Braid group actions on derived categories of coherent sheaves.

*(English)*Zbl 1092.14025Let \(X\) be a smooth complex projective variety, and let \(D^b(X)\) be the bounded derived category of coherent sheaves on \(X\). Due to several spectacular results obtained during the past decade, it has become evident that the derived category \(D^b(X)\) encodes quite a lot of information about the variety \(X\). In fact, certain invariants of \(X\) turned out to depend only on \(D^b(x)\), and certain special varieties could be even entirely reconstructed from \(D^b(X)\) and its group of self-equivalences. On the other hand, there are also examples of non-isomorphic varieties with equivalent derived categories, which seem to indicate that the structure of the group \(\text{Auteq}(D^b(X))\) of self-equivalences of \(D^b(X)\) essentially regulates the relationship between the variety \(X\) and its derived category.

The paper under review is devoted to a refined analysis of the group \(\text{Auteq}(D^b(X))\) from a particular viewpoint. The authors’ approach is motivated by the mirror symmetry phenomenon, especially by M. Kontsevich’s celebrated “Homological Mirror Conjecture”. Namely, this conjecture predicts an equivalence of derived categories of coherent sheaves on a Calabi-Yau manifold, on the one hand, and Floer-Fukaya categories of Lagrangians in the dual symplectic manifold, on the other hand, and one consequence of this conjecture is that, for Calabi-Yau manifolds to which mirror symmetry applies, the group \(\text{Auteq}(D(X))\) should be closely related to the symplectic automorphisms of the mirror manifold. Basically, this rather abstract and difficult conjectural relationship is the main topic of the present paper. With a view to the occurrence or certain braid group actions in symplectic geometry, the authors give a construction of braid group actions on bounded derived categories of fairly general abelian categories, and apply then this general framework to the concrete situation of derived categories of coherent sheaves and their groups of self-equivalences.

As to the contents, the paper consists of four main sections, each of which is subdiveded into several subsections.

Section 1 gives a very thorough and detailed introduction explaining the general problem, its genesis, its overall significance, the authors’ original strategy of tackling it, and the main results of the present work.

Section 2 develops a theory of spherical objects and twist functors for derived categories of general abelian categories satisfying certain (axiomatic) properties. These properties are always satisfied by the categories of (quasi-)coherent sheaves on a noetherian scheme over a ground field \(k\). It is then shown that special configurations of spherical objects in \(D^b(\text{Coh}(X))\) generate homomorphisms from braid groups to \(\text{Auteq}(D^b(\text{Coh}(X))\), where \(X\) is a noetherian \(k\)-scheme, \(\text{Coh}(X)\) is its category of coherent sheaves, and \(\text{Auteq}(D^b(\text{Coh} (X))\) stands for the group of self-equivalences of the derived category \(D^b (\text{Coh}(X))\). Having established various braid group actions on derived categories of coherent sheaves in such a way the authors study their concrete applications to different classes of (quasi-) projective varieties in Section 3. This includes singular varieties, varieties with finite group action, Fano varieties, and the well-understood case of elliptic curves as handy testing grounds. Moreover, the authors present a systematic way of producing spherical objects in the respective derived categories of coherent sheaves, they show how groups of symplectic automorphisms and groups of categorical self-equivalences can be compared in an explicit way, and they finally put their results in the context of mirror symmetry for singularities of threefolds, thereby returning to the main motivation for their work.

Section 4 is devoted to one of the main results of the paper. It contains the proof of the following theorem (Theorem 2.18): Let \(X\) be a noetherian scheme over an algebraically closed field \(k\), and assume that \(\dim X\geq 2\). Then, for any integer \(m<0\), the constructed group homomorphism \(B_{m+1}\to\text{Auteq}(D^b(\text{Coh}(X))\) from the braid group \(B_{m+1}\) to the self-equivalence group \(D^b(\text{Coh}(X))\) is injective.

