Abstract
These lectures are devoted to the study of various contemporary problems of algebraic geometry, using fundamental tools from complex potential theory, namely plurisubharmonic functions, positive currents and Monge-Ampère operators. Since their inception by Oka and Lelong in the mid 1940s, plurisubharmonic functions have been used extensively in many areas of algebraic and analytic geometry, as they are the function theoretic counterpart of pseudoconvexity, the complexified version of convexity. One such application is the theory of L 2 estimates via the Bochner-Kodaira-Hörmander technique, which provides very strong existence theorems for sections of holomorphic vector bundles with positive curvature. One can mention here the foundational work achieved by Bochner, Kodaira, Nakano, Morrey, Kohn, Andreotti-Vesentini, Grauert, Hörmander, Bombieri, Skoda and Ohsawa-Takegoshi in the course of more than four decades. Another development is the theory of holomorphic Morse inequalities (1985), which relate certain curvature integrals with the asymptotic cohomology of large tensor powers of line or vector bundles, and bring a useful complement to the Riemann-Roch formula.We describe here the main techniques involved in the proof of holomorphic Morse inequalities (Sect. 1) and their link with Monge-Ampère operators and intersection theory. Section 2, especially, gives a fundamental approximation theorem for closed (1, 1)-currents, using a Bergman kernel technique in combination with the Ohsawa-Takegoshi theorem. As an application, we study the geometric properties of positives cones of an algebraic variety (nef and pseudo-effective cone), and derive from there some results about asymptotic cohomology functionals in Sect. 3. The last Sect. 4 provides an application to the study of the Green-Griffiths-Lang conjecture. The latter conjecture asserts that every entire curve drawn on a projective variety of general type should satisfy a global algebraic equation; via a probabilistic curvature estimate, holomorphic Morse inequalities imply that entire curves must at least satisfy a global algebraic differential equation.
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References
M.F. Atiyah, R. Bott, V.K. Patodi, On the heat equation and the index theorem. Invent. Math. 19, 279–330 (1973)
Y. Akizuki, S. Nakano, Note on Kodaira-Spencer’s proof of Lefschetz theorems. Proc. Jpn. Acad. 30, 266–272 (1954)
A. Andreotti, H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr. 90, 193–259 (1962)
A. Andreotti, E. Vesentini, Carleman estimates for the Laplace-Beltrami equation in complex manifolds. Publ. Math. I.H.E.S. 25, 81–130 (1965)
F. Angelini, An algebraic version of Demailly’s asymptotic Morse inequalities. Proc. Am. Math. Soc. 124, 3265–3269 (1996)
G. Bérczi, Thom Polynomials and the Green-Griffiths Conjecture. arXiv: 1011.4710, Contributions to Algebraic Geometry (EMS Series of Congress Reports – European Mathematical Society, Zürich, 2012), pp. 141–167
R. Berman, J.-P. Demailly, Regularity of plurisubharmonic upper envelopes in big cohomology classes | arXiv:math.CV/0905.1246v1, in Proceedings of the Symposium “Perspectives in Analysis, Geometry and Topology” in honor of Oleg Viro (Stockholm University, May 2008), Progress in Math., 296 (Birkhäuser/Springer, New York, 2012), pp. 39–66
J.-M. Bismut, Demailly’s asymptotic inequalities: a heat equation proof. J. Funct. Anal. 72, 263–278 (1987)
