Let L be a (semi)-positive line bundle over Kähler manifold, X, fibered complex manifold Y .Assuming the fibers are compact and nonsingular we prove that hermitian vector E whose points y spaces of global sections X to ⊗ K X/Y , endowed with 2 -metric, is in sense Nakano.We also discuss various applications, among them partial result on conjecture Griffiths positivity ample bundles.

The main result of this article is a (practically optimal) criterion for the pseudoeffectivity twisted relative canonical bundles surjective projective maps. Our theorem has several applications in algebraic geometry; to start with, we obtain natural analytic generalization some semipositivity results due E. Viehweg [40], [41] and F. Campana [6]. As byproduct, give simple direct proof recent C. Hacon J. McKernan [16] S. Takayama [29], [30] concerning extension pluricanonical forms. More will...

We construct a generalization of the Henkin-Ramírez (or Cauchy-Leray) kernels for ∂ ¯-equation. The consists in multiplication by weight factor and addition suitable lower order terms, is found via representation as an "oscillating integral". As special cases we consider weights which behave like power distance to boundary, exp-φ with φ convex, polynomial decrease C n . also briefly singularities on subvarieties domains

We give an elementary proof of the existence asymptotic expansion in powers k Bergman kernel associated to Lk, where L is a positive line bundle over compact complex manifold. also algorithm for computing coefficients expansion.

We establish the convexity of Mabuchi’s <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-energy functional along weak geodesics in space Kähler potentials on a compact manifold, thus confirming conjecture Chen, and give some applications geometry, including proof uniqueness...

We give a proof of the Ohsawa–Takegoshi extension theorem with sharp estimates. The is based on ideas Błocki to use variations domains simplify his Suita conjecture, and also uses positivity properties direct image bundles.

We show, using a direct variational approach, that the second boundary value problem for Monge-Ampère equation in ℝ n with exponential non-linearity and target convex body P is solvable iff 0 barycenter of P. Combined some toric geometry this confirms, particular, (generalized) Yau-Tian-Donaldson conjecture log Fano varieties (X,Δ) saying admits (singular) Kähler-Einstein metric it K-stable algebro-geometric sense. thus obtain new proof extend to setting seminal result Wang-Zhou concerning...

We give a proof of the openness conjecture Demailly and Kollár.

We give a short proof of the extension theorem Ohsawa-Takegoshi. The same method also gives generalization ∂ ¯-theorem Donnelly and Fefferman for case (n,1)-forms.

For $\phi$ a metric on the anticanonical bundle, $-K_X$, of Fano manifold $X$ we consider volume $$ \int_X e^{-\phi}. We prove that logarithm is concave along continuous geodesics in space positively curved metrics $-K_X$ and concavity strict unless geodesic comes from flow holomorphic vector field $X$. As consequences get simplified proof Bando-Mabuchi uniqueness theorem for K\"ahler - Einstein generalization this to 'twisted' K\"ahler-Einstein metrics.

We develop some results from [4] on the positivity of direct image bundles in particular case a trivial vibration over one-dimensional base. also apply to study variations tions Kähler metrics.

In this article we are interested in the differential geometric properties of certain higher direct images exterior powers sheaf relative differentials twisted with a line bundle. We obtain explicit curvature formulas, especially case where said bundle satisfies natural assumption. Several applications obtained, including proof result by Viehweg-Zuo context canonically polarized family maximal variation.

Abstract We establish a new extension result for twisted canonical forms defined on hypersurface with simple normal crossings of projective manifold. Some the examples presented in appendix show that bounds we obtain are sharp.

We develop the calculus of superforms as a tool for convex geometry. The formalism is applied to valuations on bodies, Alexandrov-Fenchel inequalities and Monge- Amp\`ere equations boundary bodies.

We prove Aubin's "Hypothese fondamentale" concerning the existence of Moser-Trudinger type inequalities on any integral compact K\"ahler manifold X. In case anti-canonical class a Fano constants in are shown to only depend dimension X (but there counterexamples precise value proposed by Aubin). different setting pseudoconvex domains complex space we also obtain quasi-sharp version and relate it Brezis-Merle inequalities. The be sharp for S^{1}-invariant functions unit-ball. give applications...

We associate certain probability measures on $\R$ to geodesics in the space $\H_L$ of positively curved metrics a line bundle $L$, and finite dimensional symmetric hermitian norms $H^0(X, kL)$. prove that associated spaces converge weakly related as $k$ goes infinity. The convergence second order moments implies recent result Chen Sun geodesic distances respective spaces, while first gives Donaldson's $Z$-functional Aubin-Yau energy. also include approximation infinite by Bergman kernels...

In this article our main result is a more complete version of the statements obtained in {\rm [6]}. One important technical point proof an $\displaystyle L^{2\over m}$ extension theorem Ohsawa-Takegoshi type, which derived from original by simple fixed method. Moreover, we show that these techniques combined with appropriate form the"invariance plurigenera" can be used order to obtain new celebrated Y. Kawamata subadjunction theorem.

Abstract We establish here several “invariance of plurigenera type” theorems for twisted pluricanonical forms and metrics adjoint ℝ-bundles.

Abstract We show that the complex projective space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msup><m:mi>ℙ</m:mi><m:mi>n</m:mi></m:msup></m:math> ${\mathbb{P}^{n}}$ has maximal degree (volume) among all n -dimensional Kähler–Einstein Fano manifolds admitting a non-trivial holomorphic xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msup><m:mi>ℂ</m:mi><m:mo>*</m:mo></m:msup></m:math> ${\mathbb{C}^{*}}$ -action with finite number of fixed points. The toric version this result,...