This establishes the fact that the various braid group actions constructed in Section 2 (via configurations of \(m\) spherical objects in \(D^b(\text{Coh} (X))\); cf. Theorem 2.17) are even faithful. The intricate proof uses techniques from the theory of differential graded algebras and modules, their derived categories, and their Hochschild cohomology. All together, this is a highly substantial paper, bursting with a wealth of pioneering ideas and constructions related to Kontsevich’s Homological Mirror Conjecture. As the authors point out, several of the basic ideas are due to their collaborator Mikhail Khovanov, although he does not figure as a co-author. Despite its utmost advanced character, the present paper is written in a very lucid, detailed, enlightening, and nearly self-contained manner, with a plentiful supply of clarifying remarks, hints, and strategic outlooks.

The paper under review is devoted to a refined analysis of the group \(\text{Auteq}(D^b(X))\) from a particular viewpoint. The authors’ approach is motivated by the mirror symmetry phenomenon, especially by M. Kontsevich’s celebrated “Homological Mirror Conjecture”. Namely, this conjecture predicts an equivalence of derived categories of coherent sheaves on a Calabi-Yau manifold, on the one hand, and Floer-Fukaya categories of Lagrangians in the dual symplectic manifold, on the other hand, and one consequence of this conjecture is that, for Calabi-Yau manifolds to which mirror symmetry applies, the group \(\text{Auteq}(D(X))\) should be closely related to the symplectic automorphisms of the mirror manifold. Basically, this rather abstract and difficult conjectural relationship is the main topic of the present paper. With a view to the occurrence or certain braid group actions in symplectic geometry, the authors give a construction of braid group actions on bounded derived categories of fairly general abelian categories, and apply then this general framework to the concrete situation of derived categories of coherent sheaves and their groups of self-equivalences.

As to the contents, the paper consists of four main sections, each of which is subdiveded into several subsections.

Section 1 gives a very thorough and detailed introduction explaining the general problem, its genesis, its overall significance, the authors’ original strategy of tackling it, and the main results of the present work.

Section 2 develops a theory of spherical objects and twist functors for derived categories of general abelian categories satisfying certain (axiomatic) properties. These properties are always satisfied by the categories of (quasi-)coherent sheaves on a noetherian scheme over a ground field \(k\). It is then shown that special configurations of spherical objects in \(D^b(\text{Coh}(X))\) generate homomorphisms from braid groups to \(\text{Auteq}(D^b(\text{Coh}(X))\), where \(X\) is a noetherian \(k\)-scheme, \(\text{Coh}(X)\) is its category of coherent sheaves, and \(\text{Auteq}(D^b(\text{Coh} (X))\) stands for the group of self-equivalences of the derived category \(D^b (\text{Coh}(X))\). Having established various braid group actions on derived categories of coherent sheaves in such a way the authors study their concrete applications to different classes of (quasi-) projective varieties in Section 3. This includes singular varieties, varieties with finite group action, Fano varieties, and the well-understood case of elliptic curves as handy testing grounds. Moreover, the authors present a systematic way of producing spherical objects in the respective derived categories of coherent sheaves, they show how groups of symplectic automorphisms and groups of categorical self-equivalences can be compared in an explicit way, and they finally put their results in the context of mirror symmetry for singularities of threefolds, thereby returning to the main motivation for their work.

Section 4 is devoted to one of the main results of the paper. It contains the proof of the following theorem (Theorem 2.18): Let \(X\) be a noetherian scheme over an algebraically closed field \(k\), and assume that \(\dim X\geq 2\). Then, for any integer \(m<0\), the constructed group homomorphism \(B_{m+1}\to\text{Auteq}(D^b(\text{Coh}(X))\) from the braid group \(B_{m+1}\) to the self-equivalence group \(D^b(\text{Coh}(X))\) is injective.

