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Birational Geometry via Moduli Spaces

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Birational Geometry, Rational Curves, and Arithmetic

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.

Mathematics Subject Classification codes (2000): 14M25, 14H10, 14Q15, 14D07

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Notes

  1. 1.

    Recall that toric n-dimensional Fano varieties with canonical Gorenstein singularities up to isomorphism are in one-two-one correspondence with reflexive n-dimensional lattice polytopes in \({\mathbb{R}}^{n}\) up to \(\mathrm{SL}_{n}(\mathbb{Z})\) action.

References

  1. M. Abouzaid, D. Auroux, and L. Katzarkov, Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces, arXiv:1205.0053, (2008).

    Google Scholar 

  2. V. Alexeev, Complete moduli in the presence of semiabelian group action, Ann. of Math. (2) 155, no. 3, 611–708, (2002).

    Google Scholar 

  3. V. Alexeev and D. Orlov, Derived categories of Burniat surfaces and exceptional collections, arXiv:1208.4348, (2012).

    Google Scholar 

  4. M. Ballard, D. Favero, andL. Katzarkov, Orlov spectra: bounds and gaps, Invent. Math. 189 (2012), no. 2, 359–430, arXiv:1012.0864 (submitted to Inventiones

    Google Scholar 

  5. M. Ballard, D. Favero, and L. Katzarkov, A category of kernels for graded matrix factorizations and its implications towards Hodge theory. arXiv:1105.3177 (accepted to Publications mathématiques de l’IHÉS), (2010).

    Google Scholar 

  6. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties, Journal of Algebraic Geometry 3, 493–535, (1994).

    Google Scholar 

  7. V. Batyrev, Toric degenerations of Fano varieties and constructing mirror manifolds, Prooceedings of the Fano conference, University of Torino (2004), 109–122.

    Google Scholar 

  8. V. Batyrev, Conifold degenerations of Fano 3-folds as hypersurfaces in toric varieties, arXiv:1203.6058, (2012).

    Google Scholar 

  9. V. Batyrev and L. Borisov, Dual Cones and Mirror Symmetry for Generalized Calabi–Yau Manifolds, Mirror symmetry, II, 7186, AMS/IP Stud. Adv. Math., 1, Amer. Math. Soc., Providence, RI, 1997.

    Google Scholar 

  10. C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, J. Am. Math. Soc. 23, no. 2, 405–468 (2010).

    Google Scholar 

  11. J. Blanc, S. Lamy, Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links, Proceedings of the London Mathematical Society, to appear.

    Google Scholar 

  12. C. Böhning, H.-C. von Bothmer, and P. Sosna, On the derived category of the classical Godeaux surface, arXiv:1206.1830, (2012).

    Google Scholar 

  13. T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166, no. 2, 317–345, (2007).

    Google Scholar 

  14. G. Brown, A database of polarized K3 surfaces, Experimental Mathematics 16, 7–20, (2007).

    Google Scholar 

  15. I. Cheltsov, Three-dimensional algebraic manifolds having a divisor with a numerically trivial canonical class, Russian Mathematical Surveys 51, 140–141, (1996).

    Google Scholar 

  16. I. Cheltsov, Anticanonical models of three-dimensional Fano varieties of degree four, Sbornik: Mathematics 194, 147–172, (2003).

    Google Scholar 

  17. I. Cheltsov, V. Przyjalkowski, andC. Shramov, Hyperelliptic and trigonal Fano threefolds, Izvestiya: Mathematics 69, 365–421, (2005).

    Google Scholar 

  18. J. Christophersen and N. Ilten, Stanley–Reisner degenerations of Mukai varieties, arXiv:1102.4521, (2011).

    Google Scholar 

  19. J. Christophersen and N. Ilten, Toric degenerations of low degree Fano threefolds, arXiv:1202.0510, (2012).

    Google Scholar 

  20. S. Cecotti and C. Vafa, Topological–anti-topological fusion, Nuclear Phys. B 367, no. 2, 359–461, (1991).

    Google Scholar 

  21. T. Coates, A. Corti, S. Galkin, V. Golyshev, and A. Kasprzyk, Fano varieties and extremal Laurent polynomials, a collaborative research blog, http://coates.ma.ic.ac.uk/fanosearch.

  22. A. Corti, Factorizing birational maps of threefolds after Sarkisov, Journal of Algebraic Geometry, 4, 223–254, (1995).

    Google Scholar 

  23. A. Corti and V. Golyshev, Hypergeometric equations and weighted projective spaces, Sci. China, Math. 54, no. 8, 1577–1590, (2011).

