About the Calabi problem: a finite-dimensional approach

H.-D. Cao; J. Keller

Displaying similar documents to “About the Calabi problem: a finite-dimensional approach”

The Kähler Ricci flow on Fano manifolds (I)

Xiuxiong Chen, Bing Wang (2012)

Journal of the European Mathematical Society

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We study the evolution of pluri-anticanonical line bundles $K_{M}^{- ν}$ along the Kähler Ricci flow on a Fano manifold $M$ . Under some special conditions, we show that the convergence of this flow is determined by the properties of the pluri-anticanonical divisors of $M$ . For example, the Kähler Ricci flow on $M$ converges when $M$ is a Fano surface satisfying $c_{1}^{2} (M) = 1$ or $c_{1}^{2} (M) = 3$ . Combined with the works in [CW1] and [CW2], this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism...

Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties

Robert J. Berman, Bo Berndtsson (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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

The gradient flow of Higgs pairs

Jiayu Li, Xi Zhang (2011)

Journal of the European Mathematical Society

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We consider the gradient flow of the Yang–Mills–Higgs functional of Higgs pairs on a Hermitian vector bundle $(E, H_{0})$ over a Kähler surface $(M, ω)$ , and study the asymptotic behavior of the heat flow for Higgs pairs at infinity. The main result is that the gradient flow with initial condition $(A_{0}, φ_{0})$ converges, in an appropriate sense which takes into account bubbling phenomena, to a critical point $(A_{\infty}, φ_{\infty})$ of this functional. We also prove that the limiting Higgs pair $(A_{\infty}, φ_{\infty})$ can be extended smoothly to a vector bundle...

Deformations of Kähler manifolds with nonvanishing holomorphic vector fields

Jaume Amorós, Mònica Manjarín, Marcel Nicolau (2012)

Journal of the European Mathematical Society

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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...

Towards a Mori theory on compact Kähler threefolds III

Thomas Peternell (2001)

Bulletin de la Société Mathématique de France

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Based on the results of the first two parts to this paper, we prove that the canonical bundle of a minimal Kähler threefold ( $K_{X}$ is nef) is good,its Kodaira dimension equals the numerical Kodaira dimension, (in particular some multiple of $K_{X}$ is generated by global sections); unless $X$ is simple. “Simple“ means that there is no compact subvariety through the very general point of $X$ and $X$ not Kummer. Moreover we show that a compact Kähler threefold with only terminal singularities...

Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

Henri Guenancia (2014)

Annales de l’institut Fourier

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Let $X$ be a compact Kähler manifold and $Δ$ be a $ℝ$ -divisor with simple normal crossing support and coefficients between $1 / 2$ and $1$ . Assuming that $K_{X} + Δ$ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on $X ∖ Supp (Δ)$ having mixed Poincaré and cone singularities according to the coefficients of $Δ$ . As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair $(X, Δ)$ .

Hölder continuous solutions to Monge–Ampère equations

Jean-Pierre Demailly, Sławomir Dinew, Vincent Guedj, Pham Hoang Hiep, Sławomir Kołodziej, Ahmed Zeriahi (2014)

Journal of the European Mathematical Society

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Let $(X, ω)$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^{p}$ right hand side, $p > 1$ . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X, ω)$ of the complex Monge-Ampère operator acting on $ω$ -plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with $L^{p}$ -density belong to $(X, ω)$ and proving that...

Approximately Einstein ACH metrics, volume renormalization, and an invariant for contact manifolds

Neil Seshadri (2009)

Bulletin de la Société Mathématique de France

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To any smooth compact manifold $M$ endowed with a contact structure $H$ and partially integrable almost CR structure $J$ , we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on $M \times (- 1, 0)$ . We consider the asymptotic expansion, in powers of a special defining function, of the volume of $M \times (- 1, 0)$ with respect to $g$ and prove that the log term coefficient is independent of $J$ (and any choice...

