Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 The supremum of first eigenvalues of conformally covariant operators in a conformal class
- 2 K-Destabilizing test configurations with smooth central fiber
- 3 Explicit constructions of Ricci solitons
- 4 Open Iwasawa cells and applications to surface theory
- 5 Multiplier ideal sheaves and geometric problems
- 6 Multisymplectic formalism and the covariant phase space
- 7 Nonnegative curvature on disk bundles
- 8 Morse theory and stable pairs
- 9 Manifolds with k-positive Ricci curvature
- References
5 - Multiplier ideal sheaves and geometric problems
Published online by Cambridge University Press: 05 November 2011
Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 The supremum of first eigenvalues of conformally covariant operators in a conformal class
- 2 K-Destabilizing test configurations with smooth central fiber
- 3 Explicit constructions of Ricci solitons
- 4 Open Iwasawa cells and applications to surface theory
- 5 Multiplier ideal sheaves and geometric problems
- 6 Multisymplectic formalism and the covariant phase space
- 7 Nonnegative curvature on disk bundles
- 8 Morse theory and stable pairs
- 9 Manifolds with k-positive Ricci curvature
- References
Summary
Abstract
In this expository article we first give an overview on multiplier ideal sheaves and geometric problems in Kählerian and Sasakian geometries. Then we review our recent results on the relationship between the support of the subschemes cut out by multiplier ideal sheaves and the invariant whose non-vanishing obstructs the existence of Kähler-Einstein metrics on Fano manifolds.
Introduction
One of the main problems in Kählerian and Sasakian geometries is the existence problem of Einstein metrics. An obvious necessary condition for the existence of a Kähler-Einstein metric on a compact Kähler manifold M is that the first Chern class c1(M) is negative, zero or positive since the Ricci form represents the first Chern class. This existence problem in Kählerian geometry was settled by Aubin [1] and Yau [58] in the negative case and by Yau [58] in the zero case. In the remaining case when the manifold has positive first Chern class, in which case the manifold is called a Fano manifold in algebraic geometry, there are two known obstructions. One is due to Matsushima [29] which says that the Lie algebra ɧ(M) of all holomorphic vector fields on a compact Kähler-Einstein manifold M is reductive. The other one is due to the first author [16] which is given by a Lie algebra character F : ɧ(M) → ℂ with the property that if M admits a Kähler-Einstein metric then F vanishes identically. Besides, it has been conjectured by Yau [59] that a more subtle condition related to geometric invariant theory (GIT) should be equivalent to the existence of Kähler-Einstein metrics.
- Type
- Chapter
- Information
- Variational Problems in Differential Geometry , pp. 68 - 93Publisher: Cambridge University PressPrint publication year: 2011
References
[1] : Equations du type de Monge-Ampère sur les variétés käahlériennes compactes, C. R. Acad. Sci. Paris, 283, 119–121 (1976)Google Scholar
[3] : Extremal Käahler metrics II, Differential geometry and complex analysis, ( and eds.), 95–114, Springer-Verlag, Berline-Heidelberg-New York, (1985).Google Scholar
[4] : Deformation of Käahler metrics to Käahler-Einstein metrics on compact Käahler manifolds, Invent. Math. 81 (1985) 359–372.Google Scholar
[5] and : Log canonical thresholds of smooth Fano threefolds (with an appendix by ), Uspekhi Mat. Nauk 63 (2008), no. 5 (383), 73–180; translation in Russian Math. Surveys 63 (2008), no. 5, 859–958.Google Scholar
[6] and : Geometry of Käahler metrics and foliations by holomorphic discs, Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 1–107. math.DG/0409433Google Scholar
[7] and : Remarks on Käahler Ricci flow, arXiv:0809.3963 (2008).
