Rational curves on complex manifolds. (English) Zbl 1286.14020
Let \(X\) be a complex compact manifold. A rational curve is a non-constant morphism \(\mathbb P^1 \rightarrow X\). Since the classification of smooth projective surfaces by the Italian school and, more recently, since the seminal papers of S. Mori [Ann. Math. (2) 110, 593–606 (1979; Zbl 0423.14006)], Ann. Math. (2) 116, 133–176 (1982; Zbl 0557.14021)] rational curves have played a prominent role in the classification theory of projective manifolds. In this survey, which is interesting for both algebraic and analytic geometers, the author gives an overview of the techniques that have been developed to prove the existence of rational curves and to study their impact on the global geometry. Starting with Mori’s bend-and-break lemma and the technique of reduction modulo \(p\), the author discusses the notions of uniruled and rationally connected manifolds and explains their role in the minimal model program (MMP). In the second part of the paper he moves on to complex manifolds which are not necessarily projective and the tools which are available in this generalised setting, notably Gromov-Witten invariants, the Kähler-Ricci flow and the bimeromorphic geometry of Kähler threefolds.
Reviewer: Andreas Höring (Nice)
MSC:
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 14J45 | Fano varieties |
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |
| 32J27 | Compact Kähler manifolds: generalizations, classification |
References:
| [1] | Abhyankar S.: On the valuations centered in a local domain. Amer. J. Math., 78, 321–348 (1956) · Zbl 0074.26301 · doi:10.2307/2372519 |
| [2] | Araujo C.: The cone of pseudo-effective divisors of log varieties after Batyrev. Math. Z., 264(1), 179–193 (2010) · Zbl 1189.14006 · doi:10.1007/s00209-008-0457-8 |
| [3] | C. Araujo, S. Druel, and S.Kovács. Cohomological characterizations of projective spaces and hyperquadrics. Invent. Math., 174(2):233–253, 2008. · Zbl 1162.14037 |
| [4] | S. Bando. On the classification of three-dimensional compact Kaehler manifolds of nonnegative bisectional curvature. J. Differential Geom., 19(2):283–297, 1984. · Zbl 0547.53034 |
| [5] | V. Batyrev. The cone of effective divisors of threefolds. In Proc. Intl. Conf. on Algebra, pages 337–352, 1989. · Zbl 0781.14027 |
| [6] | A. Beauville. Complex algebraic surfaces, volume 68 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1983. · Zbl 0512.14020 |
| [7] | M. Berger. Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. Bull. Soc. Math. France, 83:279–330, 1955. · Zbl 0068.36002 |
| [8] | C. Birkar, P. Cascini, C. Hacon, and J. McKernan. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., 23(2):405–468, 2010. · Zbl 1210.14019 |
| [9] | A. Blanchard. Sur les variétés analytiques complexes. Ann. Sci. Ecole Norm. Sup. (3), 73:157–202, 1956. · Zbl 0073.37503 |
| [10] | S. Boucksom, J.-P. Demailly, M. Păun, and T. Peternell. The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebraic Geom., 22(2):201–248, 2013. · Zbl 1267.32017 |
| [11] | Brunella M.: A positivity property for foliations on compact Kähler manifolds. Internat. J. Math., 17(1), 35–43 (2006) · Zbl 1097.37041 · doi:10.1142/S0129167X06003333 |
| [12] | G.T. Buzzard and S.Y. Lu. Algebraic surfaces holomorphically dominable by C 2. Invent. Math., 139(3):617–659, 2000. · Zbl 0967.14025 |
| [13] | F. Campana. On twistor spaces of the class C. J. Diff. Geom., 33:541–549, 1991. · Zbl 0694.32017 |
| [14] | F. Campana. Connexité rationnelle des variétés de Fano. Ann. Sci. École Norm. Sup. (4), 25(5):539–545, 1992. · Zbl 0783.14022 |
| [15] | P. Cascini and V. Lazić. The Minimal Model Program Revisited. Contributions to Algebraic Geometry, pages 169–187, 2012. · Zbl 1360.14051 |
| [16] | P. Cascini and V. Lazić. New outlook on the Minimal Model Program, I. Duke Math. J., 161(12):2415–2467, 2012. · Zbl 1261.14007 |
| [17] | P. Cascini, H. Tanaka, and C. Xu. On base point freeness in positive characteristic. Duke Math J. (under review), 2013. · Zbl 1408.14020 |
