A category of kernels for equivariant factorizations and its implications for Hodge theory. (English) Zbl 1401.14086

Ballard, Matthew; Favero, David; Katzarkov, Ludmil

Publ. Math., Inst. Hautes Étud. Sci. 120, 1-111 (2014).

In the paper under review the authors prove that, roughly speaking, the continuous internal Hom between dg-categories of equivariant factorizations in the homotopy category of \(k\)-linear dg-categories is again a category of factorizations.
Let \(X\) be a smooth variety over an algebraically closed field \(k\) of characteristic zero equipped with an action of an affine algebraic group \(G\). Write \(\pi. G\times X\to X\) for the projection and \(\sigma: G\times X\to X\) for the action. Recall that a \(G\)-equivariant quasi-coherent sheaf on \(X\) is a quasi-coherent sheaf \(\mathcal{F}\) together with an isomorphism \(\theta: \sigma^*\mathcal{F}\to \pi^*\mathcal{F}\) satisfying a cocycle condition. This gives an abelian category \(\mathrm{Qcoh}_GX\). If \(\mathcal{F}\) is coherent or locally free, we accordingly call \((\mathcal{F},\theta)\) coherent or locally free. Let \(w\) be a \(G\)-invariant global section of an invertible equivariant sheaf \(\mathcal{L}\). A quasi-coherent matrix factorization of \(w\) is given by equivariant quasi-coherent sheaves \(\mathcal{E}_{-1}\), \(\mathcal{E}_0\) together with maps \(\phi_0^{\mathcal E}: \mathcal E_{-1}\to \mathcal{E}_{0}\) and \(\phi_{-1}^{\mathcal E}: \mathcal E_{0}\to \mathcal{E}_{-1}\otimes \mathcal{L}\) satisfying \[ \phi_{-1}^{\mathcal E}\circ\phi_{0}^{\mathcal E}=w=(\phi_{0}^{\mathcal E}\otimes\mathcal{L})\circ\phi_{-1}^{\mathcal E}. \] Denote such an object by \(\mathcal{E}\). The morphisms between two matrix factorizations form a complex, where, for instance, \[ \operatorname{Hom}^{2l}(\mathcal{E},\mathcal{F}):=\mathrm{Hom}(\mathcal{E}_{-1},\mathcal{F}_{-1}\otimes\mathcal{L}^l)\oplus {\operatorname{Hom}}(\mathcal{E}_{0},\mathcal{F}_{0}\otimes\mathcal{L}^l) \] for \(l\in \mathbb{Z}\) and the differential is given by an explicit formula. Hence, matrix factorizations form a dg-category, denoted by \(\mathrm{Fact}(X,G,w)\). Considering matrix factorizations with injective components gives a dg-category \(\mathrm{Inj}(X,G,w)\).
The main result of the paper under review can be described as follows. Let \((X,G,w)\) be as above. Let \(H\) be an affine algebraic group acting on a smooth variety \(Y\) and let \(v\) be a \(G\)-invariant section of an \(H\)-equivariant line bundle \(\mathcal{L}'\) on \(Y\). Let \(U(\mathcal{L})\) be the complement of the zero section in the geometric vector bundle corresponding to \(\mathcal{L}\), and denote the same construction for \(\mathcal{L}'\) by \(U(\mathcal{L}')\). Denote the functions induced by \(w\) and \(v\) on \(U(\mathcal{L})\) and \(U(\mathcal{L}')\) by \(f_w\) and \(f_v\), respectively. Set \(-f_w\boxplus f_v=-f_w\otimes_k 1+1\otimes_k f_v\). Note that \(G\times H\) acts on \(U(\mathcal{L})\times U(\mathcal{L}')\) and allow \(\mathbb{G}_m\) to scale the fibres of \(U(\mathcal{L})\times U(\mathcal{L}')\) diagonally. The authors prove that there is an equivalence \[ \mathrm{RHom}_c(\mathrm{Inj}(X,G,w),\mathrm{Inj}(Y,H,v))\cong\mathrm{Inj}(U(\mathcal{L})\times U(\mathcal{L}'), G\times H\times \mathbb{G}_m,-f_w\boxplus f_v) \] in the homotopy category of \(k\)-linear dg-categories. So, roughly speaking, the functors between categories of matrix factorizations are again given by matrix factorizations. The above result is used to compute the (extended) Hochschild cohomology of \((X,G,w)\) when \(X\) is affine and \(G\), \(w\) satisfy some technical assumptions.
