scholarly journals On the index of a holomorphic vector field tangent to a singular variety

1995 ◽  
Vol 26 (2) ◽  
pp. 183-199 ◽  
Author(s):  
Daniel Lehmann ◽  
Marcio Soares ◽  
Tatsuo Suwa
Keyword(s):  
Vector Field ◽  
Holomorphic Vector Field ◽  
Singular Variety

scholarly journals ON THE MODIFIED FUTAKI INVARIANT OF COMPLETE INTERSECTIONS IN PROJECTIVE SPACES

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.

Vector Fields and the Cohomology Ring of Toric Varieties

2005 ◽  
Vol 48 (3) ◽  
pp. 414-427 ◽  
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.

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.

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

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.

scholarly journals A Note on Infinite Type Germs of a Real Hypersurface in

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.  

A Fundamental Solution for a Nonelliptic Partial Differential Operator

1979 ◽  
Vol 31 (5) ◽  
pp. 1107-1120 ◽  
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].

scholarly journals Complete scalar-flat Kähler metrics on affine algebraic manifolds

2021 ◽  
Author(s):  
Takahiro Aoi
Keyword(s):  
Vector Field ◽  
Line Bundle ◽  
Scalar Curvature ◽  
Holomorphic Section ◽  
Positive Scalar Curvature ◽  
Holomorphic Vector Field ◽  
Kähler Metric ◽  
Algebraic Manifolds ◽  
Kahler Metric ◽  
Positive Integers

AbstractLet $$(X,L_{X})$$ ( X , L X ) be an n-dimensional polarized manifold. Let D be a smooth hypersurface defined by a holomorphic section of $$L_{X}$$ L X . We prove that if D has a constant positive scalar curvature Kähler metric, $$X {\setminus } D$$ X \ D admits a complete scalar-flat Kähler metric, under the following three conditions: (i) $$n \ge 6$$ n ≥ 6 and there is no nonzero holomorphic vector field on X vanishing on D, (ii) the average of a scalar curvature on D denoted by $${\hat{S}}_{D}$$ S ^ D satisfies the inequality $$0< 3 {\hat{S}}_{D} < n(n-1)$$ 0 < 3 S ^ D < n ( n - 1 ) , (iii) there are positive integers $$l(>n),m$$ l ( > n ) , m such that the line bundle $$K_{X}^{-l} \otimes L_{X}^{m}$$ K X - l ⊗ L X m is very ample and the ratio m/l is sufficiently small.

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