| [18] | A. Corti and V. Lazić. New outlook on the Minimal Model Program, II. Math. Ann., 356(2):617–633, 2013. · Zbl 1273.14033 |
| [19] | O. Debarre. Higher-Dimensional Algebraic Geometry. Universitext. Springer-Verlag, New York, 2001. · Zbl 0978.14001 |
| [20] | J.-P. Demailly. Multiplier ideal sheaves and analytic methods in algebraic geometry. In School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), volume 6 of ICTP Lect. Notes, pages 1–148. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. |
| [21] | J.-P. Demailly and M. Pǎun. Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. of Math. (2), 159(3):1247–1274, 2004. |
| [22] | M.P. do Carmo. Riemannian geometry. Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston, MA, 1992. |
| [23] | W. Fulton and R. Pandharipande. Notes on stable maps and Quantum Cohomology. In Algebraic Geometry–Santa Cruz, pages 45–96. Amer. math. Soc., 1997. · Zbl 0898.14018 |
| [24] | S.I. Goldberg and S. Kobayashi. Holomorphic bisectional curvature. J. Differential Geometry, 1:225–233, 1967. · Zbl 0169.53202 |
| [25] | T. Graber, J. Harris, and J. Starr. Families of rationally connected varieties. J. Amer. Math. Soc., 16(1):57–67 (electronic), 2003. · Zbl 1092.14063 |
| [26] | P. Griffiths and J. Harris. Principles of algebraic geometry. Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. · Zbl 0408.14001 |
| [27] | Grothendieck A.: Sur la classification des fibrés holomorphes sur la sphère de Riemann. Amer. J. Math., 79, 121–138 (1957) · Zbl 0079.17001 · doi:10.2307/2372388 |
| [28] | C. Hacon and J. McKernan. On Shokurov’s rational connectedness conjecture. Duke Math. J., 138(1):119–136, 2007. · Zbl 1128.14028 |
| [29] | C. Hacon and J. McKernan. Existence of minimal models for varieties of log general type II. J. Amer. Math. Soc., 23(2):469–490, 2010. · Zbl 1210.14021 |
| [30] | R. Hartshorne. Ample subvarieties of algebraic varieties. Notes written in collaboration with C. Musili. Lecture Notes in Mathematics, Vol. 156. Springer-Verlag, Berlin, 1970. · Zbl 0208.48901 |
| [31] | W. He and S. Sun. Frankel conjecture and Sasaki geometry. arXiv:1202.2589, 2012. · Zbl 1315.53044 |
| [32] | A. Hoering and T. Peternell. Minimal models for Kaehler threefolds. arXiv:1304.4013, 2013. |
| [33] | J.-M. Hwang. Geometry of minimal rational curves on Fano manifolds. In School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), volume 6 of ICTP Lect. Notes, pages 335–393. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. |
| [34] | Y. Kawamata. The cone of curves of algebraic varieties. Ann. of Math. (2), 119(3):603– 633, 1984. · Zbl 0544.14009 |
| [35] | S. Kebekus and L. Solá Conde. Existence of rational curves on algebraic varieties, minimal rational tangents, and applications. In Global aspects of complex geometry, pages 359–416. Springer, Berlin, 2006. · Zbl 1121.14012 |
| [36] | S. Keel and J. McKernan. Rational curves on quasi-projective surfaces. Mem. Amer. Math. Soc., 140(669):viii+153, 1999. · Zbl 0955.14031 |
| [37] | S. Kobayashi and T. Ochiai. Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ., 13:31–47, 1973. · Zbl 0261.32013 |
| [38] | J. Kollár. Cone theorems and cyclic covers. In Algebraic geometry and analytic geometry (Tokyo, 1990), ICM-90 Satell. Conf. Proc., pages 101–110. Springer, Tokyo, 1991. · Zbl 0807.14010 |
| [39] | J. Kollár. Effective base point freeness. Math. Ann., 296(4):595–605, 1993. · Zbl 0818.14002 |
| [40] | J. Kollár. Rational Curves on Algebraic Varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, 1996. |
| [41] | J. Kollár. Low degree polynomial equations: arithmetic, geometry and topology. In European Congress of Mathematics, Vol. I (Budapest, 1996), volume 168 of Progr. Math., pages 255–288. Birkhäuser, Basel, 1998. · Zbl 0970.14001 |
| [42] | J. Kollár et al. Flips and abundance for algebraic threefolds. Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992). |
| [43] | J. Kollár and T. Matsusaka. Riemann-Roch type inequalities. Amer. J. Math., 105(1):229–252, 1983. · Zbl 0538.14006 |
