The Experts below are selected from a list of 10284 Experts worldwide ranked by ideXlab platform
Gordon Heier - One of the best experts on this subject based on the ideXlab platform.
-
Hermitian metrics of positive holomorphic Sectional Curvature on fibrations
Mathematische Zeitschrift, 2020Co-Authors: Ananya Chaturvedi, Gordon HeierAbstract:The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic Sectional Curvature, then so does the total space of the fibration. The proof is based on the use of a warped product metric as in the work by Cheung in case of negative holomorphic Sectional Curvature, but differs in certain key aspects, e.g., in that it does not use the subadditivity property for holomorphic Sectional Curvature due to Grauert-Reckziegel and Wu.
-
reduction of manifolds with semi negative holomorphic Sectional Curvature
Mathematische Annalen, 2018Co-Authors: Gordon Heier, Fangyang Zheng, Steven S Y Lu, Bun WongAbstract:In this note, we continue the investigation of a projective Kahler manifold M of semi-negative holomorphic Sectional Curvature H. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent truly flat directions of H in the tangent spaces. We prove that this invariant is bounded above by the nef dimension and bounded below by the numerical Kodaira dimension of M. We also prove a splitting theorem for M in terms of the nef dimension and, under some additional hypotheses, in terms of the new rank invariant.
-
on projectivized vector bundles and positive holomorphic Sectional Curvature
arXiv: Differential Geometry, 2016Co-Authors: Angelynn Alvarez, Gordon Heier, Fangyang ZhengAbstract:We generalize a construction of Hitchin to prove that, given any compact K\"ahler manifold $M$ with positive holomorphic Sectional Curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a K\"ahler metric with positive holomorphic Sectional Curvature.
-
optimal pinching for the holomorphic Sectional Curvature of hitchin s metrics on hirzebruch surfaces
arXiv: Differential Geometry, 2015Co-Authors: Angelynn Alvarez, Ananya Chaturvedi, Gordon HeierAbstract:The main result of this note is that, for each $n\in \{1,2,3,\ldots\}$, there exists a Hodge metric on the $n$-th Hirzebruch surface whose positive holomorphic Sectional Curvature is $\frac{1}{(1+2n)^2}$-pinched. The type of metric under consideration was first studied by Hitchin in this context. In order to address the case $n=0$, we prove a general result on the pinching of the holomorphic Sectional Curvature of the product metric on the product of two Hermitian manifolds $M$ and $N$ of positive holomorphic Sectional Curvature.
-
k ahler manifolds of semi negative holomorphic Sectional Curvature
arXiv: Algebraic Geometry, 2014Co-Authors: Gordon Heier, Steven S Y Lu, Bun WongAbstract:In an earlier work, we investigated some consequences of the existence of a K\"ahler metric of negative holomorphic Sectional Curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic Sectional Curvature. In doing so, we define a new invariant that records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic Sectional Curvature vanishes. Using this invariant, we establish lower bounds for the nef dimension and, under certain additional assumptions, for the Kodaira dimension of the manifold. In dimension two, a precise structure theorem is obtained.
Fangyang Zheng - One of the best experts on this subject based on the ideXlab platform.
-
reduction of manifolds with semi negative holomorphic Sectional Curvature
Mathematische Annalen, 2018Co-Authors: Gordon Heier, Fangyang Zheng, Steven S Y Lu, Bun WongAbstract:In this note, we continue the investigation of a projective Kahler manifold M of semi-negative holomorphic Sectional Curvature H. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent truly flat directions of H in the tangent spaces. We prove that this invariant is bounded above by the nef dimension and bounded below by the numerical Kodaira dimension of M. We also prove a splitting theorem for M in terms of the nef dimension and, under some additional hypotheses, in terms of the new rank invariant.
-
hirzebruch manifolds and positive holomorphic Sectional Curvature
arXiv: Differential Geometry, 2016Co-Authors: Bo Yang, Fangyang ZhengAbstract:This paper is the first step in a systematic project to study examples of Kahler manifolds with positive holomorphic Sectional Curvature ($H > 0$). Previously Hitchin proved that any compact Kahler surface with $H>0$ must be rational and he constructed such examples on Hirzebruch surfaces $M_{2, k}=\mathbb{P}(H^{k}\oplus 1_{\mathbb{CP}^1})$. We generalize Hitchin's construction and prove that any Hirzebruch manifold $M_{n, k}=\mathbb{P}(H^{k}\oplus 1_{\mathbb{CP}^{n-1}})$ admits a Kahler metric of $H>0$ in each of its Kahler classes. We demonstrate that the pinching behaviors of holomorphic Sectional Curvatures of new examples differ from those of Hitchin's which were studied in the recent work of Alvarez-Chaturvedi-Heier. Some connections to recent works on the Kahler-Ricci flow on Hirzebruch manifolds are also discussed. It seems interesting to study the space of all Kahler metrics of $H>0$ on a given Kahler manifold. We give higher dimensional examples such that some Kahler classes admit Kahler metrics with $H>0$ and some do not.
-
on projectivized vector bundles and positive holomorphic Sectional Curvature
arXiv: Differential Geometry, 2016Co-Authors: Angelynn Alvarez, Gordon Heier, Fangyang ZhengAbstract:We generalize a construction of Hitchin to prove that, given any compact K\"ahler manifold $M$ with positive holomorphic Sectional Curvature and any holomorphic vector bundle $E$ over $M$, the projectivized vector bundle ${\mathbb P}(E)$ admits a K\"ahler metric with positive holomorphic Sectional Curvature.
