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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