×

Fourier–Mukai transforms for Gorenstein schemes. (English) Zbl 1118.14022

What is now called a Fourier-Mukai functor was first studied by S. Mukai [Nagoya Math. J. 81, 153–175 (1981; Zbl 0417.14036)]. Such a functor sends an object \(F\) of a derived category on \(X\) to \(Rq_{\ast}(K \otimes p^{\ast} F)\), where the so-called kernel \(K\) is an object of a derived category on \(X\times Y\) and \(p: X\times Y \rightarrow X\) and \(q: X\times Y\rightarrow Y\) are projections in the category of schemes. Such functors were shown to be a useful tool in the study of moduli spaces of coherent sheaves and to understand the structure of the derived category of coherent sheaves on a smooth variety. More recently, attempts were made to use derived categories and Fourier-Mukai functors to tackle problems in higher dimensional birational geometry.
The relevance of Fourier-Mukai functors in the smooth case is highlighted by Orlov’s Theorem which says that any exact equivalence between bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. Fourier-Mukai functors on singular varieties have not yet been studied in depth. The paper under review is one of the first attempts to generalise results to singular varieties which were previously known in the smooth case only. One of the main results of this article is a characterisation of those kernels \(K\) which give fully faithful functors (resp. equivalences) between bounded derived categories on Gorenstein schemes. This generalises a result of A. Bondal and D. Orlov [Semiorthogonal decomposition for algebraic varieties, preprint MPIM 95/15 (1995), see also arXiv:math/9506012]. An interesting example in characteristic \(p>0\) is provided which shows that the given criterion does not hold in the case of positive characteristic. As an application of their characterisation of Fourier-Mukai equivalences, the authors show that the dimension and the order of the canonical line bundle are Fourier-Mukai invariants of projective Gorenstein schemes.
In the final section of this paper, relative Fourier-Mukai transforms are studied. As their main application, they give a new proof of a result of I. Burban and the reviewer [Manuscr. Math. 120, No. 3, 283–306 (2006; Zbl 1105.18011)] which says that the Fourier-Mukai functor, given by the relative Poincaré bundle of an elliptic fibration with irreducible fibres, is an equivalence.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14E30 Minimal model program (Mori theory, extremal rays)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andreas, B.; Hernández Ruipérez, D., Fourier-Mukai transforms and applications to string theory, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 99, 1, 29-77 (2005), Available from: · Zbl 1086.14507
[2] Andreas, B.; Yau, S.-T.; Curio, G.; Hernández Ruipérez, D., Fibrewise \(T\)-duality for D-branes on elliptic Calabi-Yau, J. High Energy Phys. (3) (2001), paper 20, 13 pp
[3] Aspinwall, P. S., D-branes on Calabi-Yau manifolds, available from: · Zbl 1084.81058
[4] Bartocci, C.; Bruzzo, U.; Hernández Ruipérez, D., A Fourier-Mukai transform for stable bundles on K3 surfaces, J. Reine Angew. Math., 486, 1-16 (1997) · Zbl 0872.14013
[5] Bartocci, C.; Bruzzo, U.; Hernández Ruipérez, D., Existence of \(μ\)-stable vector bundles on K3 surfaces and the Fourier-Mukai transform, (Algebraic Geometry. Algebraic Geometry, Catania, 1993/Barcelona, 1994. Algebraic Geometry. Algebraic Geometry, Catania, 1993/Barcelona, 1994, Lecture Notes in Pure and Appl. Math., vol. 200 (1998), Dekker: Dekker New York), 245-257 · Zbl 0961.14025
[6] Bartocci, C.; Bruzzo, U.; Hernández Ruipérez, D.; Muñoz Porras, J. M., Mirror symmetry on K3 surfaces via Fourier-Mukai transform, Comm. Math. Phys., 195, 1, 79-93 (1998) · Zbl 0930.14028
[7] Bartocci, C.; Bruzzo, U.; Hernández Ruipérez, D.; Muñoz Porras, J. M., Relatively stable bundles over elliptic fibrations, Math. Nachr., 238, 23-36 (2002) · Zbl 1033.14007
[8] C. Bartocci, U. Bruzzo, D. Hernández Ruipérez, Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics, Progr. Math. Phys., Birkhäuser, 2007, in press; C. Bartocci, U. Bruzzo, D. Hernández Ruipérez, Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics, Progr. Math. Phys., Birkhäuser, 2007, in press
[9] Bondal, A. I.; Orlov, D. O., Semi orthogonal decomposition for algebraic varieties, MPIM preprint 95/15 (1995)
[10] Bondal, A. I.; Orlov, D. O., Reconstruction of a variety from the derived category and groups of autoequivalences, Compos. Math., 125, 3, 327-344 (2001) · Zbl 0994.18007