[8] , and : Uniqueness and examples of toric Sasaki-Einstein manifolds, Comm. Math. Phys., 277 (2008), 439–458, math.DG/0701122Google Scholar
[9] and : Semi-continuity of complex singularity exponents and Käahler-Einstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) 34, no. 4 (2001) 525–556.Google Scholar
[10] : Scalar curvature and stability of toric varieties, J. Differential Geometry, 62 (2002), 289–349.Google Scholar
[11] : Lower bounds on the Calabi functional, J. Differential Geometry, 70 (2005), 453–472.Google Scholar
[12] : Käahler geometry on toric manifolds, and some other manifolds with large symmetry, Handbook of geometric analysis. No. 1, 29–75, Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA, 2008. arXiv:0803.0985 (2008).Google Scholar
[13] : Remarks on gauge theory, complex geometry and four-manifold topology, in ‘Fields Medallists Lectures’ (Atiyah, Iagolnitzer eds.), World Scientific, 1997, 384–403.Google Scholar
[14] and : The geometry of four manifolds, Oxford Mathematical Monographs, Claren Press, Oxford, 1990.Google Scholar
[15] : Moduli space of polarized algebraic manifolds and Käahler metrics, Sugaku Expositions, 5 (1992), 173–191.Google Scholar
[16] : An obstruction to the existence of Einstein Käahler metrics, Invent. Math. 73, 437–443 (1983).Google Scholar
[17] : On compact Käahler manifolds of constant scalar curvature, Proc. Japan Acad., Ser. A, 59, 401–402 (1983).Google Scholar
[18] : Käahler-Einstein metrics and integral invariants, Lecture Notes in Math., vol.1314, Springer-Verlag, Berline-Heidelberg-New York, (1988).Google Scholar
[19] : Stability, integral invariants and canonical Käahler metrics, Proc. 9-th Internat. Conf. on Differential Geometry and its Applications, 2004 Prague, (eds. et al), 2005, 45–58, MATFYZPRESS, Prague.Google Scholar
[20] , and : Transverse Käahler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds, J. Differential Geom. 83 (2009), no. 3, 585–635.Google Scholar
[21] and : Multiplier ideal sheaves and integral invariants on toric Fano manifolds, arXiv:0711.0614 (2007).
[22] : Convergence of the Käahler-Ricci flow and multiplier ideal sheaves on Del Pezzo surfaces, Michigan Math. J. 58 (2009), no. 2, 423–440. arXiv:0710.5725Google Scholar
[23] : Existence of Käahler-Einstein metrics and multiplier ideal sheaves on Del Pezzo surfaces, Math. Z. 264 (2010), no. 4, 727–743. arXiv:0710.5724.Google Scholar
[24] : Singularities of pairs. (Algebraic Geometry–Santa Cruz 1995), Proc. Sympos. Pure Math., 62, part 1 (1995) 221–287.Google Scholar
[26] : The Monge-Ampère equation on compact Käahler manifolds, Indiana Univ. Math. J. 52, no.3 (2003) 667–686.Google Scholar
[27] : Positivity in Algebraic Geometry II: Positivity for vector bundles, and multiplier ideals, (Springer-Verlag, 2004).Google Scholar
[28] : K-stability of constant scalar curvature polarization, arXiv:0812. 4093
[29] : Sur la structure du groupe d'homéomorphismes d'une certaine variété kaehlérienne, Nagoya Math. J., 11, 145-150 (1957).Google Scholar
[30] : Multiplier ideal sheaves and Käahler-Einstein metrics of positive scalar curvature, Ann. of Math. (2) 132, no.3 (1990) 549–596.Google Scholar
[31] : Multiplier ideal sheaves and Futaki's invariant, Geometric Theory of Singular Phenomena in Partial Differential Equations (Cortona, 1995), Sympos. Math., XXXVIII, Cambridge Univ. Press, Cambridge, 1998. (1995) 7–16.Google Scholar
[32] and : Slope Stability and Exceptional Divisors of High Genus, Math. Ann. 343 (2009), no. 1, 79–101. arXiv:0710.4078v1 [math.AG].Google Scholar
[33] and : CM Stability and the Generalized Futaki Invariant I., math.AG/0605278, 2006.
[34] and : CM Stability and the Generalized Futaki Invariant II. (To appear in Asterisque), math.AG/0606505, 2006.
[35] and : On stability and the convergence of the Käahler-Ricci flow, J. Differential Geom. 72 (2006), no. 1, 149–168.Google Scholar
[36] and : Lectures on stability and constant scalar curvature, Handbook of geometric analysis, No. 3, 357–436, Adv. Lect. Math. (ALM), 14, Int. Press, Somerville, MA, 2010. arXiv:0801.4179.
