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
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 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
M. Abouzaid, D. Auroux, and L. Katzarkov, Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces, arXiv:1205.0053, (2008).
V. Alexeev, Complete moduli in the presence of semiabelian group action, Ann. of Math. (2) 155, no. 3, 611–708, (2002).
V. Alexeev and D. Orlov, Derived categories of Burniat surfaces and exceptional collections, arXiv:1208.4348, (2012).
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
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).
V. Batyrev, Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties, Journal of Algebraic Geometry 3, 493–535, (1994).
V. Batyrev, Toric degenerations of Fano varieties and constructing mirror manifolds, Prooceedings of the Fano conference, University of Torino (2004), 109–122.
V. Batyrev, Conifold degenerations of Fano 3-folds as hypersurfaces in toric varieties, arXiv:1203.6058, (2012).
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.
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).
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.
C. Böhning, H.-C. von Bothmer, and P. Sosna, On the derived category of the classical Godeaux surface, arXiv:1206.1830, (2012).
T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166, no. 2, 317–345, (2007).
G. Brown, A database of polarized K3 surfaces, Experimental Mathematics 16, 7–20, (2007).
I. Cheltsov, Three-dimensional algebraic manifolds having a divisor with a numerically trivial canonical class, Russian Mathematical Surveys 51, 140–141, (1996).
I. Cheltsov, Anticanonical models of three-dimensional Fano varieties of degree four, Sbornik: Mathematics 194, 147–172, (2003).
I. Cheltsov, V. Przyjalkowski, andC. Shramov, Hyperelliptic and trigonal Fano threefolds, Izvestiya: Mathematics 69, 365–421, (2005).
J. Christophersen and N. Ilten, Stanley–Reisner degenerations of Mukai varieties, arXiv:1102.4521, (2011).
J. Christophersen and N. Ilten, Toric degenerations of low degree Fano threefolds, arXiv:1202.0510, (2012).
S. Cecotti and C. Vafa, Topological–anti-topological fusion, Nuclear Phys. B 367, no. 2, 359–461, (1991).
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.
A. Corti, Factorizing birational maps of threefolds after Sarkisov, Journal of Algebraic Geometry, 4, 223–254, (1995).
A. Corti and V. Golyshev, Hypergeometric equations and weighted projective spaces, Sci. China, Math. 54, no. 8, 1577–1590, (2011).
A. Corti, A. Pukhlikov, andM. Reid, Fano 3-fold hypersurfaces, LMS. Lecture Note Series 281, 175–258, (2000).
J. Cutrone andN. Marshburn, Towards the classification of weak Fano threefolds withρ = 2, arXiv:math.AG/1009.5036, (2010).
M. Demazure, Surfaces de Del Pezzo. I–V., Seminaire sur les singularites des surfaces, Lecture Notes in Mathematics, 777, 21–69, (1980).
C. Diemer, L. Katzarkov, and G. Kerr, Symplectic relations arising from toric degenerations, arXiv:1204.2233, (2012).
C. Diemer, L. Katzarkov, and G. Kerr, Compactifications of spaces of Landau–Ginzburg models, arXiv:1207.0042, (2012).
C. Diemer, L. Katzarkov, and G. Kerr, Stability conditions for Fukaya–Seidel categories, in preparation.
S. Donaldson, Discussion of the Kähler–Einstein problem, preprint, http://www2.imperial.ac.uk/~skdona/KENOTES.PDF.
S. Donaldson, Stability, birational transformations and the Kähler–Einstein problem, arXiv:1007.4220, (2010).
S. Donaldson, b-Stability and blow-ups, arXiv:1107.1699, (2011).
S. Donaldson and R. Thomas, Gauge theory in higher dimensions, The geometric universe (Oxford, 1996), 31-47, Oxford Univ. Press, Oxford, 1998.
