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Quotient categories, stability conditions, and birational geometry

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Abstract

This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. We prove that this category has homological dimension c. As an application, we describe the space of stability conditions on its derived category in the case c \(=\) 1. Moreover, we describe all exact equivalences between these quotient categories in this particular case, which is closely related to classification problems in birational geometry.

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Notes

  1. 1.

    There is a canonical choice given by \(E=\rho ^{-1}(j_*\Gamma _f)\) with \(j_*:U \hookrightarrow X\) and \(\rho :F\oplus G \longrightarrow j_*(F|_U\oplus G|_U)\).

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Acknowledgments

We would like to thank Daniel Huybrechts for many useful discussions and his warm support as well as Raphaël Rouquier, Henning Krause, Torsten Wedhorn, Xiao-WU Chen for a lot of useful comments and suggestions. Another big thank you goes to the referee for pointing out misprints and a lot of useful remarks. Finally we are very grateful to the following institutions: Bonn International Graduate School in Mathematics, Imperial College London and SFB/TR 45.

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Correspondence to Sven Meinhardt.

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Meinhardt, S., Partsch, H. Quotient categories, stability conditions, and birational geometry. Geom Dedicata 173, 365–392 (2014). https://doi.org/10.1007/s10711-013-9947-x

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Keywords

  • Stability conditions
  • Abelian or triangulated quotient categories
  • Birational maps

Mathematics Subject Classification

  • 18E30