This establishes the fact that the various braid group actions constructed in Section 2 (via configurations of \(m\) spherical objects in \(D^b(\text{Coh} (X))\); cf. Theorem 2.17) are even faithful. The intricate proof uses techniques from the theory of differential graded algebras and modules, their derived categories, and their Hochschild cohomology. All together, this is a highly substantial paper, bursting with a wealth of pioneering ideas and constructions related to Kontsevich’s Homological Mirror Conjecture. As the authors point out, several of the basic ideas are due to their collaborator Mikhail Khovanov, although he does not figure as a co-author. Despite its utmost advanced character, the present paper is written in a very lucid, detailed, enlightening, and nearly self-contained manner, with a plentiful supply of clarifying remarks, hints, and strategic outlooks.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

18E30 | Derived categories, triangulated categories (MSC2010) |

53D40 | Symplectic aspects of Floer homology and cohomology |

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\textit{P. Seidel} and \textit{R. Thomas}, Duke Math. J. 108, No. 1, 37--108 (2001; Zbl 1092.14025)

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##### References:

[1] | J. Bernstein and V. Lunts, Equivariant Sheaves and Functors , Lecture Notes in Math. 1578 , Springer, Berlin, 1994. MR 95k:55012 · Zbl 0808.14038 |

[2] | A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and mutations , Math. USSR-Izv. 35 (1990), 519–541. MR 91b:14013 · Zbl 0703.14011 |

[3] | –. –. –. –., Enhanced triangulated categories , Math. USSR-Sb. 70 (1991), 93–107. |

[4] | A. I. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties . · Zbl 0994.18007 |

[5] | ——–, Reconstruction of a variety from the derived category and groups of autoequivalences . · Zbl 0994.18007 |

[6] | A. I. Bondal and A. E. Polishchuk, Homological properties of associative algebras: The method of helices (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), 3–50.; English translation in Russian Acad. Sci. Izv. Math. 42 (1994), 219–260. MR 94m:16011 · Zbl 0847.16010 |

[7] | T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms , Bull. London Math. Soc. 31 (1999), 25–34. MR 99k:18014 · Zbl 0937.18012 |

[8] | T. Bridgeland, A. King, and M. Reid, Mukai implies McKay: The McKay correspondence as an equivalence of derived categories . · Zbl 0966.14028 |

[9] | C. H. Clemens, Double solids , Adv. Math. 47 (1983), 107–230. MR 85e:14058 · Zbl 0509.14045 |

[10] | A. Craw and M. Reid, How to calculate \(A\)-\(\Hilb \mathbb C^3\) . · Zbl 1080.14502 |

[11] | P. Deligne, Action du groupe des tresses sur une catégorie , Invent. Math. 128 (1997), 159–175. MR 98b:20061 · Zbl 0879.57017 |

[12] | W. Ebeling, “Strange duality, mirror symmetry, and the Leech lattice” in Singularity Theory (Liverpool, 1996) , London Math. Soc. Lecture Note Ser. 263 , Cambridge Univ. Press, Cambridge, England, 1999, xv –.xvi, 55–77. MR 2000g:14045 · Zbl 0958.14021 |

[13] | A. Fathi, F. Laudenbach, and V. Poénaru, Travaux de Thurston sur les surfaces , Astérisque 66 128 (1997), 159–175. MR 98b:20061 |

[14] | W. Ebeling, “Strange duality, mirror symmetry, and the Leech lattice” in Singularity Theory (Liverpool, 1996) , London Math. Soc. Lecture Note Ser. 263 , Cambridge Univ. Press, Cambridge, England, 1999, xv –.xvi, 55–77. MR 2000g:14045 · Zbl 0958.14021 |

[15] | A. Fathi, F. Laudenbach, and V. Poénaru, Travaux de Thurston sur les surfaces , Astérisque 66 – 67 , Soc. Math. France, Montrouge, 1979. MR 82m:57003 |

[16] | S. I. Gelfand and Yu. I. Manin, Methods of Homological Algebra , Springer, Berlin, 1996. MR 97j:18001 · Zbl 0855.18001 |