C. Doran, A. Harder, L. Katzarkov, J. Lewis, andV. Przyjalkowski, Modularity of Fano threefolds, in preparation.
M. Douglas, D-branes, categories and N = 1 supersymmetry. Strings, branes, and M-theory, J. Math. Phys. 42, 2818–2843, (2001).
G. Fano, Sulle varieta algebriche a tre dimensioni aventi tutti i generi nulu, Proc. Internat. Congress Mathematicians (Bologna), 4, Zanichelli, 115–119, (1934).
G. Fano, Su alcune varieta algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche, Commentarii Mathematici Helvetici 14, 202–211, (1942).
S. Galkin, Small toric degenerations of Fano threefolds, preprint, http://member.ipmu.jp/sergey.galkin/papers/std.pdf.
S. Galkin and E. Shinder, Exceptional collections of line bundles on the Beauville surface, arXiv:1210.3339, (2012).
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).
I. Gelfand, M. Kapranov, and A. Zelevinski, Discriminants, resultants and multidimensional determinants, Mathematics: Theory and Applications. Birkhauser Boston, Inc., Boston, MA, 1994.
V. Golyshev, The geometricity problem and modularity of some Riemann–Roch variations, Russian Academy of Sciences (Doklady, Mathematics) 386, 583–588, (2002).
V. Golyshev, Classification problems and mirror duality, LMS Lecture Note Series 338, 88–121, (2007).
C. Hacon and J. McKernan, The Sarkisov program, arXiv:0905.0946, (2009).
F. Haiden, L. Katzarkov, and M. Kontsevich, Stability conditions for Fukaya–Seidel categories, in prepation.
B. Hassett, Some rational cubic fourfolds, J. Algebraic Geom. 8, 103–114, (1999).
K. Hori, C. Vafa, Mirror symmetry, hep-th/0002222, (2000).
E. Horikawa, Algebraic surfaces of general type with small \(c_{1}^{2}\).I, Ann. Math. (2) 104, 357–387, (1976).
C. Ingalls and A. Kuznetsov, On nodal Enriques surfaces and quartic double solids, arXiv:1012.3530, (2010).
A. Iliev, L. Katzarkov, and V. Przyjalkowski, Double solids, categories and non-rationality, to appear in PEMS, Shokurov’s volume, arXiv:1102.2130, (2013).
A. Iliev, L. Katzarkov, and E. Scheidegger, Automorphic forms and cubics, in preparation.
N. Ilten, J. Lewis, and V. Przyjalkowski, Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models, Journal of Algebra 374, 104–121, (2013).
N. Ilten and R. Vollmert, Deformations of Rational T-Varieties, To appear in Journal of Algebraic Geometry, arXiv:0903.1393, (2009).
V. Iskovskikh, Fano 3-folds I, Mathematics of the USSR, Izvestija 11, 485–527, (1977).
V. Iskovskikh, Fano 3-folds II, Mathematics of the USSR, Izvestija 12, 469–506, (1978).
V. Iskovskikh and Yu. Manin, Three-dimensional quartics and counterexamples to the Lüroth problem, Matematical Sbornik 86, 140–166, (1971).
V. Iskovskikh, Yu. Prokhorov, Fano varieties, Encyclopaedia of Mathematical Sciences 47, (1999) Springer, Berlin.
P. Jahnke, T. Peternell, and I. Radloff, Threefolds with big and nef anticanonical bundles I, Mathematische Annalen 333, 569–631, (2005).
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).
I. Jahnke and I. Radloff, Gorenstein Fano threefolds with base points in the anticanonical system, Compositio Mathematica 142, 422–432, (2006).
P. Jahnke and I. Radloff, Terminal Fano threefolds and their smoothings. Mathematische Zeitschrift 269, 1129–1136, (2011).
D. Joyce and Y. Song, A theory of generalized Donaldson–Thomas invariants, Mem. Amer. Math. Soc. 217, no. 1020, (2012).
A.-S. Kaloghiros, The topology of terminal quartic 3-folds, arXiv:math.AG/0707.1852, (2007).