    Google Scholar 

  24. A. Corti, A. Pukhlikov, andM. Reid, Fano 3-fold hypersurfaces, LMS. Lecture Note Series 281, 175–258, (2000).

    Google Scholar 

  25. J. Cutrone andN. Marshburn, Towards the classification of weak Fano threefolds withρ = 2, arXiv:math.AG/1009.5036, (2010).

    Google Scholar 

  26. M. Demazure, Surfaces de Del Pezzo. I–V., Seminaire sur les singularites des surfaces, Lecture Notes in Mathematics, 777, 21–69, (1980).

    Google Scholar 

  27. C. Diemer, L. Katzarkov, and G. Kerr, Symplectic relations arising from toric degenerations, arXiv:1204.2233, (2012).

    Google Scholar 

  28. C. Diemer, L. Katzarkov, and G. Kerr, Compactifications of spaces of Landau–Ginzburg models, arXiv:1207.0042, (2012).

    Google Scholar 

  29. C. Diemer, L. Katzarkov, and G. Kerr, Stability conditions for Fukaya–Seidel categories, in preparation.

    Google Scholar 

  30. S. Donaldson, Discussion of the Kähler–Einstein problem, preprint, http://www2.imperial.ac.uk/~skdona/KENOTES.PDF.

  31. S. Donaldson, Stability, birational transformations and the Kähler–Einstein problem, arXiv:1007.4220, (2010).

    Google Scholar 

  32. S. Donaldson, b-Stability and blow-ups, arXiv:1107.1699, (2011).

    Google Scholar 

  33. S. Donaldson and R. Thomas, Gauge theory in higher dimensions, The geometric universe (Oxford, 1996), 31-47, Oxford Univ. Press, Oxford, 1998.

    Google Scholar 

  34. C. Doran, A. Harder, L. Katzarkov, J. Lewis, andV. Przyjalkowski, Modularity of Fano threefolds, in preparation.

    Google Scholar 

  35. M. Douglas, D-branes, categories and N = 1 supersymmetry. Strings, branes, and M-theory, J. Math. Phys. 42, 2818–2843, (2001).

    Google Scholar 

  36. G. Fano, Sulle varieta algebriche a tre dimensioni aventi tutti i generi nulu, Proc. Internat. Congress Mathematicians (Bologna), 4, Zanichelli, 115–119, (1934).

    Google Scholar 

  37. G. Fano, Su alcune varieta algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche, Commentarii Mathematici Helvetici 14, 202–211, (1942).

    Google Scholar 

  38. S. Galkin, Small toric degenerations of Fano threefolds, preprint, http://member.ipmu.jp/sergey.galkin/papers/std.pdf.

  39. S. Galkin and E. Shinder, Exceptional collections of line bundles on the Beauville surface, arXiv:1210.3339, (2012).

    Google Scholar 

  40. D. Gaiotto, G. Moore, and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299, no. 1, 163–224, (2010).

    Google Scholar 

  41. I. Gelfand, M. Kapranov, and A. Zelevinski, Discriminants, resultants and multidimensional determinants, Mathematics: Theory and Applications. Birkhauser Boston, Inc., Boston, MA, 1994.

    Google Scholar 

  42. V. Golyshev, The geometricity problem and modularity of some Riemann–Roch variations, Russian Academy of Sciences (Doklady, Mathematics) 386, 583–588, (2002).

    Google Scholar 

  43. V. Golyshev, Classification problems and mirror duality, LMS Lecture Note Series 338, 88–121, (2007).

    Google Scholar 

  44. C. Hacon and J. McKernan, The Sarkisov program, arXiv:0905.0946, (2009).

    Google Scholar 

  45. F. Haiden, L. Katzarkov, and M. Kontsevich, Stability conditions for Fukaya–Seidel categories, in prepation.

    Google Scholar 

  46. B. Hassett, Some rational cubic fourfolds, J. Algebraic Geom. 8, 103–114, (1999).

    Google Scholar 

  47. K. Hori, C. Vafa, Mirror symmetry, hep-th/0002222, (2000).

    Google Scholar 

  48. E. Horikawa, Algebraic surfaces of general type with small \(c_{1}^{2}\).I, Ann. Math. (2) 104, 357–387, (1976).

    Google Scholar 

  49. C. Ingalls and A. Kuznetsov, On nodal Enriques surfaces and quartic double solids, arXiv:1012.3530, (2010).

    Google Scholar 

  50. A. Iliev, L. Katzarkov, and V. Przyjalkowski, Double solids, categories and non-rationality, to appear in PEMS, Shokurov’s volume, arXiv:1102.2130, (2013).

