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Selecta Mathematica

, Volume 24, Issue 1, pp 85–143 | Cite as

Perverse schobers and birational geometry

  • Alexey Bondal
  • Mikhail Kapranov
  • Vadim Schechtman
Article
  • 78 Downloads

Abstract

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 \(\mathfrak {g}\) and study the corresponding schober-type diagram. For \(\mathfrak {g}={\mathfrak {s}\mathfrak {l}}_3\) we relate this diagram to the classical space of complete triangles studied by Schubert, Semple and others.

Mathematics Subject Classification

14E05 14F05 18D05 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Alexey Bondal
    • 1
    • 2
    • 3
  • Mikhail Kapranov
    • 1
  • Vadim Schechtman
    • 4
  1. 1.Kavli Institute for Physics and Mathematics of the Universe (WPI)Kashiwa-shiJapan
  2. 2.Steklov Institute of MathematicsMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia
  4. 4.Université Paul SabatierInstitut de Mathématiques de ToulouseToulouseFrance

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