The story begins with a thorough introduction to equivariant sheaves. The authors define the abelian category of quasi-coherent equivariant sheaves, restriction and inflation functors, pullback etc. They also study the global dimension of \(\mathrm{Qcoh}_GX\).
Section 3 is devoted to equivariant factorizations. The dg-categories mentioned above and their variants (for instance, one involving coherent sheaves) are defined and studied. For example, the authors define a dg-functor \(\mathrm{Fact}(X,G,w)\otimes_k \mathrm{Fact}(X,G,v)\to \mathrm{Fact}(X,G,w+v)\), a version of the \(\mathcal{H}om\)-functor in this setting, an appropriate notion of box product, restriction and inflation functors for the case of a subgroup \(H\) of \(G\), and establish results relating various of these functors.
In the same section the authors also introduce and study the absolute derived category of matrix factorizations, following work of Positselski. The idea of the construction is roughly as follows. To the dg-category \(\mathrm{Fact}(X,G,w)\) one can associate an abelian category having the same objects, but where the morphisms between two factorizations are given by closed degree-zero morphisms between them in \(\mathrm{Fact}(X,G,w)\). The resulting category is denoted by \(Z^0\mathrm{Fact}(X,G,w)=\mathcal{A}\). Given a complex with objects from \(\mathcal{A}\), one can define a matrix factorization, called its totalization. Considering the subcategory of \(\mathrm{Fact}(X,G,w)\) consisting of totalizations of bounded exact complexes from \(\mathcal{A}\) gives the subcategory of acyclic factorizations. The absolute derived category \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) is then defined as the Verdier quotient of \([\mathrm{Fact}(X,G,w)]\), the homotopy category of \(\mathrm{Fact}(X,G,w)\) (which is a triangulated category), by the homotopy category of acyclic factorizations. Of course, there are variants of this definition if one works with coherent or locally free factorizations.
One of the good properties \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) has is that it is a compactly-generated triangulated category and equivalent to \([\mathrm{Inj}(X,G,w)]\). Furthermore, the authors show that the idempotent completion of the coherent version \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) is equivalent to the homotopy category of those factorizations of \(\mathrm{Inj}(X,G,w)\) which are compact in \([\mathrm{Inj}(X,G,w)]\). Another advantage of the absolute derived category is the existence of an essentially surjective functor from a certain singularity category to it, allowing one to use geometry when proving statements about the absolute derived category. More precisely, let \(Y\) be the vanishing locus of \(w\). Under some technical conditions, there is an essentially surjective functor \[ \mathrm{D}^{\mathrm{sg}}_G(Y):=\mathrm{D}^{\mathrm{b}}(\mathrm{Coh}_GY)/\mathrm{Perf}_GY\to \mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)], \] where \(\mathrm{Perf}_GY\) is the subcategory of perfect complexes, that is, bounded complexes of locally free \(G\)-equivariant sheaves of finite rank. The functor actually exists in greater generality, since one can consider categories supported on a closed \(G\)-invariant subset of \(Y\).
In Section 4 generation of equivariant derived categories is studied. More precisely, one wants to find a set of generators for the bounded derived category of coherent \(G\)-equivariant sheaves \(\mathrm{D}^{\mathrm{b}}(\mathrm{Coh}_G X)\), where \(X\) is a singular variety equipped with a \(G\)-action. The rough idea is to focus on \(\mathrm{D}^{\mathrm{b}}(\mathrm{Qcoh}_G X)\). Using the essentially surjective functor mentioned above, the authors can then produce a set of generators for an appropriate absolute derived category.
In the following section the main result is proved. For this the authors need to recall some facts from the theory of dg-categories and, in particular, results by Toën concerning the construction of the internal (continuous) Hom in the homotopy category of dg-categories. One of the main steps towards the proof of the main result is an equivalence between some absolute derived category and the derived category of dg-modules over the tensor product of matrix factorizations categories, and this is where the generators from the previous section come in, since the proof involves showing that a compact generating set is sent to a compact generating set and is fully faithful on these. In the same section the authors define and study the (extended) Hochschild (co)homology and compute it in the case mentioned above.