$η$ -Ricci Solitons on $η$ -Einstein ${(L C S)}_{n}$ -Manifolds

Shyamal Kumar Hui, Debabrata Chakraborty (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The object of the present paper is to study $η$ -Ricci solitons on $η$ -Einstein ${(L C S)}_{n}$ -manifolds. It is shown that if $ξ$ is a recurrent torse forming $η$ -Ricci soliton on an $η$ -Einstein ${(L C S)}_{n}$ -manifold then $ξ$ is (i) concurrent and (ii) Killing vector field.

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

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We study the extension problem for germs of holomorphic isometries $f : (D; x_{0}) \to (Ω; f (x_{0}))$ 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 $(z, \bar{w})$ to a neighborhood of $\bar{D} \times D$ , 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...

Convergence in capacity

Pham Hoang Hiep (2008)

Annales Polonici Mathematici

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We prove that if $(Ω) ∋ u_{j} \to u \in (Ω)$ in Cₙ-capacity then $l i m i n f_{j \to \infty} {(d d^{c} u_{j})}^{n} \geq 1_{u > - \infty} {(d d^{c} u)}^{n}$ . This result is used to consider the convergence in capacity on bounded hyperconvex domains and compact Kähler manifolds.

Appendix to the article of T. Peternell: the Kodaira dimension of Kummer threefolds

Frédéric Campana, Thomas Peternell (2001)

Bulletin de la Société Mathématique de France

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We prove that Kummer threefolds $T / G$ with algebraic dimension $0$ have Kodaira dimension 0.

An attraction result and an index theorem for continuous flows on $ℝ^{n} \times [0, \infty)$

Klaudiusz Wójcik (1997)

Annales Polonici Mathematici

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We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E = ℝ^{n + 1}$ for which $\partial E = ℝ^{n} \times 0$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on $ℝ^{n} \times [0, \infty)$ .

On the volume of a pseudo-effective class and semi-positive properties of the Harder-Narasimhan filtration on a compact Hermitian manifold

Zhiwei Wang (2016)

Annales Polonici Mathematici

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This paper divides into two parts. Let (X,ω) be a compact Hermitian manifold. Firstly, if the Hermitian metric ω satisfies the assumption that $\partial \partial ̅ ω^{k} = 0$ for all k, we generalize the volume of the cohomology class in the Kähler setting to the Hermitian setting, and prove that the volume is always finite and the Grauert-Riemenschneider type criterion holds true, which is a partial answer to a conjecture posed by Boucksom. Secondly, we observe that if the anticanonical bundle $K_{X}^{- 1}$ is nef, then for...

Three examples of brownian flows on $ℝ$

Yves Le Jan, Olivier Raimond (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We show that the only flow solving the stochastic differential equation (SDE) on $ℝ$ $d X_{t} = 1_{{X_{t} g t; 0}} W^{+} (d t) + 1_{{X_{t} l t; 0}} d W^{-} (d t),$ where $W^{+}$ and $W^{-}$ are two independent white noises, is a coalescing flow we will denote by $ϕ^{\pm}$ . The flow $ϕ^{\pm}$ is a Wiener solution of the SDE. Moreover, $K^{+} = 𝖤 [δ_{ϕ^{\pm}} | W^{+}]$ is the unique solution (it is also a Wiener solution) of the SDE $K_{s, t}^{+} f (x) = f (x) + \int_{s}^{t} K_{s, u} (1_{ℝ}^{+} f^{'}) (x) W^{+} (d u) + \frac{1}{2} \int_{s}^{t} K_{s, u} f ` ` (x) d u$ for $s l t; t$ , $x \in ℝ$ and $f$ a twice continuously differentiable function. A third flow $ϕ^{+}$ can be constructed out of the $n$ -point motions of $K^{+}$ . This flow is coalescing and its $n$ -point motion...