R. K. Nagaich - One of the best experts on this subject based on the ideXlab platform.
-
Constancy of φ-Holomorphic Sectional Curvature of ( )-Sasakian Manifolds
2020Co-Authors: Rachna Rani, Rakesh Kumar, R. K. NagaichAbstract:The study of constancy of holomorphic Sectional Curvature of almost Hermitian manifolds was initiated byS. Tanno [8]. He provided an algebraic characterization for an almostHermitian manifold to reduce to a space of constant holomorphic Sectional Curvature, which he later extended for Sasakian manifolds as well. In the present paper authors considered the similar condition to characterize another class of almost contact indefinite manifolds known as ( )-Sasakian manifolds. 2000 Mathematics Subject Classification: 53C25, 53C50. DOI: 10.1134/S1995080209010107
-
Some characterization theorems on holomorphic Sectional Curvature of GCR-lightlike submanifolds
International Journal of Geometric Methods in Modern Physics, 2017Co-Authors: Varun Jain, Rachna Rani, Rakesh Kumar, R. K. NagaichAbstract:We obtain the expressions for Sectional Curvature, holomorphic Sectional Curvature and holomorphic biSectional Curvature of a GCR-lightlike submanifold of an indefinite Sasakian manifold and obtain...
-
Some characterization theorems on holomorphic Sectional Curvature of GCR-lightlike submanifolds
International Journal of Geometric Methods in Modern Physics, 2017Co-Authors: Varun Jain, Rachna Rani, Rakesh Kumar, R. K. NagaichAbstract:We obtain the expressions for Sectional Curvature, holomorphic Sectional Curvature and holomorphic biSectional Curvature of a GCR-lightlike submanifold of an indefinite Sasakian manifold and obtain some characterization theorems on holomorphic Sectional and holomorphic biSectional Curvature.
-
Constancy of ϕ -holomorphic Sectional Curvature of ( ε )-Sasakian manifolds
Lobachevskii Journal of Mathematics, 2009Co-Authors: Rachna Rani, Rakesh Kumar, R. K. NagaichAbstract:The study of constancy of holomorphic Sectional Curvature of almost Hermitian manifolds was initiated by S. Tanno [8]. He provided an algebraic characterization for an almost Hermitian manifold to reduce to a space of constant holomorphic Sectional Curvature, which he later extended for Sasakian manifolds as well. In the present paper authors considered the similar condition to characterize another class of almost contact indefinite manifolds known as (e)-Sasakian manifolds.
-
indefinite rk manifolds of constant holomorphic Sectional Curvature
2009Co-Authors: Rakesh Kumar, R. K. NagaichAbstract:The Riemannian Curvature tensor for an RK-manifolds with constant holomorphic Sectional Curvature has been derived and then discussed the Cartan’s Lemma for indefinite RK-manifolds. Mathematics Subject Classification: 53B35, 53C55
Gerard Walschap - One of the best experts on this subject based on the ideXlab platform.
-
twisted submersions in nonnegative Sectional Curvature
Archiv der Mathematik, 2013Co-Authors: Pablo Anguloardoy, Luis Guijarro, Gerard WalschapAbstract:In [16], Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative Sectional Curvature and proved that when the Curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We show that the same phenomenon often occurs for Riemannian submersions from nonnegatively curved spaces even without the strict positive Curvature assumption and irrespective of the particular metric.
-
twisted submersions in nonnegative Sectional Curvature
arXiv: Differential Geometry, 2012Co-Authors: Pablo Anguloardoy, Luis Guijarro, Gerard WalschapAbstract:B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative Sectional Curvature, and proved that when the Curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We show that the same phenomenon often occurs for nonnegatively curved Riemannian submersions even without the strict positive Curvature condition, and irrespective of the particular metric.
-
The dual foliation in open manifolds with nonnegative Sectional Curvature
Proceedings of the American Mathematical Society, 2007Co-Authors: Luis Guijarro, Gerard WalschapAbstract:We study several properties of the Sharafutdinov dual foliation in open manifolds with nonnegative Sectional Curvature.
Bun Wong - One of the best experts on this subject based on the ideXlab platform.
-
reduction of manifolds with semi negative holomorphic Sectional Curvature
Mathematische Annalen, 2018Co-Authors: Gordon Heier, Fangyang Zheng, Steven S Y Lu, Bun WongAbstract:In this note, we continue the investigation of a projective Kahler manifold M of semi-negative holomorphic Sectional Curvature H. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent truly flat directions of H in the tangent spaces. We prove that this invariant is bounded above by the nef dimension and bounded below by the numerical Kodaira dimension of M. We also prove a splitting theorem for M in terms of the nef dimension and, under some additional hypotheses, in terms of the new rank invariant.
-
k ahler manifolds of semi negative holomorphic Sectional Curvature
arXiv: Algebraic Geometry, 2014Co-Authors: Gordon Heier, Steven S Y Lu, Bun WongAbstract:In an earlier work, we investigated some consequences of the existence of a K\"ahler metric of negative holomorphic Sectional Curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic Sectional Curvature. In doing so, we define a new invariant that records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic Sectional Curvature vanishes. Using this invariant, we establish lower bounds for the nef dimension and, under certain additional assumptions, for the Kodaira dimension of the manifold. In dimension two, a precise structure theorem is obtained.
-
on the canonical line bundle and negative holomorphic Sectional Curvature
Mathematical Research Letters, 2010Co-Authors: Gordon Heier, Steven S Y Lu, Bun WongAbstract:We prove that a smooth complex projective threefold with a Kahler metric of negative holomorphic Sectional Curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, ampleness is again obtained. The methods used come from both complex differential geometry and complex algebraic geometry.