A. Bloch, Sur les systèmes de fonctions uniformes satisfaisant à l’équation d’une variété algébrique dont l’irrégularité dépasse la dimension. J. Math. 5, 19–66 (1926)
S. Bochner, Curvature and Betti numbers (I) and (II). Ann. Math. 49, 379–390 (1948); 50, 77–93 (1949)
F.A. Bogomolov, Holomorphic tensors and vector bundles on projective varieties. Math. USSR Izv. 13/3, 499–555 (1979)
E. Bombieri, Algebraic values of meromorphic maps. Invent. Math. 10, 267–287 (1970); Addendum, Invent. Math. 11, 163–166 (1970)
L. Bonavero, Inégalités de Morse holomorphes singulières. C. R. Acad. Sci. Paris Sér. I Math. 317, 1163–1166 (1993)
L. Bonavero, Inégalités de Morse holomorphes singulières. J. Geom. Anal. 8, 409–425 (1998)
D. Borthwick, A. Uribe, Nearly Kählerian embeddings of symplectic manifolds. Asian J. Math. 4, 599–620 (2000)
Th. Bouche, Inégalités de Morse pour la d′-cohomologie sur une variété holomorphe non compacte. Ann. Sci. École Norm. Sup. 22, 501–513 (1989)
Th. Bouche, Convergence de la métrique de Fubini-Study d’un fibré linéaire positif. Ann. Inst. Fourier (Grenoble) 40, 117–130 (1990)
S. Boucksom, Cônes positifs des variétés complexes compactes. Thesis, Grenoble, 2002
S. Boucksom, J.-P. Demailly, M. Păun, Th. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, manuscript May 2004 [math.AG/0405285]
R. Brody, Compact manifolds and hyperbolicity. Trans. Amer. Math. Soc. 235, 213–219 (1978)
H. Clemens, Curves on generic hypersurfaces. Ann. Sci. Éc. Norm. Sup. 19, 629–636 (1986); Erratum: Ann. Sci. Éc. Norm. Sup. 20, 281 (1987)
M. Cowen, P. Griffiths, Holomorphic curves and metrics of negative curvature. J. Anal. Math. 29, 93–153 (1976)
F. Campana, Th. Peternell, Geometric stability of the cotangent bundle and the universal cover of a projective manifold [arXiv:math.AG/0405093]
J.-P. Demailly, Estimations L 2 pour l’opérateur \(\overline{\partial }\) d’un fibré vectoriel holomorphe semi-positif au dessus d’une variété kählérienne complète. Ann. Sci. Ec. Norm. Sup. 15, 457–511 (1982)
J.-P. Demailly, Champs magnétiques et inégalités de Morse pour la d′-cohomologie. Ann. Inst. Fourier (Grenoble) 35, 189–229 (1985)
J.-P. Demailly, in Singular Hermitian Metrics on Positive Line Bundles, ed. by K. Hulek, T. Peternell, M. Schneider, F. Schreyer. Proceedings of the Conference on Complex Algebraic Varieties, Bayreuth, April 2–6, 1990. Lecture Notes in Mathematics, vol. 1507 (Springer, Berlin, 1992)
J.-P. Demailly, in Holomorphic Morse Inequalities. Lectures given at the AMS Summer Institute on Complex Analysis held in Santa Cruz, July 1989. Proceedings of Symposia in Pure Mathematics, vol. 52, Part 2 (1991), pp. 93–114
J.-P. Demailly, Regularization of closed positive currents and Intersection Theory. J. Algebr. Geom. 1, 361–409 (1992)
J.-P. Demailly, A numerical criterion for very ample line bundles. J. Differ. Geom. 37, 323–374 (1993)
J.-P. Demailly, in L 2 Vanishing Theorems for Positive Line Bundles and Adjunction Theory, ed. by F. Catanese, C. Ciliberto. Lecture Notes of the CIME Session “Transcendental methods in Algebraic Geometry”, Cetraro, Italy, July 1994. Lecture Notes in Mathematics, vol. 1646, pp. 1–97
J.-P. Demailly, in Algebraic Criteria for Kobayashi Hyperbolic Projective Varieties and Jet Differentials, ed. by J. Kollár, R. Lazarsfeld. AMS Summer School on Algebraic Geometry, Santa Cruz 1995. Proceedings of Symposia in Pure Mathematics, 76 p.