The last section is devoted to two applications. Firstly, combining the computation of the extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov relating the bounded derived category of coherent sheaves on a smooth complex hypersurface \(Z\) in projective space with equivariant matrix factorizations allows the authors to recover Griffiths’ description of the primitive cohomology of \(Z\) in terms of homogeneous pieces of the Jacobian algebra of the polynomial defining \(Z\). As a second application the authors give a new proof of the Hodge conjecture for self-products of a particular \(K3\) surface closely related to the Fermat cubic fourfold.

Reviewer: Pawel Sosna (Hamburg) (MR3270588)

MSC:

14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
18E30	Derived categories, triangulated categories (MSC2010)
18G55	Nonabelian homotopical algebra (MSC2010)

References:

[1]	S. Anantharaman, Schémas en groupes, espaces homogẽnes et espaces algébriques sur une base de dimension 1. Sur les groupes algébriques, Bull. Soc. Math. France, Mem., vol. 33, pp. 5-79, Soc. Math. France, Paris, 1973 (French). · Zbl 0286.14001
[2]	N. Aoki, On some arithmetic problems related to the Hodge cycles on the Fermat varieties, Math. Ann., 266 (1983), 23-54. · Zbl 0506.14030 · doi:10.1007/BF01458703
[3]	M. Ballard, D. Favero, and L. Katzarkov, Orlov spectra: bounds and gaps, Invent. Math., 189 (2012), 359-430. · Zbl 1266.14013 · doi:10.1007/s00222-011-0367-y
[4]	M. Ballard, D. Deliu, D. Favero, M. U. Isik, and L. Katzarkov, Resolutions in factorization categories, arXiv:1212.3264. · Zbl 1353.13016
[5]	M. Ballard, D. Favero, and L. Katzarkov, Variation of geometric invariant theory quotients and derived categories, arXiv:1203.6643. · Zbl 1400.14048
[6]	M. Ballard, D. Favero, and L. Katzarkov, A category of kernels for equivariant factorizations, II: further implications, preprint. · Zbl 1326.14036
[7]	H. Becker, Models for singularity categories, arXiv:1205.4473. · Zbl 1348.16009
[8]	D. Ben-Zvi, J. Francis, and D. Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, J. Am. Math. Soc., 23 (2010), 909-966. · Zbl 1202.14015 · doi:10.1090/S0894-0347-10-00669-7
[9]	A. Blanc, Topological K-theory and its Chern character for non-commutative spaces, arXiv:1211.7360. · Zbl 1200.18007
[10]	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. · Zbl 1269.81168
[11]	A. Bondal and M. Van den Bergh, Generators and representability of functors in commutative and non-commutative geometry, Mosc. Math. J., 3 (2003), 1-36. · Zbl 1135.18302
[12]	R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, preprint (1986). · Zbl 1218.18009
[13]	A. Căldăraru, The Mukai pairing, II: the Hochschild-Kostant-Rosenberg isomorphism, Adv. Math., 194 (2005), 34-66. · Zbl 1098.14011 · doi:10.1016/j.aim.2004.05.012
[14]	A. Căldăraru and J. Tu, Curved A∞ algebras and Landau-Ginzburg models, arXiv:1007.2679. · Zbl 1278.18022
[15]	A. Canonaco and P. Stellari, Non-uniqueness of Fourier-Mukai kernels, Math. Z., 272 (2012), 577-588. · Zbl 1282.14033 · doi:10.1007/s00209-011-0950-3
[16]	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. · Zbl 0203.23401
[17]	V. Drinfeld, DG quotients of DG categories, J. Algebra, 272 (2004), 643-691. · Zbl 1064.18009 · doi:10.1016/j.jalgebra.2003.05.001
[18]	T. Dyckerhoff, Compact generators in categories of matrix factorizations, Duke Math. J., 159 (2011), 223-274. · Zbl 1252.18026 · doi:10.1215/00127094-1415869
[19]	T. Dyckerhoff and D. Murfet, The Kapustin-Li formula revisited, Adv. Math., 231 (2012), 1858-1885. · Zbl 1269.81168 · doi:10.1016/j.aim.2012.07.021
