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About the Calabi problem: a finite-dimensional approach

H.-D. Cao, J. Keller (2013)

Journal of the European Mathematical Society

Let us consider a projective manifold M n and a smooth volume form Ω on M . We define the gradient flow associated to the problem of Ω -balanced metrics in the quantum formalism, the Ω -balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the Ω -balancing flow converges towards a natural flow in Kähler geometry, the Ω -Kähler flow. We also prove the long time existence of the Ω -Kähler flow and its convergence towards Yau’s solution to the Calabi conjecture of prescribing the...

Alpha-invariant of toric line bundles

Thibaut Delcroix (2015)

Annales Polonici Mathematici

We generalize the work of Jian Song by computing the α-invariant of any (nef and big) toric line bundle in terms of the associated polytope. We use the analytic version of the computation of the log canonical threshold of monomial ideals to give the log canonical threshold of any non-negatively curved singular hermitian metric on the line bundle, and deduce the α-invariant from this.

An obstruction to homogeneous manifolds being Kähler

Bruce Gilligan (2005)

Annales de l’institut Fourier

Let G be a connected complex Lie group, H a closed, complex subgroup of G and X : = G / H . Let R be the radical and S a maximal semisimple subgroup of G . Attempts to construct examples of noncompact manifolds X homogeneous under a nontrivial semidirect product G = S R with a not necessarily G -invariant Kähler metric motivated this paper. The S -orbit S / S H in X is Kähler. Thus S H is an algebraic subgroup of S [4]. The Kähler assumption on X ought to imply the S -action on the base Y of any homogeneous fibration X Y is algebraic...

Analytic inversion of adjunction: L 2 extension theorems with gain

Jeffery D. McNeal, Dror Varolin (2007)

Annales de l’institut Fourier

We establish new results on weighted L 2 -extension of holomorphic top forms with values in a holomorphic line bundle, from a smooth hypersurface cut out by a holomorphic function. The weights we use are determined by certain functions that we call denominators. We give a collection of examples of these denominators related to the divisor defined by the submanifold.

Canonical metrics on some domains of n

Fabio Zuddas (2008/2009)

Séminaire de théorie spectrale et géométrie

The study of the existence and uniqueness of a preferred Kähler metric on a given complex manifold M is a very important area of research. In this talk we recall the main results and open questions for the most important canonical metrics (Einstein, constant scalar curvature, extremal, Kähler-Ricci solitons) in the compact and the non-compact case, then we consider a particular class of complex domains D in n , the so-called Hartogs domains, which can be equipped with a natural Kaehler metric g ....

Classical Poincaré metric pulled back off singularities using a Chow-type theorem and desingularization

Caroline Grant Melles, Pierre Milman (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

We construct complete Kähler metrics on the nonsingular set of a subvariety X of a compact Kähler manifold. To that end, we develop (i) a constructive method for replacing a sequence of blow-ups along smooth centers, with a single blow-up along a product of coherent ideals corresponding to the centers and (ii) an explicit local formula for a Chern form associated to this ‘singular’ blow-up. Our metrics have a particularly simple local formula of a sum of the original metric and of the pull back...

Compact lcK manifolds with parallel vector fields

Andrei Moroianu (2015)

Complex Manifolds

We show that for n > 2 a compact locally conformally Kähler manifold (M2n , g, J) carrying a nontrivial parallel vector field is either Vaisman, or globally conformally Kähler, determined in an explicit way by a compact Kähler manifold of dimension 2n − 2 and a real function.

Convergence in capacity on smooth hypersurfaces of compact Kähler manifolds

Vu Viet Hung, Hoang Nhat Quy (2012)

Annales Polonici Mathematici

We study restrictions of ω-plurisubharmonic functions to a smooth hypersurface S in a compact Kähler manifold X. The result obtained and the characterization of convergence in capacity due to S. Dinew and P. H. Hiep [to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci.] are used to study convergence in capacity on S.

Deformations of Kähler manifolds with nonvanishing holomorphic vector fields

Jaume Amorós, Mònica Manjarín, Marcel Nicolau (2012)

Journal of the European Mathematical Society

We study compact Kähler manifolds X admitting nonvanishing holomorphic vector fields, extending the classical birational classification of projective varieties with tangent vector fields to a classification modulo deformation in the Kähler case, and biholomorphic in the projective case. We introduce and analyze a new class of 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛𝑠 , and show that they form a smooth subspace in the Kuranishi space of deformations of the complex structure of X . We extend Calabi’s theorem on the structure of compact Kähler...

Diastatic entropy and rigidity of complex hyperbolic manifolds

Roberto Mossa (2016)

Complex Manifolds

Let f : Y → X be a continuous map between a compact real analytic Kähler manifold (Y, g) and a compact complex hyperbolic manifold (X, g0). In this paper we give a lower bound of the diastatic entropy of (Y, g) in terms of the diastatic entropy of (X, g0) and the degree of f . When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary,when X = Y,we...

Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric

Ngaiming Mok (2012)

Journal of the European Mathematical Society

We study the extension problem for germs of holomorphic isometries f : ( D ; x 0 ) ( Ω ; f ( x 0 ) ) up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics d s D 2 on D and d s Ω 2 on Ω . Our main focus is on boundary extension for pairs of bounded domains ( D , Ω ) such that the Bergman kernel K D ( z , w ) extends meromorphically in ( z , w ¯ ) to a neighborhood of D ¯ × D , and such that the analogous statement holds true for the Bergman kernel K Ω ( ς , ξ ) on Ω . Assuming that ( D ; d s D 2 ) and ( Ω ; d s Ω 2 ) are complete Kähler manifolds, we prove that the germ...

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