On Schrödinger maps from $T^{1}$ to $S^{2}$

Robert L. Jerrard, Didier Smets (2012)

Annales scientifiques de l'École Normale Supérieure

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We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from $T^{1}$ to $S^{2} .$ This estimate yields some continuity properties of the flow map for the topology of $L^{2} (T^{1}, S^{2})$ , provided one takes its quotient by the continuous group action of $T^{1}$ given by translations. We also prove that without taking this quotient, for any $t > 0$ the flow map at time $t$ is discontinuous as a map from $𝒞^{\infty} (T^{1}, S^{2})$ , equipped with the weak topology of $H^{1 / 2},$ to the space of distributions ${(𝒞^{\infty} (T^{1}, ℝ^{3}))}^{*} .$ The argument relies...

Measure-preserving homeomorphisms of noncompact manifolds and mass flow toward ends

Tatsuhiko Yagasaki (2007)

Fundamenta Mathematicae

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Suppose M is a noncompact connected n-manifold and ω is a good Radon measure of M with ω(∂M) = 0. Let ℋ(M,ω) denote the group of ω-preserving homeomorphisms of M equipped with the compact-open topology, and $ℋ_{E} (M, ω)$ the subgroup consisting of all h ∈ ℋ(M,ω) which fix the ends of M. S. R. Alpern and V. S. Prasad introduced the topological vector space (M,ω) of end charges of M and the end charge homomorphism $c^{ω} : ℋ_{E} (M, ω) \to (M, ω)$ , which measures for each $h \in ℋ_{E} (M, ω)$ the mass flow toward ends induced by h. We show that the...

On the Picard number of divisors in Fano manifolds

Cinzia Casagrande (2012)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$ . We consider the image $𝒩_{1} (D, X)$ of $𝒩_{1} (D)$ in $𝒩_{1} (X)$ under the natural push-forward of $1$ -cycles. We show that $ρ_{X} - ρ_{D} \leq codim 𝒩_{1} (D, X) \leq 8$ . Moreover if $codim 𝒩_{1} (D, X) \geq 3$ , then either $X ≅ S \times T$ where $S$ is a Del Pezzo surface, or $codim 𝒩_{1} (D, X) = 3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that $ρ_{X} - ρ_{T} = 4$ .

Symplectic critical surfaces in Kähler surfaces

Xiaoli Han, Jiayu Li (2010)

Journal of the European Mathematical Society

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Let $M$ be a Kähler surface and $Σ$ be a closed symplectic surface which is smoothly immersed in $M$ . Let $α$ be the Kähler angle of $Σ$ in $M$ . We first deduce the Euler-Lagrange equation of the functional $L = \int_{Σ} \frac{1}{cos α} d μ$ in the class of symplectic surfaces. It is ${cos}^{3} α H = {(J {(J \nabla cos α)}^{⊤})}^{⊥}$ , where $H$ is the mean curvature vector of $Σ$ in $M$ , $J$ is the complex structure compatible with the Kähler form $ω$ in $M$ , which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface...

Generalized gradient flow and singularities of the Riemannian distance function

Piermarco Cannarsa (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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Significant information about the topology of a bounded domain $Ω$ of a Riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial Ω}$ , from the boundary of $Ω$ . We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of $d_{\partial Ω}$ , as well as applications to homotopy equivalence.

The basic construction from the conditional expectation on the quantum double of a finite group

Qiaoling Xin, Lining Jiang, Zhenhua Ma (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and $H$ a subgroup. Denote by $D (G; H)$ (or $D (G)$ ) the crossed product of $C (G)$ and $ℂ H$ (or $ℂ G$ ) with respect to the adjoint action of the latter on the former. Consider the algebra $〈 D (G), e 〉$ generated by $D (G)$ and $e$ , where we regard $E$ as an idempotent operator $e$ on $D (G)$ for a certain conditional expectation $E$ of $D (G)$ onto $D (G; H)$ . Let us call $〈 D (G), e 〉$ the basic construction from the conditional expectation $E : D (G) \to D (G; H)$ . The paper constructs a crossed product algebra $C (G / H \times G) ⋊ ℂ G$ , and proves that there is an algebra isomorphism between...