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## Perverse sheaves of categories and some applications

We study perverse sheaves of categories their connections to classical algebraic geometry. We show how perverse sheaves of categories encode naturally derived categories of coherent sheaves on P1bundles, semiorthogonal decompositions, and relate them to a recent proof of Segal that all autoequivalences of triangulated categories are spherical twists. Furthermore, we show that perverse sheaves of categories can be used to represent certain degenerate Calabi–Yau varieties.

We introduce the notions of consistent pairs and consistent chains of $ t$-structures and prove that two consistent chains of $ t$-structures generate a distributive lattice. The technique developed is then applied to the pairs of chains obtained from the standard $ t$-structure on the derived category of coherent sheaves and the dual $ t$-structure by means of the shift functor. This yields a family of $ t$-structures whose hearts are known as perverse coherent sheaves. Access this article Login options Individual login Institutional loginvia Athens/Shibboleth The computer you are using is not registered by an institution with a subscription to this article. Please log in below. Find out more about journal subscriptions at your site. Purchase this article online Buy this article £33.00 (£39.60 incl. VAT) $59.70 US Dollar price There are no additional delivery charges. By purchasing this article, you are accepting IOP's Terms and Conditions for Document Delivery. If you would like to buy this article, but not online, please contact custserv@iop.org. Make a recommendation To gain access to this content, please complete the Recommendation Form and we will follow up with your librarian or Institution on your behalf. For corporate researchers we can also follow up directly with your R&D manager, or the information management contact at your company. Recommend this journal Institutional subscribers have access to the current volume, plus a 10-year back file (where available). Subscribe to this journal The title that you are trying to access is not part of the IOP Historic Archive. You can purchase a copy of the article that you wish to view. Related Articles Hyperplane sections and derived categories Derived categories of coherent sheaves and equivalences between them PROJECTIVE BUNDLES, MONOIDAL TRANSFORMATIONS, AND DERIVED CATEGORIES OF COHERENT SHEAVES More Related Review Articles Classification of isomonodromy problems on elliptic curves Number-theoretic properties of hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves over the rational number field Geometric structures on moment-angle manifolds More

Perverse schobers are conjectural categorical analogs of perverse sheaves. We show that such structures appear naturally in Homological Minimal Model Program which studies the effect of birational transformations such as flops, on the coherent derived categories. More precisely, the flop data are analogous to hyperbolic stalks of a perverse sheaf. In the first part of the paper we study schober-type diagrams of categories corresponding to flops of relative dimension 1, in particular we determine the categorical analogs of the (compactly supported) cohomology with coefficients in such schobers. In the second part we consider the example of a “web of flops” provided by the Grothendieck resolution associated to a reductive Lie algebra g and study the corresponding schober-type diagram. For g = sl3 we relate this diagram to the classical space of complete triangles studied by Schubert, Semple and others.

abstract

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.