    Google Scholar 

  51. A. Iliev, L. Katzarkov, and E. Scheidegger, Automorphic forms and cubics, in preparation.

    Google Scholar 

  52. N. Ilten, J. Lewis,  and V. Przyjalkowski, Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models, Journal of Algebra 374, 104–121, (2013).

    Google Scholar 

  53. N. Ilten and R. Vollmert, Deformations of Rational T-Varieties, To appear in Journal of Algebraic Geometry, arXiv:0903.1393, (2009).

    Google Scholar 

  54. V. Iskovskikh, Fano 3-folds I, Mathematics of the USSR, Izvestija 11, 485–527, (1977).

    Google Scholar 

  55. V. Iskovskikh, Fano 3-folds II, Mathematics of the USSR, Izvestija 12, 469–506, (1978).

    Google Scholar 

  56. V. Iskovskikh and Yu. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem, Matematical Sbornik 86, 140–166, (1971).

    Google Scholar 

  57. V. Iskovskikh, Yu. Prokhorov, Fano varieties, Encyclopaedia of Mathematical Sciences 47, (1999) Springer, Berlin.

    Google Scholar 

  58. P. Jahnke, T. Peternell, and I. Radloff, Threefolds with big and nef anticanonical bundles I, Mathematische Annalen 333, 569–631, (2005).

    Google Scholar 

  59. P. Jahnke, and T. Peternell, I. Radloff, Threefolds with big and nef anticanonical bundles II, Central European Journal of Mathematics, Cent. Eur. J. Math. 9, no. 3, 449–488, (2011).

    Google Scholar 

  60. I. Jahnke and I. Radloff, Gorenstein Fano threefolds with base points in the anticanonical system, Compositio Mathematica 142, 422–432, (2006).

    Google Scholar 

  61. P. Jahnke and I. Radloff, Terminal Fano threefolds and their smoothings. Mathematische Zeitschrift 269, 1129–1136, (2011).

    Google Scholar 

  62. D. Joyce and Y. Song, A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 217, no. 1020, (2012).

    Google Scholar 

  63. A.-S. Kaloghiros, The topology of terminal quartic 3-folds, arXiv:math.AG/0707.1852, (2007).

    Google Scholar 

  64. A. Kapustin, L. Katzarkov, D. Orlov, andM. Yotov, Homological Mirror Symmetry for manifolds of general type, Cent. Eur. J. Math. 7, no. 4, 571–605, (2009).

    Google Scholar 

  65. I. Karzhemanov, On Fano threefolds with canonical Gorenstein singularities, Sbornik: Mathematics, 200 (2009), 111–146.

    Google Scholar 

  66. I. Karzhemanov, Fano threefolds with canonical Gorenstein singularities and big degree, arXiv:0908.1671, (2009).

    Google Scholar 

  67. Y. Kawamata, Derived categories and birational geometry, Proceedings of Symposia in Pure Mathematics, 80 Part 2, 655–665 (2009).

    Google Scholar 

  68. L. Katzarkov, M. Kontsevich, and T. Pantev, Hodge theoretic aspects of mirror symmetry 2, in preparation.

    Google Scholar 

  69. L. Katzarkov, M. Kontsevich, T. Pantev, and Y. Soibelman, Stability Hodge structures., in preparation.

    Google Scholar 

  70. L. Katzarkov and V. Przyjalkowski, Generalized Homological Mirror Symmetry and cubics, Proc. Steklov Inst. Math., vol. 264, 2009, 87–95.

    Google Scholar 

  71. L. Katzarkov and V. Przyjalkowski, Landau–Ginzburg models — old and new, Akbulut, Selman (ed.) et al., Proceedings of the 18th Gokova geometry–topology conference. Somerville, MA: International Press; Gokova: Gokova Geometry-Topology Conferences, 97–124 (2012).

    Google Scholar 

  72. M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants. Commun. Number Theory Phys. 5, no. 2, 231–352, (2011).

    Google Scholar 

  73. M. Kontsevich and Y. Soibelman, Motivic Donaldson–Thomas invariants: summary of results, Mirror symmetry and tropical geometry, 55–89, Contemp. Math. 527, Amer. Math. Soc., Providence, RI, 2010.

    Google Scholar 

  74. M. Kreuzer and H. Skarke, On the classification of reflexive polyhedra, Communications in Mathematical Physics 185, 495–508, (1997).

    Google Scholar 

  75. M. Kreuzer and H. Skarke, Classification of reflexive polyhedra in three dimensions, Advances in Theoretical and Mathematical Physics 2, 853–871, (1998).

    Google Scholar 

  76. O. Kuchle, On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians, Mathematische Zeitschrift 218, 563–575, (1995).