[11] Bridgeland, T., Fourier-Mukai transforms for elliptic surfaces, J. Reine Angew. Math., 498, 115-133 (1998) · Zbl 0905.14020
[12] Bridgeland, T., Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc., 31, 1, 25-34 (1999) · Zbl 0937.18012
[13] Bridgeland, T., Flops and derived categories, Invent. Math., 147, 3, 613-632 (2002) · Zbl 1085.14017
[14] Bridgeland, T.; Maciocia, A., Complex surfaces with equivalent derived categories, Math. Z., 236, 4, 677-697 (2001) · Zbl 1081.14023
[15] Bridgeland, T.; Maciocia, A., Fourier-Mukai transforms for \(K3\) and elliptic fibrations, J. Algebraic Geom., 11, 4, 629-657 (2002) · Zbl 1066.14047
[16] Burban, I.; Kreußler, B., On a relative Fourier-Mukai transform on genus one filtrations, preprint · Zbl 1105.18011
[17] Burban, I.; Kreußler, B., Fourier-Mukai transforms and semistable sheaves on nodal Weierstrass cubics, J. Reine Angew. Math., 564, 45-82 (2005) · Zbl 1085.14018
[18] Căldăraru, A., Derived categories of twisted sheaves on elliptic threefolds, J. Reine Angew. Math., 544, 161-179 (2002) · Zbl 0995.14012
[19] Chen, J.-C., Flops and equivalences of derived categories for threefolds with only terminal Gorenstein singularities, J. Differential Geom., 61, 2, 227-261 (2002) · Zbl 1090.14003
[20] Friedman, R.; Morgan, J. W.; Witten, E., Vector bundles over elliptic fibrations, J. Algebraic Geom., 8, 2, 279-401 (1999) · Zbl 0937.14004
[21] Hartshorne, R., Residues and Duality, Lecture Notes in Math., vol. 20 (1966), Springer-Verlag: Springer-Verlag Berlin, with an appendix by P. Deligne
[22] Hartshorne, R., Local Cohomology, A Seminar Given by Grothendieck, Harvard University, Fall 1961, Lecture Notes in Math., vol. 41 (1967), Springer-Verlag: Springer-Verlag Berlin
[23] Hernández Ruipérez, D.; Muñoz Porras, J. M., Stable sheaves on elliptic fibrations, J. Geom. Phys., 43, 2-3, 163-183 (2002) · Zbl 1068.14051
[24] Horja, R. P., Derived category automorphisms from mirror symmetry, Duke Math. J., 127, 1, 1-34 (2005) · Zbl 1075.18006
[25] Illusie, L., Frobenius et dégénérescence de Hodge, (Introduction à la théorie de Hodge. Introduction à la théorie de Hodge, Panor. Synthèses, vol. 3 (1996), Soc. Math. France: Soc. Math. France Paris), 113-168
[26] Kawamata, Y., \(D\)-equivalence and \(K\)-equivalence, J. Differential Geom., 61, 1, 147-171 (2002) · Zbl 1056.14021
[27] Kawamata, Y., Francia’s flip and derived categories, (Algebraic Geometry (2002), de Gruyter: de Gruyter Berlin), 197-215 · Zbl 1092.14023
[28] Kawamata, Y., Equivalences of derived categories of sheaves on smooth stacks, Amer. J. Math., 126, 5, 1057-1083 (2004) · Zbl 1076.14023
[29] Kuznetsov, A., Homological projective duality, available from: · Zbl 1131.14017
[30] Matsumura, H., Commutative Ring Theory, Cambridge Stud. Adv. Math., vol. 8 (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
[31] Miranda, R., The Basic Theory of Elliptic Surfaces, Dottorato di Ricerca in Matematica (Doctorate in Mathematical Research) (1989), ETS Editrice: ETS Editrice Pisa
[32] Mukai, S., Duality between \(D(X)\) and \(D(\hat{X})\) with its application to Picard sheaves, Nagoya Math. J., 81, 153-175 (1981) · Zbl 0417.14036
[33] Mukai, S., Fourier functor and its application to the moduli of bundles on an abelian variety, (Algebraic Geometry. Algebraic Geometry, Sendai, 1985. Algebraic Geometry. Algebraic Geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10 (1987), North-Holland: North-Holland Amsterdam), 515-550
[34] Mukai, S., On the moduli space of bundles on \(K3\) surfaces. I, (Vector Bundles on Algebraic Varieties. Vector Bundles on Algebraic Varieties, Bombay, 1984. Vector Bundles on Algebraic Varieties. Vector Bundles on Algebraic Varieties, Bombay, 1984, Tata Inst. Fund. Res. Stud. Math., vol. 11 (1987), Tata Inst. Fund. Res.: Tata Inst. Fund. Res. Bombay), 341-413 · Zbl 0674.14023
[35] Neeman, A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc., 9, 1, 205-236 (1996) · Zbl 0864.14008
[36] Orlov, D. O., Equivalences of derived categories and \(K3\) surfaces, J. Math. Sci. (NY), 84, 5, 1361-1381 (1997), Algebraic Geometry, 7 · Zbl 0938.14019
[37] Orlov, D. O., Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova, 246, 240-262 (2004) · Zbl 1101.81093
[38] Roberts, P. C., Multiplicities and Chern Classes in Local Algebra, Cambridge Tracts in Math., vol. 133 (1998), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0917.13007
[39] Van den Bergh, M., Three-dimensional flops and noncommutative rings, Duke Math. J., 122, 3, 423-455 (2004) · Zbl 1074.14013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.