[17] | M. Gerstenhaber and S. Schack, “Algebraic cohomology and deformation theory” in Deformation Theory of Algebras and Structures and Applications (Il Ciocco, Italy, 1986) , NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 247 , Kluwer, Dordrecht, 1988, 11–264. MR 90c:16016 · Zbl 0676.16022 |

[18] | M. Gross, Blabber about black holes , unpublished notes. · Zbl 0167.29403 |

[19] | A. Grothendieck and J. Dieudonné, Élements de géométrie algebrique, I: Le langage des schemas , Inst. Hautes Études Sci. Publ. Math. 4 (1960). MR 29:1207 |

[20] | V. K. A. M. Gugenheim, L. A. Lambe, and J. D. Stasheff, Algebraic aspects of Chen’s twisting cochain , Illinois J. Math. 34 (1990), 485–502. MR 91c:55018 · Zbl 0684.55006 |

[21] | S. Halperin and J. Stasheff, Obstructions to homotopy equivalences , Adv. Math. 32 (1979), 233–279. MR 80j:55016 · Zbl 0408.55009 |

[22] | R. Hartshorne, Residues and Duality , Lecture Notes in Math. 20 , Springer, Berlin, 1966. MR 36 #5145 · Zbl 0212.26101 |

[23] | ——–, Algebraic Geometry , Grad. Texts in Math. 52 , Springer, New York, 1977. MR 57 #3116 |

[24] | R. P. Horja, Hypergeometric functions and mirror symmetry in toric varieties . · Zbl 1075.18006 |

[25] | T. V. Kadeishvili, The structure of the \(A_\infty\) -algebra, and the Hochschild and Harrison cohomologies (in Russian), Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988), 19–27. MR 91a:18016 · Zbl 0717.55011 |

[26] | M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras , Math. Ann. 316 (2000), 565–576. MR CMP 1 752 785 · Zbl 0997.14001 |

[27] | B. Keller, A remark on tilting theory and \(DG\) algebras, Manuscripta Math. 79 (1993), 247–252. MR 94d:18016 · Zbl 0810.16006 |

[28] | –. –. –. –., On the cyclic homology of exact categories , J. Pure Appl. Algebra 136 (1999), 1–56. MR 99m:18012 · Zbl 0923.19004 |

[29] | ——–, Introduction to \(A_\infty\) -algebras and modules. |

[30] | M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions . · Zbl 1035.53122 |

[31] | M. Kobayashi, Duality of weights, mirror symmetry and Arnold’s strange duality . · Zbl 1152.14040 |

[32] | M. Kontsevich, “Homological algebra of mirror symmetry” in Proceedings of the International Congress of Mathematicians (Zürich, 1994), Vol. 1,2 , Birkhäuser, Basel, 1995, 120–139. MR 97f:32040 · Zbl 0846.53021 |

[33] | S. Kuleshov, “Exceptional bundles on \(K3\) surfaces” in Helices and Vector Bundles: Seminaire Rudakov , London Math. Soc. Lecture Note Ser. 148 , Cambridge Univ. Press, Cambridge, England, 105–114. MR 91m:14067 |

[34] | A. Maciocia, Generalized Fourier-Mukai transforms , J. Reine Angew. Math. 480 (1996), 197–211. MR 97g:14013 \enlargethispage10pt |

[35] | J. McKay, “Graphs, singularities, and finite groups” in The Santa Cruz Conference on Finite Groups (Santa Cruz, Calif., 1979) , Proc. Sympos. Pure Math. 37 , Amer. Math. Soc., Providence, 1980, 183–186. MR 82e:20014 · Zbl 0451.05026 |

[36] | J. C. Moore, Algèbre homologique et homologie des espaces classifiants , Séminaire Henri Cartan 1959/1960, exp. no. 7. MR 28 #1092 · Zbl 0115.17205 |

[37] | –. –. –. –., “Differential homological algebra” in Actes du Congrès International des Mathématiciens (Nice, 1970), Vol. 1 , Gauthier-Villars, Paris, 1971, 335–339. MR 55 #9128 |

