Complex Monge-Ampère Equations on Compact Kähler Manifolds (14w5033)

Arriving in Banff, Alberta Sunday, April 6 and departing Friday April 11, 2014


(Institut de Mathématiques de Jussieu)

(Université Joseph Fourier (Grenoble1))

(Université Paul Sabatier)


The unifying theme of the present project is analytic methods in complex algebraic and Kähler geometry, with a special emphasis on complex Monge-Ampère equations. From a PDE point of view the latter are fully non-linear and possibly degenerate second order elliptic equations, and are quite ubiquitous in complex geometry and analysis, most strikingly in the context of Kähler geometry. Indeed on the one hand the complex Monge-Ampère operator is closely related to intersection theory since it can be seen as the top degree self-intersection operator on closed positive $(1,1)$-currents. On the other hand the Ricci curvature of a Kähler metric is expressed in terms of the complex Monge-Ampère operator of the Kähler potential, which explains why the existence problem for Kähler-Einstein metrics and the study of the Kähler-Ricci flow boil down to the study of a complex Monge-Ampère equation and the associated parabolic evolution equation.An impressive number of works have been devoted to the existence, uniqueness and regularity of solutions to complex Monge-Ampère equations, both on compact manifolds and on domains. These problems were settled for Kähler-Einstein equations of negative and zero curvature by Yau's resolution of the Calabi conjecture in the late 70's [Yau78], but to get a geometric understanding of the existence of Kähler-Einstein metrics on higher dimensional Fano manifolds (and more generally of constant scalar curvature Kähler metrics on polarized manifolds) is still a widely open problem and a very active research area. Roughly at the same time as Yau's result, Bedford and Taylor's fundamental work on degenerate Monge-Ampère equations in domains [BT82] opened new research directions in several complex variables and pluripotential theory. Let us mention notably the work of Kolodziej [Kol05].It is interesting to note that until very recently the differential-geometric and potential-theoretic sides have developed rather independently from each other, mostly because of a lack of common vocabulary and interest. Complex geometry allows to build a bridge between complex analysis and pluripotential theory on the one hand, and complex differential and algebraic geometry on the other hand. This was well illustrated by [Dem93], [DP04] where Monge-Ampère equations were used to obtain the first general results in the direction of Fujita's conjecture and to get a numerical characterization of the Kähler cone respectively.Perhaps the most famous problem in Kähler geometry is the problem of finding when Kähler-Einstein metrics exist on Fano manifolds. Since Kähler-Einstein metrics can also be viewed as the stationary points of the Kähler-Ricci flow, this problem is the same as the one of convergence of this flow. Its relation with the Monge-Ampère equation is particularly strong, since the Kähler-Ricci flow is just a parabolic version of the Monge-Ampère equation.Perelman's work [SeT08] has created new tools for the study of the Kähler-Ricci flow [TZ07, PSSW08, ST09], while the recent breakthrough of Birkar, Cascini, Hacon and McKernan [BCHM10] in the Minimal Model Program has motivated the study of Kähler-Einstein metrics on singular varieties [EGZ09, BEGZ10, ST12]. Campana's birational classification scheme [Camp11], which aims at understanding the hyperbolicity properties of complex varieties, also calls for Kähler-Einstein metrics on geometric orbifolds. Birational geometry of higher dimensional varieties more generally leads to consider complex Monge-Ampère equations in more degenerate situations:-the cohomology classes involved are no longer Kaehler,-the measures to be considered are no longer volume forms,-the solutions are merely weak (non smooth, possibly unbounded). These additional complications require the use of fine tools from complex analysis, pluripotential theory and algebraic geometry.It thus appears that the study of complex Monge-Ampère equations in a context general enough to fit with the birational classification schemes calls for a whole range of very diversified techniques.The last decade has witnessed an explosive growth in the subject, which has opened up entire new venues for investigation. Let us stress important progress on -developing new methods for solving Monge-Ampère equations (algebraic approximation [PS06], variational methods [BB10,BBGZ09])-regularity properties of degenerate solutions [DZ10,EGZ11],-extensions of Bando-Mabuchi uniqueness theorem [CT08,Ber09,Ber11], -conical Kähler-Einstein metrics [Don11,CGP11,JMR11], -limits of Kähler-Einstein manifolds [DS12,BBEGZ11]. A strong motivation for all these works comes from the Yau-Tian-Donaldson