We discuss some recent estimates for Bergman kernels of spaces of holomorphic sections or harmonic forms taking values in high powers of a given line bundle. The case where Î½ is Lebesgue measure on a polytope with rational coordinates is of special interest in differential and algebraic geometry because of its relation to the theory of toric varietes. Our theorem has several applications in algebraic geometry; to start with, we obtain the natural analytic generalization of some semipositivity results due to E. In the case of the anti-canonical class on a Fano manifold the constants in the inequalities are shown to only depend on the dimension of X but there are counterexamples to the precise value proposed by Aubin. The main result of this article is a practically optimal criterion for the pseudoeffectivity of the twisted relative canonical bundles of surjective projective maps.

As a result we derive a sharp Moser-Trudinger inequality for such functions. The holomorphic functional calculus for a is extended to algebras of ultra-differentiable functions on R n , depending on the growth of parallel to exp ia. I have more than 20 years of experience in Swedish and international business law, mainly from Swedish and international corporations, but also from law firms. We also show that similar results do not hold for general balanced domains except for complex ellipsoids and discuss related questions for convex functions. Varolin, A Takayama-type extension theorem, Compos.

As a byproduct, we give a simple and direct proof of a recent result due to C. Supercurrents, as introduced by Lagerberg, were mainly motivated as a way to study tropical varieties. Bo Berndtssons hemsida Bo Berndtsson Foto: T SjÃ¶stedt Professor of Mathematics Research: My research is primarily about functions of several complex variables and complex geometry. We prove that this is so provided the twisting line bundle is stricty positive along fibers, but not in general. In the different setting of pseudoconvex domains in complex space we also obtain a quasi-sharp version of the inequalities and relate it to Brezis-Merle type inequalities. More applications will be offered in the sequel of this article.

Here we adress the question if the curvature is strictly positive when the Kodaira-Spencer class does not vanish. The estimates are applied to the asymptotic study of eigenvalues of Toeplitz operators, sampling sequences and Morse inequalities. Tsuji concerning the extension of twisted pluricanonical forms. We moreover discuss a generalization of the main result to other bundles than Formula presented. We obtain explicit curvature formulas, especially in case where the said line bundle satisfies a natural curvature assumption. This paper is a sequel to Berndtsson in Ann Math 169:531â€”560, 2009.

Kawamata's celebrated subadjunction theorem in the spirit of our previous work on Bergman kernels. Several applications are obtained, including a proof of a result by Viehweg-Zuo in the context of a canonically polarized family of maximal variation. The notes can be found ; hopefully they will be continuously updated. Examples show that the estimate gives the right order of magnitude in terms of the two spectral parameters k and Î». As a consequence we get a simplified proof of the Bandoâ€”Mabuchi uniqueness theorem for KÃ¤hlerâ€”Einstein metrics. Examples show that the estimate gives the right order of magnitude in terms of the two spectral parameters k and lambda. We proved that the curvature of a such vector bundles is always semipositive in the sense of Nakano.

We also include a result on approximation of infinite dimensional geodesics by Bergman kernels which generalizes work of Phong and Sturm. In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on Formula presented. Full-text: Access denied no subscription detected We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. We proved that the curvature of a such vector bundles is always semipositive in the sense of Nakano. We also discuss various applications, among them a partial result on a conjecture of Griffiths on the positivity of ample bundles. We illustrate the idea by giving area estimates of minimal manifolds and a relatively short proof of Weyl's tube formula. Here we address the question if the curvature is strictly positive when the Kodairaâ€”Spencer class does not vanish.

. If you have a personal subscription to this journal, then please login. The paper supersedes our previous preprint concerning the case of toric Fano manifolds. The key ingredient is a new local positivity property of weak solutions to the homogenuous Monge-Ampere equation on a product domain, whose proof uses local Bergman kernels. We prove that this is so provided the twisting line bundle is strictly positive along fibers, but not in general. Boucksom, On the volume of a line bundle, Internat.

If you are already logged in, then you may need to update your profile to register your subscription. In this article we are interested in the differential geometric properties of certain higher direct images of exterior powers of the sheaf of relative differentials twisted with a line bundle. The underlying duality in this method is given by the Legendre transform. In this way we can use techniques from complex analysis to study real submanifolds. More generally, we obtain KÃ¤hler-Ricci solitons on any log Fano variety and show that they appear as the large time limit of the KÃ¤hler-Ricci flow. In the spring 2015 I am giving a minicourse on 'Convex and complex geometry'. The proof is inspired by our recent work on sharp Moser-Trudinger and Brezis-Merle type inequalities for the complex Monge-Ampere operator, but is essentially self-contained.

I am passionate about helping companies to build systematic ways to counteract business risks and thus build long-term successful businesses. Examples show that the estimate gives the right order of magnitude in terms of the two spectral parameters k and Î». The main result of the present article is a practically optimal criterium for the pseudoeffectivity of the twisted relative canonical bundles of surjective projective maps. As a byproduct, we give a simple and direct proof of a recent result due to C. Again, the existence of this sort of expansion was well known cf.

A brief and a quasicomplete My publications since 2005 can be found on Office: H5021, fourth floor of the math building. Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on X, even with very low regularity assumptions on the geodesic. The construction is based on a construction related to the Ohsawa-Takegoshi extension theorem combined with a method to construct weighted integral representations due to M. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart's volume conjecture for a large class of rational polytopes, including duals of lattice polytopes. We prove a conjecture saying that complex projective space has maximal volume degree among all toric Kaehler-Einstein manifolds of dimension n.