[38] | D. R. Morrison, “Through the looking glass” in Mirror Symmetry (Montreal, 1995), III, AMS/IP Stud. Adv. Math. 10 , Amer. Math. Soc., Providence, 1999, 263–277. MR 2000d:14049 · Zbl 0935.32020 |

[39] | S. Mukai, Duality between \(D(X)\) and \(D(\hat{X})\) with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153–175. MR 82f:14036 · Zbl 0417.14036 |

[40] | –. –. –. –., “On the moduli space of bundles on \(K3\) surfaces, I” in Vector Bundles on Algebraic Varieties (Bombay, 1984) , Tata Inst. Fund. Res. Stud. Math. 11 , Tata Inst. Fund. Res., Bombay, 1987, 341–413. MR 88i:14036 · Zbl 0674.14023 |

[41] | I. Nakamura, Hilbert schemes of Abelian group orbits , to appear in J. Algebraic Geom. · Zbl 1104.14003 |

[42] | D. O. Orlov, “Equivalences of derived categories and \(K3\) surfaces” in Algebraic Geometry, 7 , J. Math. Sci. (New York) 84 (1997), 1361–1381. MR 99a:14054 · Zbl 0938.14019 |

[43] | ——–, On equivalences of derived categories of coherent sheaves on abelian varieties . · Zbl 1031.18007 |

[44] | H. Pinkham, Singularités exceptionnelles, la dualité étrange d’Arnold et les surfaces \(K3\) , C. R. Acad. Sci. Paris Sér. A-B 284 (1977), A615–A618. MR 55 #2886 · Zbl 0375.14004 |

[45] | A. Polishchuk, Symplectic biextensions and a generalization of the Fourier-Mukai transform , Math. Res. Lett. 3 (1996), 813–828. MR 97j:14051 · Zbl 0886.14019 |

[46] | ——–, Massey and Fukaya products on elliptic curves . (revised version, July 1999). |

[47] | A. Polishchuk and E. Zaslow, Categorical mirror symmetry: The elliptic curve , Adv. Theor. Math. Phys. 2 (1998), 443–470. MR 99j:14034 · Zbl 0947.14017 |

[48] | J. Rickard, Morita theory for derived categories , J. London Math. Soc. (2) 39 (1989), 436–456. MR 91b:18012 · Zbl 0642.16034 |

[49] | A. N. Rudakov et al., Helices and Vector Bundles: Seminaire Rudakov , London Math. Soc. Lecture Note Ser. 148 , Cambridge Univ. Press, Cambridge, England, 1990. MR 91e:14002 · Zbl 0721.14011 |

[50] | P. Seidel, Lagrangian two-spheres can be symplectically knotted , J. Differential Geom. 52 (1999), 145–171. MR CMP 1 743 463 · Zbl 1032.53068 |

[51] | –. –. –. –., Graded Lagrangian submanifolds , Bull. Soc. Math. France 128 (2000), 103–149. MR CMP 1 765 826 · Zbl 0992.53059 |

[52] | ——–, An exact sequence for symplectic Floer homology , in preparation. |

[53] | A. Strominger, S.-T. Yau, and E. Zaslow, Mirror symmetry is \(T\)-duality , Nuclear Phys. B 479 (1996), 243–259. MR 97j:32022 · Zbl 0896.14024 |

[54] | D. Tanré, “Cohomologie de Harrison et type d’homotopie rationelle” in Algebra, Algebraic Topology and Their Interactions (Stockholm, 1983) , Lecture Notes in Math. 1183 , Springer, Berlin, 1986, 361–370. MR 87m:55015 |

[55] | R. P. Thomas, “Mirror symmetry and actions of braid groups on derived categories” in Proceedings of the Harvard Winter School on Mirror Symmetry, Vector Bundles, and Lagrangian Cycles (Cambridge, Mass., 1999) , International Press, Cambridge, Mass., 2001. · Zbl 1079.14530 |

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