conjecture (see [PS10]) which states that the existence of constant scalar curvature Kähler metrics in a Hodge class is equivalent to a suitably modified version of GIT stability of the underlying polarized manifold. This influential conjecture has recently attracted great interest among algebraic geometers (see e.g. [Oda12]).The objective of the workshop is to bring together an international group of leading experts with complementary backgrounds from each above mentioned field, to report on and discuss recent progress and open problems in the area and thus foster interaction and collaboration between researchers in diverse subfields.The workshop is timely, since the bulk of the progress described in the scientific project took place in the last three or four years. It is remarkable that major contributions came from researchers from all over the world (China, France, Great Britain, Japan, Poland, Sweden, USA, etc.). Thus the workshop would also provide a unique opportunity for interaction between different groups who would normally reside in several distinct continents. [BT82] E.~Bedford, B.~A.~Taylor: A new capacity for plurisubharmonic functions. Acta Math. 149 (1982), no. 1-2, 1--40. [BB10] R.~Berman, S.~Boucksom: Growth of balls of holomorphic sections and energy at equilibrium. Invent. Math. 181 (2010).[BBGZ09] R.~Berman, S.~Boucksom, V.~Guedj, A.~Zeriahi: A variational approach to complex Monge-Ampère equations. arXiv:09[BBEGZ11] R.~Berman, S.~Boucksom, P.Eyssidieux, V.~Guedj, A.~Zeriahi: Kähler-Ricci flow and Ricci iteration on log-Fano varieties.[Ber09] B.~Berndtsson: Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. 169 (2009), no.2, 531-560. [Ber11] B.~Berndtsson: A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem. Preprint (2011) arXiv:1103.0923. [BCHM10] C.Birkar, P.Cascini, C.Hacon, J.McKernan: Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. 23 (2010), no. 2, 405-468. [BEGZ10] S.~Boucksom, P.~Eyssidieux, V.~Guedj, A.~Zeriahi: Monge-Amp{`e}re equations in big cohomology classes. Acta Math. 205 (2010), 199--262. [Camp11] F.~Campana: Special geometric orbifolds and bimeromorphic classification of compact Kähler manifolds. J. Inst. Math. Jussieu 10 (2011), no. 4, 809-934. [CGP11] F.~Campana, H.~Guenancia, M.~Paun: Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. Preprint (2011) arXiv:1104.4879v2. [CT08] X.X.Chen, G.Tian: Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. I.H.E.S. 107 (2008), 1-107.[Dem93] J.~P.~Demailly: A numerical criterion for very ample line bundles. J. Differential Geom. 37 (1993), no. 2, 323-374. [DP04] J.P.Demailly, M.Paun: Numerical characterization of the K{"a}hler cone of a compact K{"a}hler manifold. Ann. Math. (2004).[DZ10] S.~Dinew, Z.~Zhang: On stability and continuity of bounded solutions of degenerate complex Monge-Ampère equations over compact Kähler manifolds. Adv. Math. 225 (2010), 367--388. [Don11] S.~K.~Donaldson: Kähler metrics with cone singularities along a divisor. Preprint arXiv (2011).[DS12] S.~K.~Donaldson, S.~Sun: Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry. Preprint arXiv:1206.2609. [EGZ09] P.~Eyssidieux, V.~Guedj, A.~Zeriahi: Singular Kähler-Einstein metrics. J. Amer. Math. Soc. 22 (2009), 607-639. [EGZ11] P.~Eyssidieux, V.~Guedj, A.~Zeriahi: Viscosity solutions to degenerate Complex Monge-Ampère equations. Comm.Pure & Appl.Math 64 (2011), 1059--1094. [JMR11] T.D.~Jeffres, R.~Mazzeo, Y.A.~Rubinstein: Kähler-Einstein metrics with edge singularities. Preprint arXiv:1105.5216. [Kol05] S.Kolodziej: The complex Monge-Amp{`e}re equation and pluripotential theory. Mem.A.M.S. 178 (2005), no. 840, 64 pp.[Oda12] Y.~Odaka: The GIT stability of polarized varieties via discrepancy, Annals of Math., to appear.[PS06] D.~H.~Phong, J.~Sturm: The Monge-Ampère operator and geodesics in the space of Kähler potentials. Invent. Math. 166 (2006), no. 1, 125-149.[PSSW08] D.~H.~Phong, J.~Song, J.~Sturm, B.~Weinkove: The Kähler-Ricci flow with positive bisectional curvature. Invent. Math. 173 (2008), no. 3, 651-665.[PS10] D.~H.~Phong, J.~Sturm: Lectures on stability and constant scalar curvature. Handbook of geometric analysis, No. 3, 357–436, Adv. Lect. Math. (ALM), 14, Int. Press (2010).[SeT08] N.Sesum, G.Tian: Bounding scalar curvature and diameter along the Kähler-Ricci flow (after Perelman). J. Inst. Math. Jussieu 7 (2008), no. 3, 575-587. [ST12] J.~Song, G.~Tian: Canonical measures and Kähler-Ricci flow. J. Amer. Math. Soc. 25 (2012), no. 2, 303-353.[ST09] J.~Song, G.~Tian: The Kähler-Ricci flow through singularities. Preprint (2009) arXiv:0909.4898. [TZ07] G.~Tian, X.~Zhu: Convergence of Kähler-Ricci flow. J. Amer. Math. Soc. 20 (2007), no. 3, 675--699.[Yau78] S.~T.~Yau: On the Ricci curvature of a compact K{"a}hler manifold and the complex Monge-Amp{`e}re equation. I. Comm. Pure Appl. 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