scholarly journals Kähler–Einstein metrics: From cones to cusps

2020 ◽  
Vol 2020 (759) ◽  
pp. 1-27
Author(s):  
Henri Guenancia
Keyword(s):  
Einstein Metrics ◽  
Boundary Behavior ◽  
Cone Angle ◽  
Kähler Manifold ◽  
Strong Sense ◽  
Kahler Manifold ◽  
Kähler Einstein Metrics ◽  
Compact Kähler Manifold

AbstractIn this note, we prove that on a compact Kähler manifold \hskip-0.569055pt{X}\hskip-0.569055pt carrying a smooth divisor D such that {K_{X}+D} is ample, the Kähler–Einstein cusp metric is the limit (in a strong sense) of the Kähler–Einstein conic metrics when the cone angle goes to 0. We further investigate the boundary behavior of those and prove that the rescaled metrics converge to a cylindrical metric on {\mathbb{C}^{*}\times\mathbb{C}^{n-1}}.

scholarly journals Relative non-pluripolar product of currents

2021 ◽  
Author(s):  
Duc-Viet Vu
Keyword(s):  
Kähler Manifold ◽  
Monotonicity Property ◽  
Necessary Condition ◽  
Positive Current ◽  
Lelong Numbers ◽  
Kahler Manifold ◽  
Closed Positive Current ◽  
Positive Currents ◽  
Compact Kähler Manifold

AbstractLet X be a compact Kähler manifold. Let $$T_1, \ldots , T_m$$ T 1 , … , T m be closed positive currents of bi-degree (1, 1) on X and T an arbitrary closed positive current on X. We introduce the non-pluripolar product relative to T of $$T_1, \ldots , T_m$$ T 1 , … , T m . We recover the well-known non-pluripolar product of $$T_1, \ldots , T_m$$ T 1 , … , T m when T is the current of integration along X. Our main results are a monotonicity property of relative non-pluripolar products, a necessary condition for currents to be of relative full mass intersection in terms of Lelong numbers, and the convexity of weighted classes of currents of relative full mass intersection. The former two results are new even when T is the current of integration along X.

A POSITIVITY PROPERTY FOR FOLIATIONS ON COMPACT KÄHLER MANIFOLDS

2006 ◽  
Vol 17 (01) ◽  
pp. 35-43 ◽  
Author(s):  
MARCO BRUNELLA
Keyword(s):  
Kähler Manifold ◽  
Rational Curves ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Canonical Bundle ◽  
Positivity Property ◽  
Kahler Manifold ◽  
Compact Kähler Manifold

We prove that the canonical bundle of a foliation by curves on a compact Kähler manifold is pseudoeffective, unless the foliation is a (special) foliation by rational curves.

scholarly journals Zeros of Holomorphic One-Forms and Topology of Kähler Manifolds

2020 ◽  
Author(s):  
Stefan Schreieder
Keyword(s):  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Joint Paper ◽  
And Topology ◽  
Kahler Manifold ◽  
Compact Kähler Manifold

Abstract A conjecture of Kotschick predicts that a compact Kähler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao [ 10], we use our approach to prove Kotschick’s conjecture for smooth projective three-folds.

scholarly journals TWISTING OF N=1 SUSY GAUGE THEORIES AND HETEROTIC TOPOLOGICAL THEORIES

1995 ◽  
Vol 10 (30) ◽  
pp. 4325-4357 ◽  
Author(s):  
A. JOHANSEN
Keyword(s):  
Complex Structure ◽  
Kähler Manifold ◽  
Dolbeault Cohomology ◽  
Kähler Metric ◽  
Yang Mills ◽  
Kahler Manifold ◽  
Brst Charge ◽  
Kahler Metric ◽  
Compact Kähler Manifold ◽  
Topological Theories

It is shown that D=4N=1 SUSY Yang-Mills theory with an appropriate supermultiplet of matter can be twisted on a compact Kähler manifold. The conditions for cancellation of anomalies of BRST charge are found. The twisted theory has an appropriate BRST charge. We find a nontrivial set of physical operators defined as classes of the cohomology of this BRST operator. We prove that the physical correlators are independent of the external Kähler metric up to a power of a ratio of two Ray-Singer torsions for the Dolbeault cohomology complex on a Kähler manifold. The correlators of local physical operators turn out to be independent of antiholomorphic coordinates defined with a complex structure on the Kähler manifold. However, a dependence of the correlators on holomorphic coordinates can still remain. For a hyper-Kähler metric the physical correlators turn out to be independent of all coordinates of insertions of local physical operators.

