Geometric and categorical nonabelian duality in complex geometry

Kosarew, Siegmund

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 4, pp. 769-797.

The Leitmotiv of this work is to find suitable notions of dual varieties in a general sense. We develop the basic elements of a duality theory for varieties and complex spaces, by adopting a geometric and a categorical point of view. One main feature is to prove a biduality property for each notion which is achieved in most cases.

Classification : 14F05, 14M17, 32C37, 32G08

[1] J. F. Adams, “Stable homotopy and generalized homology”, The University of Chicago Press, Chicago, 1974. | MR | Zbl

[2] A. Altman - S. Kleiman, Compactifying the Picard Scheme, Adv. Math. 35 (1980), 50-112. | MR | Zbl

[3] V. Balaji - L. Brambila-Paz - P. Newstead, Stability of the Poincaré bundle, Math. Nachr. 188 (1997), 5-15. | MR | Zbl

[4] A. Bondal - D. Orlov., Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125 (2001), 327-344. | MR | Zbl

[5] A. Bondal - D. Orlov, Semiorthogonal decomposition for algebraic varieties, MPI Bonn, Preprint 95-15.

[6] T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), 25-34. | MR | Zbl

[7] I. Ciocan-Fontanine - M. M. Kapranov, Derived Quot schemes, Ann. Sci. Ecole Norm. Sup. (4) 34 (2001), 403-440. | Numdam | MR | Zbl

[8] I. Ciocan-Fontanine - M. M. Kapranov, Derived Hilbert schemes, J. Amer. Math. Soc. 15 (2002), 787-815. | MR | Zbl

[9] A. Dold - D. Puppe, Duality, trace and transfer, In: “Proc. Steklov Inst.” (4); Topology, A collection of papers, P. S. Alexandrov (ed.) (AMS translation), 1984, pp. 85-103. | MR | Zbl

[10] A. A. Kirillov, “Elements of the theory of representations”, Grundl. math. Wiss. 220, Springer-Verlag, Berlin-Heidelberg-New York, 1976. | MR | Zbl

[11] S. Kosarew, Nonabelian duality on Stein spaces, Amer. J. Math. 120 (1998), 637-648. | MR | Zbl

[12] S. Kosarew - C. Okonek, Global Moduli Spaces and Simple Holomorphic bundles, Publ. RIMS, Kyoto Univ. 25 (1989), 1-19. | MR | Zbl

[13] A. Maciocia, Generalized Fourier-Mukai transforms, J. Reine Angew. Math. 480 (1996), 197-211. | MR | Zbl

[14] S. Mac Lane, “Categories for the Working Mathematician”, Grad. Texts in Math. 5, Springer-Verlag, Berlin-Heidelberg-New York, 1971. | MR | Zbl

[15] S. Mukai, On the moduli space of bundles on a K3-surface, In: “Vector Bundles on Algebraic Varieties” Bombay Coll.,1984, Tata Inst., Oxford University Press, 1987, pp. 341-413. | MR | Zbl

[16] M. S. Narasimhan - S. Ramanan, Deformations of the moduli spaces of vector bundles over an algebraic curve, Ann. Math. 101 (1975), 391-417. | MR | Zbl

[17] A. Rosenberg, “Noncommutative Algebraic Geometry and Representations of Quantized Algebras”, Kluwer Acad. Publ., Dordrecht-London-Boston, 1995. | MR | Zbl

[18] C. S. Seshadri, “Fibrés vectoriels sur les courbes algébriques”, Astérisque 96 (1982). | Numdam | Zbl

[19] C. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Publ. Math. IHES 79 (1994), 47-129. | Numdam | MR | Zbl

[20] C. Simpson, A closed model structure for $n$-categories, internal $\mathrm{Hom}$, $n$-stacks and generalized Seifert-Van Kampen, Preprint math. AG 9704006.

[21] C. Simpson, Algebraic aspects of higher nonabelian Hodge theory, Preprint math. AG 9902067. | MR | Zbl

[22] B. Toen, Dualité de Tannaka supérieure I: Structures monoidales, MPI Bonn, Preprint June 10, 2000.

[23] G. W. Whitehead, “Elements of homotopy theory”, Grad. Texts in Math. 61, Springer-Verlag, Berlin-Heidelberg-New York, 1978. | MR | Zbl