J.-P. Demailly, Variétés hyperboliques et équations différentielles algébriques. Gaz. Math. 73, 3–23 (juillet 1997)
J.-P. Demailly, in Multiplier Ideal Sheaves and Analytic Methods in Algebraic Geometry. Lecture Notes, School on “Vanishing theorems and effective results in Algebraic Geometry, ICTP Trieste, April 2000 (Publications of ICTP, 2001)
J.-P. Demailly, Holomorphic Morse inequalities and asymptotic cohomology groups: a tribute to Bernhard Riemann. Milan J. Math. 78, 265–277 (2010) [arXiv: math.CV/1003.5067]
J.-P. Demailly, A converse to the Andreotti-Grauert theorem. Ann. Faculté des Sciences de Toulouse, Volume spécial en l’honneur de Nguyen Thanh Van. 20, 123–135 (2011)
J.-P. Demailly, J. El Goul, Hyperbolicity of generic surfaces of high degree in projective 3-space. Am. J. Math. 122, 515–546 (2000)
J.-P. Demailly, L. Ein, R. Lazarsfeld, A subadditivity property of multiplier ideals math. AG/0002035 ; Michigan Math. J. (special volume in honor of William Fulton) 48, 137–156 (2000)
J.-P. Demailly, J. Kollár, Semicontinuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Ec. Norm. Sup. 34, 525–556 (2001) [math.AG/9910118]
S. Diverio, Differential equations on complex projective hypersurfaces of low dimension. Compos. Math. 144, 920–932 (2008)
S. Diverio, Existence of global invariant jet differentials on projective hypersurfaces of high degree. Math. Ann. 344, 293–315 (2009)
S. Diverio, J. Merker, E. Rousseau, Effective algebraic degeneracy. Invent. Math. 180, 161–223 (2010)
J.-P. Demailly, M. Pǎun, Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. 159, 1247–1274 (2004) [arXiv: math.AG/0105176]
J.-P. Demailly, Th. Peternell, M. Schneider, Compact complex manifolds with numerically effective tangent bundles. J. Algebr. Geometry 3, 295–345 (1994)
J.-P. Demailly, Th. Peternell, M. Schneider, Pseudo-effective line bundles on compact Kähler manifolds. Int. J. Math. 12, 689–741 (2001)
S. Diverio, S. Trapani, A remark on the codimension of the Green-Griffiths locus of generic projective hypersurfaces of high degree. J. Reine Angew. Math. 649, 55–61 (2010)
T. de Fernex, A. Küronya, R. Lazarsfeld, Higher cohomology of divisors on a projective variety. Math. Ann. 337, 443–455 (2007)
T. Fujita, Approximating Zariski decomposition of big line bundles. Kodai Math. J. 17, 1–3 (1994)
E. Getzler, Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem. Commun. Math. Phys. 92, 167–178 (1983)
E. Getzler, An analogue of Demailly’s inequality for strictly pseudoconvex CR manifolds. J. Differ. Geom. 29, 231–244 (1989)
H. Grauert, O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen. Invent. Math. 11, 263–292 (1970)
M. Green, P. Griffiths, in Two Applications of Algebraic Geometry to Entire Holomorphic Mappings. The Chern Symposium 1979, Proceedings of the International Symposium, Berkeley, CA, 1979 (Springer, New York, 1980), pp. 41–74