    Google Scholar 

  77. A. Kuznetsov, Derived categories of cubic fourfolds, Cohomological and geometric approaches to rationality problems, Progress in Mathematics 282, Birkhauser Boston, 219–243, (2010).

    Google Scholar 

  78. L. Lafforgue, Une compactification des champs classifiant les chtoucas de Drinfeld, J. Amer. Math. Soc. 11, no. 4, 1001–1036, (1998).

    Google Scholar 

  79. D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory. I, Compos. Math. 142, no. 5, 1263–1285, (2006).

    Google Scholar 

  80. D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson–Thomas theory. II, Compos. Math. 142, no. 5, 1286–1304, (2006).

    Google Scholar 

  81. S. Mukai, New developments in the theory of Fano threefolds: vector bundle method and moduli problems, Sugaku Expositions 15, 125–150, (2002).

    Google Scholar 

  82. Y. Namikawa, Smoothing Fano 3-folds, Journal of Algebraic Geomemtry 6, 307–324, (1997).

    Google Scholar 

  83. D. Orlov, Remarks on generators and dimensions of triangulated categories, Mosc. Math. J. 9, no. 1, 153–159, (2009).

    Google Scholar 

  84. P. del Pezzo, Sulle superficie dell’nmo ordine immerse nello spazio di n dimensioni, Rend. del circolo matematico di Palermo 1 (1): 241–271, (1887).

    Google Scholar 

  85. Yu. Prokhorov, The degree of Fano threefolds with canonical Gorenstein singularities, Sbornik: Mathematics 196, 77–114, (2005).

    Google Scholar 

  86. V. Przyjalkowski, On Landau–Ginzburg models for Fano varieties, Communications in Number Theory and Physics 1, 713–728, (2008).

    Google Scholar 

  87. V. Przyjalkowski, Weak Landau–Ginzburg models for smooth Fano threefolds, to appear in Izv. Math. 77 (4), (2013).

    Google Scholar 

  88. M. Reid, Young person’s guide to canonical singularities, Proceedings of Symposia in Pure Mathematics, 46, 345–414, (1987).

    Google Scholar 

  89. R. Rouquier, Dimensions of triangulated categories, Journal of K-theory 1, no. 2, 193–256, (2008).

    Google Scholar 

  90. V. Sarkisov, On conic bundle structures, Izv. Akad. Nauk SSSR Ser. Mat. 46:2, 371–408, (1982).

    Google Scholar 

  91. N. Seiberg; E. Witten, Electric–magnetic duality, monopole condensation, and confinement inN = 2 supersymmetric Yang–Mills theory, Nuclear Phys. B 426, no. 1, 19–52, (1994).

    Google Scholar 

  92. C. Simpson, Higgs bundles and local systems, Publ. Math., Inst. Hautes Etud. Sci. 75, 5–95, (1992).

    Google Scholar 

  93. D. Stepanov, Combinatorial structure of exceptional sets in resolutions of singularities, arXiv:math/0611903, (2006).

    Google Scholar 

  94. K. Takeuchi, Some birational maps of Fano 3-folds, Compositio Mathematica 71, 265–283, (1989).

    Google Scholar 

  95. G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Inventiones Mathematicae 101, 101–172, (1990).

    Google Scholar 

  96. G. Tian, Kähler–Einstein metrics with positive scalar curvature, Inventiones Mathematicae 130, 1–37, (1997).

    Google Scholar 

  97. X. Wang and X. Zhu, Kähler–Ricci solitons on toric manifolds with positive first Chern class, Advances in Mathematics 188, 87–103, (2004).

    Google Scholar 

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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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Authors and Affiliations

  1. University of Edinburgh, Edinburgh, EH1 2QL, UK

    Ivan Cheltsov

  2. University of Miami, Coral Gables, FL, USA

    Ludmil Katzarkov

  3. University of Vienna, Vienna, Austria

    Ludmil Katzarkov

  4. Steklov Mathematical Institute, Moscow, Russia

    Victor Przyjalkowski

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  1. Ivan Cheltsov
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Editors and Affiliations

  1. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Fedor Bogomolov

  2. , Department of Mathematics, Rice University, S. Main St. 6100, Houston, 77251-1892, Texas, USA

    Brendan Hassett

  3. Department of Mathematics, Courant Institute of Math. Sciences, New York University, Mercer St. 251, New York, 10012, New York, USA

    Yuri Tschinkel

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Cheltsov, I., Katzarkov, L., Przyjalkowski, V. (2013). Birational Geometry via Moduli Spaces. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds) Birational Geometry, Rational Curves, and Arithmetic. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6482-2_5

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