Bridgeland, Tom; Maciocia, Antony Complex surfaces with equivalent derived categories. (English) Zbl 1081.14023 Math. Z. 236, No. 4, 677-697 (2001). For a smooth complex projective variety \(X\), let \(D(X)\) denote the triangulated category of bounded complexes of coherent sheaves on \(X\). \(D(X)\) is called the bounded derived category of \(X\). If \(Y\) is another smooth complex projective variety, then any equivalence of categories \(\Phi: D(Y)\to D(X)\) preserving the triangles is called a Fourier-Mukai transform. Due to the very fact that Fourier-Mukai transforms have shown themselves to be important tools in both moduli theory of sheaves and geometric duality theory for varieties, it is natural to attempt to classify them in a suitable manner. In this context, one crucial problem is to find, for a given variety \(X\), the set of Fourier-Mukai partners of \(X\) that is the set of varieties \(Y\) for which there exists a Fourier-Mukai transform relating \(X\) and \(Y\). In the paper under review, the authors provide a solution to this problem in the special case when \(X\) is a minimal complex projective surface. More precisely, their main theorem states the following: Let \(X\) be a minimal complex projective surface. Then any Fourier-Mukai partner \(Y\) is isomorphic to \(X\), with two basic exceptions:(1) \(X\) is an elliptic surface, and \(Y\) is another elliptic surface obtained by taking a relative Picard scheme of some elliptic fibration of \(X\); (2) \(X\) is a \(K3\) surface or an abelian surface, and \(Y\) is of the same type, respectively, with Hodge-isometric transcendental lattice. The proof of this classification theorem is rather long and subtle, since each type of surface appearing in the Enriques classification must be analyzed separately. As for surfaces of Kodaira dimension \(0\), the problem is mostly of lattice-theoretic nature, and the authors can use earlier results of V. V. Nikulin [Math. USSR, Izv. 14, 103–167 (1980; Zbl 0427.10014)]. Other surfaces are treated more geometrically, in particular by classifying curves on a surface with non-positive self-intersection which do not meet the canonical divisor. In the course of their fine analysis, the authors rely on related previous (partial) results of S. Mukai [Nagoya Math. J. 81, 153–175 (1981; Zbl 0417.14036)], T. Bridgeland [J. Reine Angew. Math. 498, 115–133 (1998; Zbl 0905.14020)], T. Bridgeland and A. Maciocia [Fourier-Mukai transforms for quotient varieties, Preprint http://arxiv.org/math.A6/9811101], and D. O. Orlov [J. Math. Sci., New York 84, No. 5, 1361–1381 (1997; Zbl 0938.14019)]. Reviewer: Werner Kleinert (Berlin) Cited in 7 ReviewsCited in 45 Documents MSC: 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 18E30 Derived categories, triangulated categories (MSC2010) Keywords:algebraic sheaves; Fourier-Mukai transform; Enriques classification of surfaces; fibrations Citations:Zbl 0427.10014; Zbl 0417.14036; Zbl 0905.14020; Zbl 0938.14019 × Cite Format Result Cite Review PDF Full Text: DOI arXiv