P.A. Griffiths, in Hermitian Differential Geometry, Chern Classes and Positive Vector Bundles. Global Analysis, papers in honor of K. Kodaira (Princeton University Press, Princeton, 1969), pp. 181–251
R. Hartshorne, in Ample Subvarieties of Algebraic Varieties. Lecture Notes in Mathematics, vol. 156 (Springer, Berlin 1970)
H. Hess, R. Schrader, D.A. Uhlenbock, Kato’s inequality and the spectral distribution of Laplacians on compact Riemannian manifolds. J. Differ. Geom. 15 (1980), 27–38
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79, 109–326 (1964)
L. Hörmander, L 2 estimates and existence theorems for the \(\overline{\partial }\) operator. Acta Math. 113, 89–152 (1965)
A.G. Hovanski, Geometry of convex bodies and algebraic geometry. Uspehi Mat. Nau 34(4), 160–161 (1979)
S. Ji, Inequality for distortion function of invertible sheaves on abelian varieties. Duke Math. J. 58, 657–667 (1989)
S. Ji, B. Shiffman, Properties of compact complex manifolds carrying closed positive currents. J. Geom. Anal. 3, 37–61 (1993)
G. Kempf, in Metrics on Invertible Sheaves on Abelian Varieties. Topics in Algebraic Geometry, Guanajuato, 1989. Aportaciones Mat. Notas Investigación, vol. 5 (Soc. Mat. Mexicana, México 1992), pp. 107–108
K. Kodaira, On a differential geometric method in the theory of analytic stacks. Proc. Nat. Acad. Sci. USA 39, 1268–1273 (1953)
K. Kodaira, On Kähler varieties of restricted type. Ann. Math. 60, 28–48 (1954)
A. Küronya, Asymptotic cohomological functions on projective varieties. Am. J. Math. 128, 1475–1519 (2006)
L. Laeng, Estimations spectrales asymptotiques en géométrie hermitienne. Thèse de Doctorat de l’Université de Grenoble I, October 2002, http://www-fourier.ujf-grenoble.fr/THESE/ps/laeng.ps.gz and http://tel.archives-ouvertes.fr/tel-00002098/en/
S. Lang, Hyperbolic and diophantine analysis. Bull. Amer. Math. Soc. 14, 159–205 (1986)
S. Lang, Introduction to Complex Hyperbolic Spaces (Springer, New York, 1987)
R. Lazarsfeld, inPositivity in Algebraic Geometry I.-II. Ergebnisse der Mathematik und ihrer Grenzgebiete, vols. 48–49 (Springer, Berlin, 2004)
P. Lelong, Intégration sur un ensemble analytique complexe. Bull. Soc. Math. Fr. 85, 239–262 (1957)
P. Lelong, Plurisubharmonic Functions and Positive Differential Forms (Gordon and Breach, New York/Dunod, Paris, 1969)
L. Manivel, Un théorème de prolongement L2 de sections holomorphes d’un fibré vectoriel. Math. Z. 212, 107–122 (1993)
M. McQuillan, Diophantine approximation and foliations. Inst. Hautes Études Sci. Publ. Math. 87, 121–174 (1998)
M. McQuillan, Holomorphic curves on hyperplane sections of 3-folds. Geom. Funct. Anal. 9, 370–392 (1999)
J. Merker, Algebraic differential equations for entire holomorphic curves in projective hypersurfaces of general type, 89 p. arXiv:1005.0405
J. Milnor, in Morse Theory. Based on Lecture Notes by M. Spivak and R. Wells. Annals of Mathematics Studies, vol. 51 (Princeton University Press, Princeton, 1963) 153 pp.
