. Math., Volume 189 (2012), pp. 359-430 | DOI | MR | Zbl

[BDFIK12] M. Ballard, D. Deliu, D. Favero, M. U. Isik, and L. Katzarkov, Resolutions in factorization categories, | arXiv

[BFK12] M. Ballard, D. Favero, and L. Katzarkov, Variation of geometric invariant theory quotients and derived categories, | arXiv

[BFK13] M. Ballard, D. Favero, and L. Katzarkov, A category of kernels for equivariant factorizations, II: further implications, preprint. | MR

[Bec12] H. Becker, Models for singularity categories, | arXiv

[BFN10] Ben-Zvi, D.; Francis, J.; Nadler, D. Integral transforms and Drinfeld centers in derived algebraic geometry, J. Am. Math. Soc., Volume 23 (2010), pp. 909-966 | DOI | MR | Zbl

[Bla12] A. Blanc, Topological K-theory and its Chern character for non-commutative spaces, | arXiv

[Blu07] M. Blume, McKay Correspondence and G-Hilbert Schemes, Ph.D. thesis, Tübingen, 2007. Currently available at http://tobias-lib.uni-tuebingen.de/volltexte/2007/2941/pdf/diss.pdf.

[BV03] Bondal, A.; Van den Bergh, M. Generators and representability of functors in commutative and non-commutative geometry, Mosc. Math. J., Volume 3 (2003), pp. 1-36 | MR | Zbl

[Buc86] R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, preprint (1986).

[Cal05] Căldăraru, A. The Mukai pairing, II: the Hochschild-Kostant-Rosenberg isomorphism, Adv. Math., Volume 194 (2005), pp. 34-66 | DOI | MR | Zbl

[CT10] A. Căldăraru and J. Tu, Curved A_∞ algebras and Landau-Ginzburg models, | arXiv | Zbl

[CS10] Canonaco, A.; Stellari, P. Non-uniqueness of Fourier-Mukai kernels, Math. Z., Volume 272 (2012), pp. 577-588 | DOI | MR | Zbl

[DeGa70] M. Demazure and P. Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. (French) Avec un appendice Corps de classes local par Michiel Hazewinkel. Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970. | MR | Zbl

[Dri04] Drinfeld, V. DG quotients of DG categories, J. Algebra, Volume 272 (2004), pp. 643-691 | DOI | MR | Zbl

[Dyc11] Dyckerhoff, T. Compact generators in categories of matrix factorizations, Duke Math. J., Volume 159 (2011), pp. 223-274 | DOI | MR | Zbl

[DM12] Dyckerhoff, T.; Murfet, D. The Kapustin-Li formula revisited, Adv. Math., Volume 231 (2012), pp. 1858-1885 | DOI | MR | Zbl

[Eis80] Eisenbud, D. Homological algebra on a complete intersection, with an application to group representations, Trans. Am. Math. Soc., Volume 260 (1980), pp. 35-64 | DOI | MR | Zbl

[Ela11] Elagin, A. Cohomological descent theory for a morphism of stacks and for equivariant derived categories, Mat. Sb., Volume 202 (2011), pp. 31-64 | DOI | MR | Zbl

[FJR07] Fan, H.; Jarvis, T.; Ruan, Y. The Witten equation, mirror symmetry and quantum singularity theory, Ann. Math., Volume 178 (2013), pp. 1-106 | DOI | MR | Zbl

[FHT01] Félix, Y.; Halperin, S.; Thomas, J.-C. Rational Homotopy Theory (2001) | MR | Zbl

[Gri69] Griffiths, P. On the periods of certain rational integrals, Ann. Math., Volume 90 (1969), pp. 460-541 | DOI | MR | Zbl

[EGA IV.2] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II (1965) | Numdam | MR | Zbl

[HKR62] Hochschild, G.; Kostant, B.; Rosenberg, A. Differential forms on regular affine algebras, Trans. Am. Math. Soc., Volume 102 (1962), pp. 383-408 | DOI | MR | Zbl

[HLOY04] Hosono, S.; Lian, B.; Oguiso, K.; Yau, S.-T. Fourier-Mukai number of a K3 surface, Algebraic Structures and Moduli Spaces (2004), pp. 177-192 | MR | Zbl

[Huy05] Huybrechts, D. Fourier-Mukai Transforms in Algebraic Geometry (2006) | MR | Zbl

[Ill71] L. Illusie, Existence de résolutions globales, in Théorie des intersections et théorème de Riemann-Roch. Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6). Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. Lecture Notes in Mathematics, vol. 225. Springer, Berlin, 1971. | MR | Zbl

[KL03a] Kapustin, A.; Li, Y. D-branes in Landau-Ginzburg models and algebraic geometry, J. High Energy Phys., Volume 5 (2003) | MR

[KL03b] Kapustin, A.; Li, Y. Topological correlators in Landau-Ginzburg models with boundaries, Adv. Theor. Math. Phys., Volume 7 (2003), pp. 727-749 | DOI | MR | Zbl

[KKP08] Katzarkov, L.; Kontsevich, M.; Pantev, T. Hodge theoretic aspects of mirror symmetry, From Hodge Theory to Integrability and TQFT tt ^∗-Geometry (2008), pp. 87-174 | MR | Zbl

[Kel06] Keller, B. On differential graded categories, International Congress of Mathematicians, vol. II (2006), pp. 151-190 | MR | Zbl

[KR08a] Khovanov, M.; Rozansky, L. Matrix factorizations and link homology, Fundam. Math., Volume 199 (2008), pp. 1-91 | DOI | MR | Zbl