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).
I. Karzhemanov, On Fano threefolds with canonical Gorenstein singularities, Sbornik: Mathematics, 200 (2009), 111–146.
I. Karzhemanov, Fano threefolds with canonical Gorenstein singularities and big degree, arXiv:0908.1671, (2009).
Y. Kawamata, Derived categories and birational geometry, Proceedings of Symposia in Pure Mathematics, 80 Part 2, 655–665 (2009).
L. Katzarkov, M. Kontsevich, and T. Pantev, Hodge theoretic aspects of mirror symmetry 2, in preparation.
L. Katzarkov, M. Kontsevich, T. Pantev, and Y. Soibelman, Stability Hodge structures., in preparation.
L. Katzarkov and V. Przyjalkowski, Generalized Homological Mirror Symmetry and cubics, Proc. Steklov Inst. Math., vol. 264, 2009, 87–95.
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).
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).
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.
M. Kreuzer and H. Skarke, On the classification of reflexive polyhedra, Communications in Mathematical Physics 185, 495–508, (1997).
M. Kreuzer and H. Skarke, Classification of reflexive polyhedra in three dimensions, Advances in Theoretical and Mathematical Physics 2, 853–871, (1998).
O. Kuchle, On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians, Mathematische Zeitschrift 218, 563–575, (1995).
A. Kuznetsov, Derived categories of cubic fourfolds, Cohomological and geometric approaches to rationality problems, Progress in Mathematics 282, Birkhauser Boston, 219–243, (2010).
L. Lafforgue, Une compactification des champs classifiant les chtoucas de Drinfeld, J. Amer. Math. Soc. 11, no. 4, 1001–1036, (1998).
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).
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).
S. Mukai, New developments in the theory of Fano threefolds: vector bundle method and moduli problems, Sugaku Expositions 15, 125–150, (2002).
Y. Namikawa, Smoothing Fano 3-folds, Journal of Algebraic Geomemtry 6, 307–324, (1997).
D. Orlov, Remarks on generators and dimensions of triangulated categories, Mosc. Math. J. 9, no. 1, 153–159, (2009).
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).
Yu. Prokhorov, The degree of Fano threefolds with canonical Gorenstein singularities, Sbornik: Mathematics 196, 77–114, (2005).
V. Przyjalkowski, On Landau–Ginzburg models for Fano varieties, Communications in Number Theory and Physics 1, 713–728, (2008).
V. Przyjalkowski, Weak Landau–Ginzburg models for smooth Fano threefolds, to appear in Izv. Math. 77 (4), (2013).
M. Reid, Young person’s guide to canonical singularities, Proceedings of Symposia in Pure Mathematics, 46, 345–414, (1987).
R. Rouquier, Dimensions of triangulated categories, Journal of K-theory 1, no. 2, 193–256, (2008).
V. Sarkisov, On conic bundle structures, Izv. Akad. Nauk SSSR Ser. Mat. 46:2, 371–408, (1982).
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).
C. Simpson, Higgs bundles and local systems, Publ. Math., Inst. Hautes Etud. Sci. 75, 5–95, (1992).
D. Stepanov, Combinatorial structure of exceptional sets in resolutions of singularities, arXiv:math/0611903, (2006).
K. Takeuchi, Some birational maps of Fano 3-folds, Compositio Mathematica 71, 265–283, (1989).
G. Tian, On Calabi’s conjecture for complex surfaces with positive first Chern class, Inventiones Mathematicae 101, 101–172, (1990).
G. Tian, Kähler–Einstein metrics with positive scalar curvature, Inventiones Mathematicae 130, 1–37, (1997).
X. Wang and X. Zhu, Kähler–Ricci solitons on toric manifolds with positive first Chern class, Advances in Mathematics 188, 87–103, (2004).
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-1-4614-6482-2_5
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-6481-5
Online ISBN: 978-1-4614-6482-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)