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Birational Geometry via Moduli Spaces

  • Ivan Cheltsov
  • Ludmil Katzarkov
  • Victor PrzyjalkowskiEmail author
Chapter

Abstract

In this paper we connect degenerations of Fano threefolds by projections. Using mirror symmetry we transfer these connections to the side of Landau–Ginzburg models. Based on that we suggest a generalization of Kawamata’s categorical approach to birational geometry enhancing it via geometry of moduli spaces of Landau–Ginzburg models. We suggest a conjectural application to the Hassett–Kuznetsov–Tschinkel program, based on new nonrationality “invariants”—gaps and phantom categories. We formulate several conjectures about these invariants in the case of surfaces of general type and quadric bundles.

Keywords

Modulus Space Pezzo Surface Smooth Point Higgs Bundle Fano Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We are very grateful to A. Bondal, M. Ballard, C. Diemer, and D. Favero, F. Haiden, A. Iliev, A. Kasprzyck, G. Kerr, M. Kontsevich, A. Kuznetsov, T. Pantev, Y. Soibelman, D. Stepanov for the useful discussions. I. C. is grateful to Hausdorff Research Institute of Mathematics (Bonn) and was funded by RFFI grants 11-01-00336-a and AG Laboratory GU-HSE and RF government grant ag. 11 11.G34.31.0023. L. K was funded by grants NSF DMS0600800, NSF FRG DMS-0652633, NSF FRG DMS-0854977, NSF DMS-0854977, NSF DMS-0901330, grants FWF P 24572-N25 and FWF P20778, and an ERC grant—GEMIS, V.P. was funded by Dynasty Foundation, grants NSF FRG DMS-0854977, NSF DMS-0854977, and NSF DMS-0901330; grants FWF P 24572-N25 and FWF P20778; RFFI grants 11-01-00336-a, 11-01-00185-a, 12-01-33024, and 12-01-31012; grants MK − 1192. 2012. 1, NSh − 5139. 2012. 1, and AG Laboratory GU-HSE and RF government grant ag. 11 11.G34.31.0023.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ivan Cheltsov
    • 1
  • Ludmil Katzarkov
    • 2
    • 3
  • Victor Przyjalkowski
    • 4
    Email author
  1. 1.University of EdinburghEdinburghUK
  2. 2.University of MiamiCoral GablesUSA
  3. 3.University of ViennaViennaAustria
  4. 4.Steklov Mathematical InstituteMoscowRussia

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