A.M. Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Proc. Nat. Acad. Sci. USA 86, 7299–7300, (1989); Ann. Math. 132, 549–596 (1990)
S. Nakano, On complex analytic vector bundles. J. Math. Soc. Jpn. 7, 1–12 (1955)
T. Ohsawa, On the extension of L2 holomorphic functions, II. Publ. RIMS, Kyoto Univ. 24, 265–275 (1988)
T. Ohsawa, K. Takegoshi, On the extension of L2 holomorphic functions. Math. Z. 195, 197–204 (1987)
D. Popovici, Regularization of currents with mass control and singular Morse inequalities. J. Differ. Geom. 80, 281–326 (2008)
J.-P. Serre, Fonctions automorphes: quelques majorations dans le cas où X ∕ G est compact. Sém. Cartan (1953–1954), 2–1 à 2–9
J.-P. Serre, Un théorème de dualité. Comment. Math. 29, 9–26 (1955)
B. Shiffman, S. Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds. J. Reine Angew. Math. 544, 181–222 (2002)
C.L. Siegel, Meromorphic Funktionen auf kompakten Mannigfaltigkeiten. Nachrichten der Akademie der Wissenschaften in Göttingen. Math. Phys. Klasse 4, 71–77 (1955)
Y.T. Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)
Y.T. Siu, A vanishing theorem for semi-positive line bundles over non-Kähler manifolds. J. Differ. Geom. 19, 431–452 (1984)
Y.T. Siu, in Some Recent Results in Complex Manifold Theory Related to Vanishing Theorems for the Semi-positive Case. Proceedings of the Math. Arbeitstagung 1984. Lecture Notes in Mathematics, vol. 1111 (Springer, Berlin, 1985), pp. 169–192
Y.T. Siu, Calculus inequalities derived from holomorphic Morse inequalities. Math. Ann. 286, 549–558 (1990)
Y.T. Siu, An effective Matsusaka big theorem. Ann. Inst. Fourier 43, 1387–1405 (1993)
Y.T. Siu, A proof of the general schwarz lemma using the logarithmic derivative lemma. Communication personnelle, avril 1997
Y.T. Siu, in Some Recent Transcendental Techniques in Algebraic and Complex Geometry. Proceedings of the International Congress of Mathematicians, vol. I, Beijing, 2002 (Higher Ed. Press, Beijing, 2002), pp. 439–448
Y.T. Siu, in Hyperbolicity in Complex Geometry. The Legacy of Niels Henrik Abel (Springer, Berlin, 2004), pp. 543–566
Y.T. Siu, S.K. Yeung, Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane. Invent. Math. 124, 573–618 (1996)
Y.T. Siu, S.K. Yeung, Defects for ample divisors of Abelian varieties, Schwarz lemma and hyperbolic surfaces of low degree. Am. J. Math. 119, 1139–1172 (1997)
H. Skoda, Sous-ensembles analytiques d’ordre fini ou infini dans \({\mathbb{C}}^{n}\). Bull. Soc. Math. Fr. 100, 353–408 (1972)
H. Skoda, in EstimationsL 2 Pour L’opérateur \(\bar{\partial }\) et Applications Arithmétiques. Séminaire P. Lelong (Analyse), année 1975/1976. Lecture Notes in Mathematics, vol. 538 (Springer, Berlin, 1977), pp. 314–323
B. Teissier, Du théorème de l’index de Hodge aux inégalités isopérimétriques. C. R. Acad. Sc. Paris, sér. A 288, 287–289 (1979)
B. Teissier, Bonnesen-type inequalities in algebraic geometry, in Seminar on Differential Geometry, ed. by S. T. Yau (Princeton University Press, Princeton, 1982), pp. 85–105
G. Tian, On a set of polarized Kähler metrics on algebraic manifolds. J. Differ. Geom. 32, 99–130 (1990)
B. Totaro, Line bundles with partially vanishing cohomology, July 2010 [arXiv: math.AG/1007.3955]
S. Trapani, Numerical criteria for the positivity of the difference of ample divisors. Math. Z. 219, 387–401 (1995)
C. Voisin, On a conjecture of Clemens on rational curves on hypersurfaces. J. Differ. Geom. 44, 200–213 (1996); Correction: J. Differ. Geom. 49, 601–611 (1998)
E. Witten, Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)
Acknowledgements
The author expresses his warm thanks to the organizers of the CIME School in Pluripotential Theory held in Cetraro in July 2011, Filippo Bracci and John Erik Fornæss, for their invitation and the opportunity to deliver these lectures to an audience of young researchers. The author is also grateful to the referee for his (her) suggestions, and for a very careful reading of the manuscript.
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Demailly, JP. (2013). Applications of Pluripotential Theory to Algebraic Geometry. In: Pluripotential Theory. Lecture Notes in Mathematics(), vol 2075. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36421-1_3
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