Existence of a complete holomorphic vector field via the Kähler–Einstein metric

Abstract.In this paper, we consider a generalized Kähler–Einstein equation on a Kähler manifold M. Using the twisted 𝒦–energy introduced by Song and Tian, we show that the existence of generalized Kähler–Einstein metrics with semi–positive twisting (1, 1)–form θ is also closely related to the properness of the twisted 𝒦-energy functional. Under the condition that the twisting form θ is strictly positive at a point or M admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized Kähler–Einstein metric implies a Moser–Trudinger type inequality.

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ON THE MODIFIED FUTAKI INVARIANT OF COMPLETE INTERSECTIONS IN PROJECTIVE SPACES

Nagoya Mathematical Journal ◽

10.1017/nmj.2016.16 ◽

2016 ◽

Vol 222 (1) ◽

pp. 186-209

Author(s):

RYOSUKE TAKAHASHI

Keyword(s):

Vector Field ◽

Recent Work ◽

Projective Spaces ◽

Ricci Soliton ◽

Complete Intersections ◽

Necessary Condition ◽

Ricci Solitons ◽

Fano Varieties ◽

Holomorphic Vector Field ◽

Fano Manifold

Let $M$ be a Fano manifold. We call a Kähler metric ${\it\omega}\in c_{1}(M)$ a Kähler–Ricci soliton if it satisfies the equation $\text{Ric}({\it\omega})-{\it\omega}=L_{V}{\it\omega}$ for some holomorphic vector field $V$ on $M$. It is known that a necessary condition for the existence of Kähler–Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian and Zhu. In a recent work of Berman and Nyström, it was generalized for (possibly singular) Fano varieties, and the notion of algebrogeometric stability of the pair $(M,V)$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.

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Vector Fields and the Cohomology Ring of Toric Varieties

Canadian Mathematical Bulletin ◽

10.4153/cmb-2005-039-1 ◽

2005 ◽

Vol 48 (3) ◽

pp. 414-427 ◽

Cited By ~ 1

Author(s):

Kiumars Kaveh

Keyword(s):

Vector Field ◽

Vector Fields ◽

Projective Variety ◽

Cohomology Ring ◽

Holomorphic Vector Field ◽

Zero Set ◽

Graded Algebras ◽

Smooth Complex ◽

Complex Projective Variety ◽

Complex Projective

AbstractLetXbe a smooth complex projective variety with a holomorphic vector field with isolated zero setZ. From the results of Carrell and Lieberman there exists a filtrationF0⊂F1⊂ · · · ofA(Z), the ring of ℂ-valued functions onZ, such thatas graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra ofX.

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Vector Field Energies and Critical Metrics on Kähler Manifolds

Nagoya Mathematical Journal ◽

10.1017/s0027763000007790 ◽

2001 ◽

Vol 162 ◽

pp. 41-63 ◽

Cited By ~ 3

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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On holomorphic flows on Stein surfaces: Transversality, dicriticalness and stability

International Journal of Mathematics ◽

10.1142/s0129167x14500931 ◽

2014 ◽

Vol 25 (10) ◽

pp. 1450093

Author(s):

T. Ito ◽

B. Scárdua ◽

Y. Yamagishi

Keyword(s):

Vector Field ◽

Periodic Orbit ◽

Finite Number ◽

Sufficient Conditions ◽

First Integral ◽

Holomorphic Vector Field ◽

Necessary And Sufficient ◽

Holonomy Map

We study the classification of the pairs (N, X) where N is a Stein surface and X is a ℂ-complete holomorphic vector field with isolated singularities on N. We describe the role of transverse sections in the classification of X and give necessary and sufficient conditions on X in order to have N biholomorphic to ℂ2. As a sample of our results, we prove that N is biholomorphic to ℂ2 if H2(N, ℤ) = 0, X has a finite number of singularities and exhibits a singularity with three separatrices or, equivalently, a singularity with first jet of the form [Formula: see text] where λ1/λ2 ∈ ℚ+. We also study flows with many periodic orbits (i.e. orbits diffeomorphic to ℂ*), in a sense we will make clear, proving they admit a meromorphic first integral or they exhibit some special periodic orbit, whose holonomy map is a non-resonant nonlinearizable diffeomorphism map.

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A Note on Infinite Type Germs of a Real Hypersurface in

VNU Journal of Science Mathematics - Physics ◽

10.25073/2588-1124/vnumap.4345 ◽

2019 ◽

Vol 35 (2) ◽

Author(s):

Nguyen Thi Kim Son ◽

Chu Van Tiep

Keyword(s):

Vector Field ◽

Automorphism Group ◽

Real Hypersurface ◽

Infinite Order ◽

Holomorphic Vector Field ◽

Key Words And Phrases ◽

Holomorphic Curve ◽

Infinite Type ◽

Field Automorphism ◽

Order Contact

Abstract: The purpose of this article is to show that there exists a smooth real hypersurface germ of D'Angelo infinite type in such that it does not admit any (singular) holomorphic curve that has infinite order contact with at . 2010 Mathematics Subject Classification. Primary 32T25; Secondary 32C25. Key words and phrases: Holomorphic vector field, automorphism group, real hypersurface, infinite type point.

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A Fundamental Solution for a Nonelliptic Partial Differential Operator

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-101-3 ◽

1979 ◽

Vol 31 (5) ◽

pp. 1107-1120 ◽

Cited By ~ 22

Author(s):

Peter C. Greiner

Keyword(s):

Vector Field ◽

Differential Operator ◽

Fundamental Solution ◽

Half Plane ◽

Partial Differential Operator ◽

Holomorphic Vector Field ◽

Partial Differential ◽

Upper Half Plane ◽

Cauchy Riemann ◽

Image Position

Let(1)and set(2)Here . Z is the “unique” (modulo multiplication by nonzero functions) holomorphic vector-field which is tangent to the boundary of the “degenerate generalized upper half-plane”(3)In our terminology t = Re z1. We note that ℒ is nowhere elliptic. To put it into context, ℒ is of the type □b, i.e. operators like ℒ occur in the study of the boundary Cauchy-Riemann complex. For more information concerning this connection the reader should consult [1] and [2].

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