[20]	D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Am. Math. Soc., 260 (1980), 35-64. · Zbl 0444.13006 · doi:10.1090/S0002-9947-1980-0570778-7
[21]	A. Elagin, Cohomological descent theory for a morphism of stacks and for equivariant derived categories, Mat. Sb., 202 (2011), 31-64 (Russian); translation in Sb. Math., 202 (2011), 495-526. · Zbl 1234.18006 · doi:10.4213/sm7729
[22]	H. Fan, T. Jarvis, and Y. Ruan, The Witten equation, mirror symmetry and quantum singularity theory, Ann. Math., 178 (2013), 1-106 · Zbl 1310.32032 · doi:10.4007/annals.2013.178.1.1
[23]	Y. Félix, S. Halperin, and J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, vol. 205, Springer, New York, 2001. · Zbl 0961.55002
[24]	P. Griffiths, On the periods of certain rational integrals, Ann. Math., 90 (1969), 460-541. · Zbl 0215.08103 · doi:10.2307/1970746
[25]	A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math., vol. 24, 1965 (French). · Zbl 0135.39701
[26]	G. Hochschild, B. Kostant, and A. Rosenberg, Differential forms on regular affine algebras, Trans. Am. Math. Soc., 102 (1962), 383-408. · Zbl 0102.27701 · doi:10.1090/S0002-9947-1962-0142598-8
[27]	Hosono, S.; Lian, B.; Oguiso, K.; Yau, S.-T., Fourier-Mukai number of a K3 surface, No. 38, 177-192 (2004), Providence · Zbl 1076.14045
[28]	D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. · Zbl 1095.14002 · doi:10.1093/acprof:oso/9780199296866.001.0001
[29]	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. · Zbl 1058.81061
[30]	A. Kapustin and Y. Li, D-branes in Landau-Ginzburg models and algebraic geometry, J. High Energy Phys., 5 (2003) 44 (electronic).
[31]	A. Kapustin and Y. Li, Topological correlators in Landau-Ginzburg models with boundaries, Adv. Theor. Math. Phys., 7 (2003), 727-749. · Zbl 1058.81061 · doi:10.4310/ATMP.2003.v7.n4.a5
[32]	Katzarkov, L.; Kontsevich, M.; Pantev, T., Hodge theoretic aspects of mirror symmetry, No. 78, 87-174 (2008), Providence · Zbl 1206.14009 · doi:10.1090/pspum/078/2483750
[33]	Keller, B., On differential graded categories, 151-190 (2006), Zürich · Zbl 1140.18008
[34]	M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fundam. Math., 199 (2008), 1-91. · Zbl 1145.57009 · doi:10.4064/fm199-1-1
[35]	M. Khovanov and L. Rozansky, Matrix factorizations and link homology. II, Geom. Topol., 12 (2008), 1387-1425. · Zbl 1146.57018 · doi:10.2140/gt.2008.12.1387
[36]	M. Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys., 66 (2003), 157-216. · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf
[37]	H. Krause, The stable derived category of a Noetherian scheme, Compos. Math., 141 (2005), 1128-1162. · Zbl 1090.18006 · doi:10.1112/S0010437X05001375
[38]	A. Kuznetsov, Derived categories of cubic fourfolds. Cohomological and geometric approaches to rationality problems, in Progr. Math., vol. 282, pp. 219-243, Birkhäuser, Boston, 2010. · Zbl 1202.14012 · doi:10.1007/978-0-8176-4934-0_9
[39]	A. Kuznetsov, Base change for semiorthogonal decompositions, Compos. Math., 147 (2011), 852-876. · Zbl 1218.18009 · doi:10.1112/S0010437X10005166
[40]	A. Kuznetsov, Hochschild homology and semiorthogonal decompositions, arXiv:0904.4330. · Zbl 0102.27701
[41]	K. Lin and D. Pomerleano, Global matrix factorizations, arXiv:1101.5847. · Zbl 1285.14019
[42]	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. · Zbl 1135.18302
[43]	D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer, Berlin, 1994. · Zbl 0797.14004 · doi:10.1007/978-3-642-57916-5
[44]	D. Murfet, Residues and duality for singularity categories of isolated Gorenstein singularities, arXiv:0912.1629. · Zbl 1326.13007
[45]	A. Neeman, 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., 25 (1992), 547-566. · Zbl 0868.19001
