arXiv subjects

Reza Seyyedali

Publications and source records attributed to Reza Seyyedali.

13 recordsLinked to original sources

arXiv2026

Boundary Estimates for the Monge-Ampère Equation in the Polygons with Guillemin Boundary Conditions

We establish a Schauder-type boundary regularity result for a two-dimensional singular Monge--Ampère equation on convex polytopes subject to the Guillemin boundary condition. Our result extends the work of Rubin and Huang to the case where the right-hand side is merely Hölder continuous. In particular, we obtain Euclidean $C^{1,α/2}$ regularity up to the boundary, including along edges and at the vertices of the polytope, and then refine this to the sharp Euclidean $C^{1,α}$ regularity. The analysis combines techniques introduced by Donaldson in his study of the Abreu equation with refined blow-up and localization arguments adapted to the degenerate boundary geometry.

math.AP↗

arXiv2023

On compactness of Kähler metrics with bounded entropy and bounded $L^{2n+1}$ scalar curvature

In their seminal work (\cite{CC}, \cite{CC2}), Chen and Cheng proved apriori estimates for the constant scalar curvature metrics on compact Kähler manifolds. They also proved $C^{3,α}$ estimate for the potential of the \ka metrics under boundedness assumption on the scalar curvature and the entropy. The goal of this short note is to slightly relax the boundedness condition on the scalar curvature.

math.DG↗

arXiv2023

Remarks on a result of Chen-Cheng

In their seminal work (\cite{CC}, \cite{CC2}), Chen and Cheng proved apriori estimates for the constant scalar curvature metrics on compact Kähler manifolds. They also prove $C^{3,α}$-estimate for the potential of the Kähler metrics under boundedness assumption on the scalar curvature and the entropy. The goal of this paper is to replace the uniform boundedness of the scalar curvature to the $L^p$-boundedness of the scalar curvature.

math.DG↗

arXiv2017

Relative Chow stability and extremal metrics

We prove that the existence of extremal metrics implies asymptotically relative Chow stability. An application of this is the uniqueness, up to automorphisms, of extremal metrics in any polarization.

math.DG↗

arXiv2016

Extremal metrics on blowups along submanifolds

We give conditions under which the blowup of an extremal Kähler manifold along a submanifold of codimension greater than two admits an extremal metric. This generalizes work of Arezzo-Pacard-Singer, who considered blowups in points.

math.DG↗

arXiv2015

Quantization of the Laplacian operator on vector bundles I

Let $(E,h)$ be a holomorphic Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of $E$. If $E$ is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian.

math.DG↗

arXiv2014

A Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic bundles

In this paper, we introduce the notions of $α$-Hermitian-Einstein metric and $α$-stability for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi-Hitchin correspondence for $I_\pm$-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include generalized holomorphic bundles on generalized Kähler manifolds. We also show that $α$-stability of a vector bundle, in this sense, can depend on the parameter $α$.

math.DG↗

arXiv2014

Quantization of Donaldson's heat flow over projective manifolds

Consider $E$ a holomorphic vector bundle over a projective manifold $X$ polarized by an ample line bundle $L$. Fix $k$ large enough, the holomorphic sections $H^0(E\otimes L^k)$ provide embeddings of $X$ in a Grassmanian space. We define the \textit{balancing flow for bundles} as a flow on the space of projectively equivalent embeddings of $X$. This flow can be seen as a flow of algebraic type hermitian metrics on $E$. At the quantum limit $k\to \infty$, we prove the convergence of the balancing flow towards the Donaldson heat flow, up to a conformal change. As a by-product, we obtain a numerical scheme to approximate the Yang-Mills flow in that context.

math.DG↗

arXiv2013

Extremal Metrics On Ruled Manifolds

In this paper, we consider a compact Kahler manifold with extremal Kahler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kahler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kahler metrics on the projectivization of the dual vector bundle.

math.DG↗

arXiv2011

Balanced Metrics and Chow Stability of Projective Bundles over Kähler Manifolds II

In the previous article (\cite{S}), we proved that slope stability of a holomorphic vector bundle $E$ over a polarized manifold $(X,L)$ implies Chow stability of $(\mathbb{P}E^*,\mathcal{O}_{\mathbb{P}E^*}(1)\otimes π^* L^k)$ for $k \gg 0$ if the base manifold has no nontrivial holomorphic vector field and admits a constant scalar curvature metric in the class of $2πc_{1}(L)$. In this article using asymptotic expansions of Bergman kernel on $\textrm{Sym}^d E$, we generalize the main theorem of \cite{S} to polarizations $(\mathbb{P}E^*,\mathcal{O}_{\mathbb{P}E^*}(d)\otimes π^* L^k)$ for $k \gg 0$, where $d$ is a positive integer.

math.DG↗

arXiv2010

Balanced metrics and chow stability of projective bundles over Riemann surfaces

In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. We generalized Morrison's result to higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group. In this article, we give a simple proof for polarizations $\mathcal{O}_{\mathbb{P}E^*}(d)\otimes π^* L^k$, where $d$ is a positive integer, $k \gg 0$ and the base manifold is a compact Riemann surface of genus $g \geq 2$.

math.DG↗

arXiv2009

Balanced Metrics and Chow Stability of Projective Bundles over Kähler Manifolds

In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using the notion of balanced metrics and recent work of Donaldson, Wang, and Phong-Sturm, we show that the statement holds for higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group.

math.DG↗

arXiv2008

Numerical Algorithms for Finding Balanced Metrics on Vector Bundles

In \cite{D3}, Donaldson defines a dynamical system on the space of Fubini-Study metrics on a polarized compact Kähler manifold. Sano proved that if there exists a balanced metric for the polarization, then this dynamical system always converges to the balanced metric (\cite{S}). In \cite{DKLR}, Douglas, et. al., conjecture that the same holds in the case of vector bundles. In this paper, we give an affirmative answer to their conjecture.

math.DG↗