| [44] | J. Kollár, Y. Miyaoka, and S. Mori. Rational curves on Fano varieties. J. Diff. Geom., 3:765–779, 1992. · Zbl 0759.14032 |
| [45] | J. Kollár, Y. Miyaoka, and S. Mori. Rationally connected varieties. J. Algebraic Geometry, 1:429–448, 1992. · Zbl 0780.14026 |
| [46] | J. Kollár, Y. Miyaoka, S. Mori, and H. Takagi. Boundedness of canonical Q-Fano 3-folds. Proc. Japan Acad. Ser. A Math. Sci., 76(5):73–77, 2000. · Zbl 0981.14016 |
| [47] | J. Kollár and S. Mori. Birational Geometry of Algebraic Varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, 1998. |
| [48] | R. Lazarsfeld. Positivity in Algebraic Geometry. II, volume 49 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 2004. |
| [49] | B. Lehmann. A cone theorem for nef curves. J. Algebraic Geom., 21(3):473–493, 2012. · Zbl 1253.14014 |
| [50] | T.-J. Li and Y. Ruan. Symplectic birational geometry. In New perspectives and challenges in symplectic field theory, volume 49 of CRM Proc. Lecture Notes, pages 307–326. Amer. Math. Soc., Providence, RI, 2009. · Zbl 1183.53079 |
| [51] | K. Matsuki. Introduction to the Mori program. Universitext. Springer-Verlag, New York, 2002. · Zbl 0988.14007 |
| [52] | B. G. Moĭšzon. On n-dimensional compact complex manifolds having n algebraically independent meromorphic functions. I. Izv. Akad. Nauk SSSR Ser. Mat., 30:133–174, 1966. · Zbl 0161.17802 |
| [53] | N. Mok. The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J. Differential Geom., 27(2):179–214, 1988. · Zbl 0642.53071 |
| [54] | S. Mori. Projective manifolds with ample tangent bundles. Ann. of Math. (2), 110(3):593–606, 1979. · Zbl 0423.14006 |
| [55] | S. Mori. Threefolds whose canonical bundles are not numerically effective. Ann. of Math. (2), 116(1):133–176, 1982. · Zbl 0557.14021 |
| [56] | A.M. Nadel. Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature. Proc. Nat. Acad. Sci. U.S.A., 86(19):7299–7300, 1989. · Zbl 0711.53056 |
| [57] | T. Peternell. Towards a Mori theory on compact Kähler threefolds. II. Math. Ann., 311(4):729–764, 1998. · Zbl 0919.32016 |
| [58] | M. Reid. Lines on fano 3-folds according to Shokurov. 1980. |
| [59] | Y. Ruan. Symplectic topology and extremal rays. Geom. Funct. Anal., 3(4):395–430, 1993. · Zbl 0806.53032 |
| [60] | Y. Ruan. Virtual neighborhoods and pseudo-holomorphic curves. In Proceedings of 6th Gökova Geometry-Topology Conference, volume 23, pages 161–231, 1999. · Zbl 0967.53055 |
| [61] | W.-X. Shi. Ricci flow and the uniformization on complete noncompact Kähler manifolds. J. Differential Geom., 45(1):94–220, 1997. · Zbl 0954.53043 |
| [62] | V.V. Shokurov. The existence of a line on Fano varieties. Izv. Akad. Nauk SSSR Ser. Mat., 43(4):922–964, 968, 1979. · Zbl 0422.14019 |
| [63] | V.V. Shokurov. Letters of a bi-rationalist. I. A projectivity criterion. In Birational algebraic geometry (Baltimore, MD, 1996), volume 207 of Contemp. Math., pages 143–152. Amer. Math. Soc., Providence, RI, 1997. · Zbl 0918.14007 |
| [64] | Y.-T. Siu. Dynamic multiplier ideal sheaves and the construction of rational curves in Fano manifolds. In Complex analysis and digital geometry, volume 86 of Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., pages 323–360. Uppsala Universitet, Uppsala, 2009. · Zbl 1201.14013 |
| [65] | Y.-T. Siu and S.-T. Yau. Compact Kähler manifolds of positive bisectional curvature. Invent. Math., 59(2):189–204, 1980. · Zbl 0442.53056 |
| [66] | Z. Tian. Symplectic geometry of rationally connected threefolds. Duke Math. J., 161(5):803–843, 2012. · Zbl 1244.14041 |
| [67] | C. Voisin. Rationally connected 3-folds and symplectic geometry. Astérisque, (322):1– 21, 2008. Géométrie différentielle, physique mathématique, mathématiques et société. II. |
| [68] | S.-T. Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Amp’ere equation. I. Comm. Pure Appl. Math., 31(3):339–411, 1978. · Zbl 0369.53059 |
| [69] | Q. Zhang. Rational connectedness of log Q-Fano varieties. J. Reine Angew. Math., 590:131–142, 2006. · Zbl 1093.14059 |
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