[46]	D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova, 246 (2004), 240-262. · Zbl 1101.81093
[47]	D. Orlov, Triangulated categories of singularities, and equivalences between Landau-Ginzburg models, Mat. Sb., 197 (2006), 117-132 (Russian); translation in Sb. Math., 197 (2006), 1827-1840. · Zbl 1161.14301 · doi:10.4213/sm1128
[48]	Orlov, D., Derived categories of coherent sheaves and triangulated categories of singularities, No. 270, 503-531 (2009), Boston · Zbl 1200.18007 · doi:10.1007/978-0-8176-4747-6_16
[49]	D. Orlov, Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math., 226 (2011), 206-217. · Zbl 1216.18012 · doi:10.1016/j.aim.2010.06.016
[50]	D. Orlov, Matrix factorizations for nonaffine LG-models, Math. Ann., 353 (2012), 95-108. · Zbl 1243.81178 · doi:10.1007/s00208-011-0676-x
[51]	A. Polishchuk and A. Vaintrob, Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations, Duke Math. J., 161 (2012), 1863-1926. · Zbl 1249.14001 · doi:10.1215/00127094-1645540
[52]	A. Polishchuk and A. Vaintrob, Matrix factorizations and singularity categories for stacks, arXiv:1011.4544. · Zbl 1278.13014
[53]	A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, arXiv:1105.2903. · Zbl 1357.14024
[54]	L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, arXiv:0905.2621. · Zbl 1275.18002
[55]	L. Positselski, Coherent analogues of matrix factorizations and relative singularity categories, arXiv:1102.0261. · Zbl 1333.14018
[56]	A. Preygel, Thom-Sebastiani and duality for matrix factorizations, arXiv:1101.5834. · Zbl 0624.14025
[57]	A. Ramadoss, The Mukai pairing and integral transforms in Hochschild homology, Mosc. Math. J., 10 (2010), 629-645, 662-663. · Zbl 1208.14013
[58]	J. J. Ramón Marí, On the Hodge conjecture for products of certain surfaces, Collect. Math., 59 (2008), 1-26. · Zbl 1188.14004 · doi:10.1007/BF03191179
[59]	Z. Ran, Cycles on Fermat hypersurfaces, Compos. Math., 42 (1980/81), 121-142. · Zbl 0463.14003
[60]	R. Rouquier, Dimensions of triangulated categories, J. K-Theory, 1 (2008), 193-256. · Zbl 1165.18008 · doi:10.1017/is007011012jkt010
[61]	E. Segal, The closed state space of affine Landau-Ginzburg B-models, arXiv:0904.1339. · Zbl 1286.14003
[62]	T. Shioda, The Hodge conjecture and the Tate conjecture for Fermat varieties, Proc. Jpn. Acad., Ser. A, Math. Sci., 55 (1979), 111-114. · Zbl 0444.14017 · doi:10.3792/pjaa.55.111
[63]	D. Shklyarov, Hirzebruch-Riemman-Roch for DG algebras, arXiv:0710.1937. · Zbl 1330.16007
[64]	R. Swan, Hochschild cohomology of quasiprojective schemes, J. Pure Appl. Algebra, 110 (1996), 57-80. · Zbl 0865.18010 · doi:10.1016/0022-4049(95)00091-7
[65]	R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math., 65 (1987), 16-34. · Zbl 0624.14025 · doi:10.1016/0001-8708(87)90016-8
[66]	B. Toën, The homotopy theory of dg-categories and derived Morita theory, Invent. Math., 167 (2007), 615-667. · Zbl 1118.18010 · doi:10.1007/s00222-006-0025-y
[67]	B. Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math., 577 (2004), 1-22. · Zbl 1077.14004 · doi:10.1515/crll.2004.2004.577.1
[68]	J. Tu, Matrix factorizations via Koszul duality, arXiv:1009.4151. · Zbl 1305.18063
[69]	C. Vafa, Topological Landau-Ginzburg models, Mod. Phys. Lett. A, 6 (1991), 337-346. · Zbl 1020.81886 · doi:10.1142/S0217732391000324
[70]	W. Waterhouse, Introduction to Affine Group Schemes, Graduate Texts in Mathematics, vol. 66, Springer, New York, 1979. · Zbl 0442.14017
[71]	A. Yekutieli, The continuous Hochschild cochain complex of a scheme, Can. J. Math., 54 (2002), 1319-1337. · Zbl 1047.16004 · doi:10.4153/CJM-2002-051-8