Coherent Sheaves, Chern Classes, and Superconnections on compact complex-analytic manifolds
- Autores: Bondal A.I.1,2,3, Rosly A.A.4,5,6
-
Afiliações:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Moscow Institute of Physics and Technology (National Research University)
- Kavli Institute for the Physics and Mathematics of the Universe
- Skolkovo Institute of Science and Technology
- Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute)
- HSE University
- Edição: Volume 87, Nº 3 (2023)
- Páginas: 23-55
- Seção: Articles
- URL: https://ogarev-online.ru/1607-0046/article/view/133907
- DOI: https://doi.org/10.4213/im9386
- ID: 133907
Citar
Acesso aberto
Acesso está concedido
Somente assinantes
Resumo
We construct a twist-closed enhancement of the category $\mathcal D^b_{coh}(X)$, the boundedderived category of complexes of $\mathcal O_X$-modules with coherent cohomology, by meansof the DG-category of $\bar\partial$-superconnections. Then we apply the techniques of $\bar\partial$-superconnections to dene Chern classes and Bott–Chern classes of objects in the category, in particular, of coherent sheaves.
Palavras-chave
Sobre autores
Alexey Bondal
Steklov Mathematical Institute of Russian Academy of Sciences; Moscow Institute of Physics and Technology (National Research University); Kavli Institute for the Physics and Mathematics of the Universe
Email: bondal@mi-ras.ru
Doctor of physico-mathematical sciences
Alexei Rosly
Skolkovo Institute of Science and Technology; Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute); HSE University
Email: rosly@itep.ru
Candidate of physico-mathematical sciences
Bibliografia
- M. Verbitsky, “Coherent sheaves on general $K3$ surfaces and tori”, Pure Appl. Math. Q., 4:3 (2008), 651–714
- A. Bondal, D. Orlov, “Reconstruction of a variety from the derived category and groups of autoequivalences”, Compositio Math., 125:3 (2001), 327–344
- Д. О. Орлов, “Производные категории когерентных пучков на абелевых многообразиях и эквивалентности между ними”, Изв. РАН. Сер. матем., 66:3 (2002), 131–158
- M. Anel, B. Toën, “Denombrabilite des classes d'equivalences derivees de varietes algebriques”, J. Algebraic Geom., 18:2 (2009), 257–277
- J. Lesieutre, “Derived-equivalent rational threefolds”, Int. Math. Res. Not. IMRN, 2015:15 (2015), 6011–6020
- А. И. Бондал, М. М. Капранов, “Представимые функторы, функторы Серра и перестройки”, Изв. АН СССР. Сер. матем., 53:6 (1989), 1183–1205
- A. I. Bondal, M. van den Bergh, “Generators and representability of functors in commutative and noncommutative geometry”, Mosc. Math. J., 3:1 (2003), 1–36
- B. Toën, M. Vaquie, “Algebrisation des varietes analytiques complexes et categories derivees”, Math. Ann., 342:4 (2008), 789–831
- А. И. Бондал, М. М. Капранов, “Оснащенные триангулированные категории”, Матем. сб., 181:5 (1990), 669–683
- J. Block, “Duality and equivalence of module categories in noncommutative geometry”, A celebration of the mathematical legacy of Raoul Bott, CRM Proc. Lecture Notes, 50, Amer. Math. Soc., Providence, RI, 2010, 311–339
- C. Voisin, “A counterexample to the Hodge conjecture extended to Kähler varieties”, Int. Math. Res. Not. IMRN, 2002:20 (2002), 1057–1075
- N. Pali, “Faisceaux $overlinepartial$-coherents sur les varietes complexes”, Math. Ann., 336:3 (2006), 571–615
- N. Pali, Une caracterisation differentielle des faisceaux analytiques coherents sur une variete complexe
- D. Quillen, “Superconnections and the Chern character”, Topology, 24:1 (1985), 89–95
- A. Bondal, A. Rosly, Derived categories for complex-analytic manifolds, IPMU11-0117, IPMU, Kashiwa, Japan, 2011, 16 pp.
- J.-M. Bismut, Shu Shen, Zhaoting Wei, Coherent sheaves, superconnections, and RRG
- A. I. Bondal, M. Larsen, V.Ȧ. Lunts, “Grothendieck ring of pretriangulated categories”, Int. Math. Res. Not. IMRN, 2004:29 (2004), 1461–1495
- M. M. Kapranov, “On DG-modules over the de Rham complex and the vanishing cycles functor”, Algebraic geometry (Chicago, IL, 1989), Lecture Notes in Math., 1479, Springer, Berlin, 1991, 57–86
- C. Sabbah, Introduction to the theory of $mathscr D$-modules, Lecture notes (Nakai, 2011), 58 pp.
- L. Illusie, “Existence de resolutions globales”, Theorie des intersections et theorème de Riemann–Roch, Lecture Notes in Math., 225, Springer-Verlag, Berlin–New York, 1971, 160–221
- H. W. Schuster, “Locally free resolutions of coherent sheaves on surfaces”, J. Reine Angew. Math., 1982:337 (1982), 159–165
- M. Kashiwara, P. Schapira, Sheaves on manifolds, Grundlehren Math. Wiss., 292, Springer-Verlag, Berlin, 1994, x+512 pp.
- M. F. Atiyah, F. Hirzebruch, “Analytic cycles on complex manifolds”, Topology, 1:1 (1962), 25–45
- H. Grauert, “On Levi's problem and the imbedding of real-analytic manifolds”, Ann. of Math. (2), 68:2 (1958), 460–472
- Б. Мальгранж, Идеалы дифференцируемых функций, Мир, М., 1968, 131 с.
- J. Grivaux, “Chern classes in Deligne cohomology for coherent analytic sheaves”, Math. Ann., 347:2 (2010), 249–284
- D. S. Freed, Geometry of Dirac operators, unpublished notes, 1987
- D. Angella, A. Tomassini, “On the $partialoverlinepartial$-lemma and Bott–Chern cohomology”, Invent. Math., 192:1 (2013), 71–81
- М. Атья, И. Макдональд, Введение в коммутативную алгебру, Мир, М., 1972, 160 с.
- M. F. Atiyah, F. Hirzebruch, “The Riemann–Roch theorem for analytic embeddings”, Topology, 1:2 (1962), 151–166
- Н. Бурбаки, Коммутативная алгебра, Элементы математики, M., Мир, 1971, 708 с.
- Hua Qiang, On the Bott–Chern characteristic classes for coherent sheaves
Arquivos suplementares