[KR08b] Khovanov, M.; Rozansky, L. Matrix factorizations and link homology. II, Geom. Topol., Volume 12 (2008), pp. 1387-1425 | DOI | MR | Zbl

[Kon03] Kontsevich, M. Deformation quantization of Poisson manifolds, I, Lett. Math. Phys., Volume 66 (2003), pp. 157-216 | DOI | MR | Zbl

[Kra05] Krause, H. The stable derived category of a Noetherian scheme, Compos. Math., Volume 141 (2005), pp. 1128-1162 | DOI | MR | Zbl

[Kuz10] Kuznetsov, A. Derived categories of cubic fourfolds. Cohomological and geometric approaches to rationality problems (2010), pp. 219-243 | MR | Zbl

[Kuz11] Kuznetsov, A. Base change for semiorthogonal decompositions, Compos. Math., Volume 147 (2011), pp. 852-876 | DOI | MR | Zbl

[Kuz09] A. Kuznetsov, Hochschild homology and semiorthogonal decompositions, | arXiv

[LP11] K. Lin and D. Pomerleano, Global matrix factorizations, | arXiv | Zbl

[Mar01] N. Markarian, Poincaré-Birkhoff-Witt isomorphism, Hochschild homology and Riemann-Roch theorem, MPI 2001-52 preprint (2001). Currently available at http://www.mpim-bonn.mpg.de/preblob/1208.

[MFK94] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric Invariant Theory (1994) | MR | Zbl

[Mur09] D. Murfet, Residues and duality for singularity categories of isolated Gorenstein singularities, | arXiv

[Nee92] Neeman, A. The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. Éc. Norm. Super., Volume 25 (1992), pp. 547-566 | Numdam | MR | Zbl

[Orl04] Orlov, D. Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova, Volume 246 (2004), pp. 240-262 | MR | Zbl

[Orl06] Orlov, D. Triangulated categories of singularities, and equivalences between Landau-Ginzburg models, Mat. Sb., Volume 197 (2006), pp. 117-132 | DOI | MR | Zbl

[Orl09] Orlov, D. Derived categories of coherent sheaves and triangulated categories of singularities, Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, vol. II (2009), pp. 503-531 | MR | Zbl

[Orl11] Orlov, D. Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math., Volume 226 (2011), pp. 206-217 | DOI | MR | Zbl

[Orl12] Orlov, D. Matrix factorizations for nonaffine LG-models, Math. Ann., Volume 353 (2012), pp. 95-108 | DOI | MR | Zbl

[PV12] Polishchuk, A.; Vaintrob, A. Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations, Duke Math. J., Volume 161 (2012), pp. 1863-1926 | DOI | MR | Zbl

[PV10] A. Polishchuk and A. Vaintrob, Matrix factorizations and singularity categories for stacks, | arXiv | Numdam | Zbl

[PV11] A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, | arXiv

[Pos09] L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, | arXiv | Zbl

[Pos11] L. Positselski, Coherent analogues of matrix factorizations and relative singularity categories, | arXiv

[Pre11] A. Preygel, Thom-Sebastiani and duality for matrix factorizations, | arXiv

[Ram10] Ramadoss, A. The Mukai pairing and integral transforms in Hochschild homology, Mosc. Math. J., Volume 10 (2010), pp. 629-645 | MR | Zbl

[RM08] Ramón Marí, J. J. On the Hodge conjecture for products of certain surfaces, Collect. Math., Volume 59 (2008), pp. 1-26 | DOI | MR | Zbl

[Ran80] Ran, Z. Cycles on Fermat hypersurfaces, Compos. Math., Volume 42 (1980/81), pp. 121-142 | Numdam | MR | Zbl

[Rou08] Rouquier, R. Dimensions of triangulated categories, J. K-Theory, Volume 1 (2008), pp. 193-256 | DOI | MR | Zbl

[Seg09] E. Segal, The closed state space of affine Landau-Ginzburg B-models, | arXiv | Zbl

[Shi79] Shioda, T. The Hodge conjecture and the Tate conjecture for Fermat varieties, Proc. Jpn. Acad., Ser. A, Math. Sci., Volume 55 (1979), pp. 111-114 | DOI | MR | Zbl

[Shk07] D. Shklyarov, Hirzebruch-Riemman-Roch for DG algebras, | arXiv

[Swa96] Swan, R. Hochschild cohomology of quasiprojective schemes, J. Pure Appl. Algebra, Volume 110 (1996), pp. 57-80 | DOI | MR | Zbl

[Tho97] Thomason, R. W. Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math., Volume 65 (1987), pp. 16-34 | DOI | MR | Zbl

[Toë07] Toën, B. The homotopy theory of dg-categories and derived Morita theory, Invent. Math., Volume 167 (2007), pp. 615-667 | DOI | MR | Zbl

[Tot04] Totaro, B. The resolution property for schemes and stacks, J. Reine Angew. Math., Volume 577 (2004), pp. 1-22 | DOI | MR | Zbl

[Tu10] J. Tu, Matrix factorizations via Koszul duality, | arXiv | Zbl

[Vaf91] Vafa, C. Topological Landau-Ginzburg models, Mod. Phys. Lett. A, Volume 6 (1991), pp. 337-346 | DOI | MR | Zbl

[Wat79] Waterhouse, W. Introduction to Affine Group Schemes (1979) | MR | Zbl

[Yek02] Yekutieli, A. The continuous Hochschild cochain complex of a scheme, Can. J. Math., Volume 54 (2002), pp. 1319-1337 | DOI | MR | Zbl