Differential forms on a Kähler manifold

1951 ◽  
Vol 47 (3) ◽  
pp. 504-517 ◽  
Author(s):  
W. V. D. Hodge
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Complex Manifold ◽  
Differential Forms ◽  
Kähler Manifold ◽  
Systematic Account ◽  
General Plan ◽  
Kahler Manifold ◽  
Frequent Use ◽  
Compact Kähler Manifold

While a number of special properties of differential forms on a Kähler manifold have been mentioned in the literature on complex manifolds, no systematic account has yet been given of the theory of differential forms on a compact Kähler manifold. The purpose of this paper is to show how a general theory of these forms can be developed. It follows the general plan of de Rham's paper (2) on differential forms on real manifolds, and frequent use will be made of results contained in that paper. For convenience we begin by giving a brief account of the theory of complex tensors on a complex manifold, and of the differential geometry associated with a Hermitian, and in particular a Kählerian, metric on such a manifold.

scholarly journals An Extension of BCOV Invariant

2020 ◽  
Author(s):  
Yeping Zhang
Keyword(s):  
Kähler Manifold ◽  
Pezzo Surface ◽  
Del Pezzo Surface ◽  
Kahler Manifold ◽  
Del Pezzo ◽  
Compact Kähler Manifold ◽  
Calabi Yau Manifolds

Abstract Bershadsky, Cecotti, Ooguri, and Vafa constructed a real-valued invariant for Calabi–Yau manifolds, which is called the BCOV invariant. In this paper, we consider a pair $(X,Y)$, where $X$ is a compact Kähler manifold and $Y\in \big |K_X^m\big |$ with $m\in{\mathbb{Z}}\backslash \{0,-1\}$. We extend the BCOV invariant to such pairs. If $m=-2$ and $X$ is a rigid del Pezzo surface, the extended BCOV invariant is equivalent to Yoshikawa’s equivariant BCOV invariant. If $m=1$, the extended BCOV invariant is well behaved under blowup. It was conjectured that birational Calabi–Yau three-folds have the same BCOV invariant. As an application of our extended BCOV invariant, we show that this conjecture holds for Atiyah flops.

Type II compact almost homogeneous manifolds of cohomogeneity one-II

2019 ◽  
Vol 30 (13) ◽  
pp. 1940002
Author(s):  
Daniel Guan
Keyword(s):  
Moduli Space ◽  
Kähler Manifold ◽  
Type Ii ◽  
Homogeneous Manifolds ◽  
Kähler Metrics ◽  
Cohomogeneity One ◽  
Kahler Metrics ◽  
Kahler Manifold ◽  
Blowing Up ◽  
Compact Kähler Manifold

In this paper, we start the program of the existence of the smooth equivariant geodesics in the equivariant Mabuchi moduli space of Kähler metrics on type II cohomogeneity one compact Kähler manifold. In this paper, we deal with the manifolds [Formula: see text] obtained by blowing up the diagonal of the product of two copies of a [Formula: see text].

scholarly journals Vector Field Energies and Critical Metrics on Kähler Manifolds

2001 ◽  
Vol 162 ◽  
pp. 41-63 ◽  
Author(s):  
Toshiki Mabuchi
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Bilinear Pairing ◽  
Fano Varieties ◽  
Holomorphic Vector Field ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Extremal Kähler Metrics ◽  
Kähler Einstein Metrics ◽  
Compact Kähler Manifold

Associated with a Hamiltonian holomorphic vector field on a compact Kähler manifold, a nice functional on a space of Kähler metrics will be constructed as an integration of the bilinear pairing in [FM] contracted with the Hamiltonian holomorphic vector field. As applications, we have functionals whose critical points are extremal Kähler metrics or “Kähler-Einstein metrics” in the sense of [M4], respectively. Finally, the same method as used by [G1] allows us to obtain, from the convexity of , the uniqueness of “Kähler-Einstein metrics” on nonsingular toric Fano varieties possibly with nonvanishing Futaki character.

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