Anti-self-dual orbifolds with cyclic quotient singularities

Michael T. Lock; Jeff A. Viaclovsky

Displaying similar documents to “Anti-self-dual orbifolds with cyclic quotient singularities”

Semilinear equations, the $γ_{k}$ function, and generalized Gauduchon metrics

Jixiang Fu, Zhizhang Wang, Damin Wu (2013)

Journal of the European Mathematical Society

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In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the $γ_{k}$ functions on the space of its hermitian metrics.

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...

About the Calabi problem: a finite-dimensional approach

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

Journal of the European Mathematical Society

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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...

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 $(X, Δ)$ .

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...

Some critical almost Kähler structures

Takashi Oguro, Kouei Sekigawa (2008)

Colloquium Mathematicae

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We consider the set of all almost Kähler structures (g,J) on a 2n-dimensional compact orientable manifold M and study a critical point of the functional $ℱ_{λ, μ} (J, g) = \int_{M} (λ τ + μ τ *) d M_{g}$ with respect to the scalar curvature τ and the *-scalar curvature τ*. We show that an almost Kähler structure (J,g) is a critical point of $ℱ_{- 1, 1}$ if and only if (J,g) is a Kähler structure on M.

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.

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...

Which 3-manifold groups are Kähler groups?

Alexandru Dimca, Alexander Suciu (2009)

Journal of the European Mathematical Society

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The question in the title, first raised by Goldman and Donaldson, was partially answered by Reznikov. We give a complete answer, as follows: if $G$ can be realized as both the fundamental group of a closed 3-manifold and of a compact Kähler manifold, then $G$ must be finite—and thus belongs to the well-known list of finite subgroups of $O (4)$ , acting freely on $S^{3}$ .

Three dimensional near-horizon metrics that are Einstein-Weyl

Matthew Randall (2017)

Archivum Mathematicum

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We investigate which three dimensional near-horizon metrics $g_{N H}$ admit a compatible 1-form $X$ such that $(X, [g_{N H}])$ defines an Einstein-Weyl structure. We find explicit examples and see that some of the solutions give rise to Einstein-Weyl structures of dispersionless KP type and dispersionless Hirota (aka hyperCR) type.

Holomorphic actions, Kummer examples, and Zimmer program

Serge Cantat, Abdelghani Zeghib (2012)

Annales scientifiques de l'École Normale Supérieure

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We classify compact Kähler manifolds $M$ of dimension $n \geq 3$ on which acts a lattice of an almost simple real Lie group of rank $\geq n - 1$ . This provides a new line in the so-called Zimmer program, and characterizes certain complex tori as compact Kähler manifolds with large automorphisms groups.

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...

On real Kähler Euclidean submanifolds with non-negative Ricci curvature

Luis A. Florit, Wing San Hui, F. Zheng (2005)

Journal of the European Mathematical Society

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We show that any real Kähler Euclidean submanifold $f : M^{2 n} \to ℝ^{2 n + p}$ with either non-negative Ricci curvature or non-negative holomorphic sectional curvature has index of relative nullity greater than or equal to $2 n - 2 p$ . Moreover, if equality holds everywhere, then the submanifold must be a product of Euclidean hypersurfaces almost everywhere, and the splitting is global provided that $M^{2 n}$ is complete. In particular, we conclude that the only real Kähler submanifolds $M^{2 n}$ in $ℝ^{3 n}$ that have either positive Ricci curvature...

Toric Hermitian surfaces and almost Kähler structures

Włodzimierz Jelonek (2007)

Annales Polonici Mathematici

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The aim of this paper is to investigate the class of compact Hermitian surfaces (M,g,J) admitting an action of the 2-torus T² by holomorphic isometries. We prove that if b₁(M) is even and (M,g,J) is locally conformally Kähler and χ(M) ≠ 0 then there exists an open and dense subset U ⊂ M such that $(U, g_{| U})$ is conformally equivalent to a 4-manifold which is almost Kähler in both orientations. We also prove that the class of Calabi Ricci flat Kähler metrics related with the real Monge-Ampère equation...

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.

An obstruction to homogeneous manifolds being Kähler

Bruce Gilligan (2005)

Annales de l’institut Fourier

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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 $S / S \cap H$ in $X$ is Kähler. Thus $S \cap H$ 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...

A nonlinear Poisson transform for Einstein metrics on product spaces

Olivier Biquard, Rafe Mazzeo (2011)

Journal of the European Mathematical Society

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We consider the Einstein deformations of the reducible rank two symmetric spaces of noncompact type. If $M$ is the product of any two real, complex, quaternionic or octonionic hyperbolic spaces, we prove that the family of nearby Einstein metrics is parametrized by certain new geometric structures on the Furstenberg boundary of $M$ .

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...

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...

Partial integrability on Thurston manifolds

Hyeseon Kim (2013)

Annales Polonici Mathematici

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We determine the maximal number of independent holomorphic functions on the Thurston manifolds $M^{2 r + 2}$ , r ≥ 1, which are the first discovered compact non-Kähler almost Kähler manifolds. We follow the method which involves analyzing the torsion tensor dθ modθ, where $θ = (θ ¹, . . ., θ^{r + 1})$ are independent (1,0)-forms.

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...

Lie description of higher obstructions to deforming submanifolds

Marco Manetti (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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To every morphism $χ : L \to M$ of differential graded Lie algebras we associate a functors of artin rings ${Def}_{χ}$ whose tangent and obstruction spaces are respectively the first and second cohomology group of the suspension of the mapping cone of $χ$ . Such construction applies to Hilbert and Brill-Noether functors and allow to prove with ease that every higher obstruction to deforming a smooth submanifold of a Kähler manifold is annihilated by the semiregularity map.

On the Kähler Rank of Compact Complex Surfaces

Matei Toma (2008)

Bulletin de la Société Mathématique de France

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Harvey and Lawson introduced the Kähler rank and computed it in connection to the cone of positive exact currents of bidimension $(1, 1)$ for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions $(ℂ^{2}, 0) \to (ℂ^{2}, 0)$ .