[37] , and : Multiplier ideal sheaves and the Käahler-Ricci flow, Comm. Anal. Geom. 15, no. 3 (2007), 613–632.Google Scholar
[38] and : An obstruction to the existence of constant scalar curvature Käahler metrics, J. Differential Geom. 72 (2006), no. 3, 429–466.Google Scholar
[39] : The Ricci iteration and its applications, C. R. Acad. Sci. Paris, Ser. I, 345 (2007), 445-448, arXiv:0706.2777.Google Scholar
[40] : On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5839–5850. arXive:math/0708.1590.Google Scholar
[41] : Some discretizations of geometric evolution equations and the Ricci iteration on the space of Käahler metrics, Adv. Math. 218 (2008), no. 5, 1526–1565. arXive:math/0709.0990.Google Scholar
[42] : Multiplier ideal sheaves and the Käahler-Ricci flow on toric Fano manifolds with large symmetry, preprint, arXiv:0811.1455.
[43] and : Bounding scalar curvature and diameter along the Käahler-Ricci flow (after Perelman), J. Inst. Math. Jussieu 7 (2008), no. 3, 575–587.Google Scholar
[44] : The existence of Käahler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group, Annals of Math. 127 (1988) 585–627.Google Scholar
[45] Y-T. Siu : Dynamical multiplier ideal sheaves and the construction of rational curves in Fano manifolds, Complex analysis and digital geometry, 323–360, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 86, Uppsala Universitet, Uppsala, 2009. arXiv:0902.2809.
[46] : The α-invariant on toric Fano manifolds, Amer. J. Math. 127, No.6 (2005) 1247–1259.Google Scholar
[47] : K-stability of constant scalar curvature Käahler manifolds, Adv. Math. 221 (2009), no. 4, 1397–1408. arXiv:0803.4095.Google Scholar
[48] : The Käahler-Ricci flow and K-polystability, Amer. J. Math. 132 (2010), no. 4, 1077–1090. arXiv:0803.1613.Google Scholar
[49] : On Käahler-Einstein metrics on certain Käahler manifolds with C1(M) > 0, Invent. Math. 89, no.2 (1987) 225–246.Google Scholar
[50] , Käahler-Einstein metrics with positive scalar curvature, Invent. Math. 130, no.1 (1997) 1–37.Google Scholar
[51] and , Käahler-Einstein metrics on complex surfaces with C1(M) positive, Comm. Math. Phys. 112 (1987) 175–203.Google Scholar
[52] and : A new holomorphic invariant and uniqueness of Käahler-Ricci solitons, Comment. Math. Helv 77, No.2 (2002) 297–325.Google Scholar
[53] and : Convergence of Käahler-Ricci flow, J. Amer. Math. Soc., 20, No.3 (2007), 675–699.Google Scholar
[54] : Käahler-Ricci flow on stable Fano manifolds, J. Reine Angew. Math. 640 (2010), 67–84. arXiv:0810.1895.Google Scholar
[55] : Moment maps, Futaki invariant and stability of projective manifolds, Comm. Anal. Geom. 12 (2004), no. 5, 1009–1037.Google Scholar
[56] and : Käahler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188, No.1 (2004) 87–103.Google Scholar
[57] , A complex Frobenius theorem, multiplier ideal sheaves and Hermitian-Einstein metrics on stable bundles, Trans. Amer. Math. Soc. 359, no.4 (2007) 1577–1592.Google Scholar
[58] : On the Ricci curvature of a compact Käahler manifold and the complex Monge-Ampère equation I, Comm. Pure Appl. Math. 31 (1978), 339–441.Google Scholar
[60] : The logarithmic Sobolev inequality along the Ricci flow, arXiv:0707.2424 (2007).
[61] : A uniform Sobolev inequality under Ricci flow, arXiv:0706.1594 (2007).
[62] , Käahler-Ricci soliton type equations on compact complex manifolds with C1(M) > 0, J. Geom. Anal., 10 (2000), 759–774.Google Scholar
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