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Harish Seshadri

Publications and source records attributed to Harish Seshadri.

At least 19 recordsLinked to original sources
arXiv2026

Volume comparison and rigidity for positive holomorphic sectional curvature

We prove that the volume of a compact connected K\"ahler manifold with holomorphic sectional curvature at least 2 is bounded above by the volume of the Fubini-Study metric of constant holomorphic sectional curvature 2 on the complex projective space of the same dimension. Moreover equality holds if and only if the manifold is biholomorphically isometric to complex projective space. This answers a question posed by Xiong and Yang. Our approach also yields a different proof of Zhang's sharp volume estimate and Liu's rigidity theorem for compact K\"ahler manifolds with positive Ricci curvature. In fact, our main result states that the same sharp volume estimate holds under a new curvature positivity condition (mean RC curvature positivity), which is implied by both positive Ricci curvature and positive holomorphic sectional curvature. The definition of this condition was inspired by the work of Yang. The proofs in this paper are due to ChatGPT 5.6 Sol Pro, and the paper is merely an exposition of its output. The proofs has been verified by the authors and they take full responsibility for any errors.

math.DG↗
arXiv2026

The top Yau--Yang conjecture for K\"ahler manifolds with positive sectional curvature

We prove that the top wedge power of the Ricci form of a complete non-compact K\"ahler manifold with positive sectional curvature has finite integral. Using a result of Chen-Zhu, an immediate consequence is the quasiprojectivity of such manifolds under the assumption of bounded sectional curvature. A key new idea to prove B\'ezout estimates along with a Lipschitz weight with finite Monge-Amp\`ere mass is used in the proof of the main result.

math.DG↗
arXiv2026

Uniformisation of complete K\"ahler surfaces with positive sectional curvature

We prove that any complete non-compact K\"ahler surface with positive sectional curvature is biholomorphic to $\mathbb{C}^2$, establishing the two dimensional case of the weaker form of Yau's uniformisation conjecture. In contrast to all previous results, no assumptions are made on the geometry at infinity. The proof introduces a new approach towards Yau-type uniformisation problems, based on uniformly Lipschitz plurisubharmonic weight functions with finite Monge-Amp\`ere mass, and weighted $L^p$ holomorphic functions. A central difficulty is that these weights are neither smooth nor proper. As a consequence of the method, we also obtain B\'ezout-type intersection and multiplicity estimates in considerable generality. In a different direction, we also prove a new obstruction to the existence of complete K\"ahler metrics with non-negative bisectional curvature on non-compact K\"ahler manifolds, and use it to construct new examples admitting no such metrics. We conclude by discussing possible extensions of our methods to higher dimensions and related open problems.

math.DG↗
arXiv2025

The complex Monge-Ampere equation and an application to uniformisation of surfaces

We prove that a complete noncompact Kähler surface with positive and bounded sectional curvature is biholomorphic to $\mathbb{C}^2$. This result confirms a special case of Yau's conjecture that a complete noncompact Kähler $n$-manifold with positive holomorphic bisectional curvature is biholomorphic to $\mathbb{C}^n$. In contrast to all known results on Yau's conjecture, we do not need additional assumptions on the global/asymptotic geometry of the Kähler surface apart from completeness. Towards this end, we prove that the integral of the square of the Ricci form of a complete Kähler surface with positive sectional curvature is finite. The work of Chen and Zhu shows that this latter result implies that the surface is biholomorphic to $\mathbb{C}^2$ . The main new idea is the construction of a Lipschitz continuous plurisubharmonic weight function with finite Monge-Ampère mass. This weight function is obtained by solving a complex Monge-Ampère equation.

math.DG↗
arXiv2020

Metric rigidity of Kahler manifolds with lower Ricci bounds and almost maximal volume

In this short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results of Kewei Zhang and Yuchen Liu on holomorphic rigidity of such Kahler manifolds with the structure theorem of Tian-Wang for almost Einstein manifolds. This can be regarded as the complex analog of the result on Colding on the shape of Riemannian manifolds with almost maximal volume

math.DG↗
arXiv2018

Chern scalar curvature and symmetric products of compact Riemann surfaces

Let $X$ be a compact connected Riemann surface of genus $g\geq 0$, and let ${\rm Sym}^d(X)$, $d \ge 1$, denote the $d$-fold symmetric product of $X$. We show that ${\rm Sym}^d(X)$ admits a Hermitian metric with negative Chern scalar curvature if and only if $g \geq 2$, and positive Chern scalar curvature if and only if $d > g$.

math.DG↗
arXiv2018

A gap theorem for positive Einstein metrics on the four-sphere

We show that there exists a universal positive constant $\varepsilon_0 > 0$ with the following property: Let $g$ be a positive Einstein metric on $S^4$. If the Yamabe constant of the conformal class $[g]$ satisfies $$ Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon_0 $$ where $g_{\mathbb S}$ denotes the standard round metric on $S^4$, then, up to rescaling, $g$ is isometric to $g_{\mathbb S}$. This is an extension of Gursky's gap theorem for positive Einstein metrics on the four-sphere.

math.DG↗
arXiv2017

On domains biholomorphic to Teichmüller spaces

We prove that the Teichmüller space $\mathscr{T}$ of a closed surface of genus $g \ge 2$ cannot be biholomorphic to any domain which is locally strictly convex at some boundary point.

math.DG↗
arXiv2017

Quot schemes and Ricci semipositivity

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${\mathcal Q}_X(r,d)$ be the quot scheme that parametrizes all the torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. This ${\mathcal Q}_X(r,d)$ is also a moduli space of vortices on $X$. Its geometric properties have been extensively studied. Here we prove that the anticanonical line bundle of ${\mathcal Q}_X(r,d)$ is not nef. Equivalently, ${\mathcal Q}_X(r,d)$ does not admit any Kähler metric whose Ricci curvature is semipositive.

math.DG↗
arXiv2015

On the Kähler structures over Quot schemes, II

Let $X$ be a compact connected Riemann surface of genus $g$, with $g \geq 2$, and let ${\mathcal O}_X$ denote the sheaf of holomorphic functions on $X$. Fix positive integers $r$ and $d$ and let ${\mathcal Q}(r,d)$ be the Quot scheme parametrizing all torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. We prove that ${\mathcal Q}(r,d)$ does not admit a Kähler metric whose holomorphic bisectional curvatures are all nonnegative.

math.DG↗
arXiv2014

Positive isotropic curvature and self-duality in dimension 4

We study a positivity condition for the curvature of oriented Riemannian 4-manifolds: The half-$PIC$ condition. It is a slight weakening of the positive isotropic curvature ($PIC$) condition introduced by M. Micallef and J. Moore. We observe that the half-$PIC$ condition is preserved by the Ricci flow and satisfies a maximality property among all Ricci flow invariant positivity conditions on the curvature of oriented 4-manifolds. We also study some geometric and topological aspects of half-$PIC$ manifolds.

math.DG↗
arXiv2014

On the Kähler structures over Quot schemes

Let $S^n(X)$ be the $n$-fold symmetric product of a compact connected Riemann surface $X$ of genus $g$ and gonality $d$. We prove that $S^n(X)$ admits a Kähler structure such that all the holomorphic bisectional curvatures are nonpositive if and only if $n < d$. Let ${\mathcal Q}_X(r,n)$ be the Quot scheme parametrizing the torsion quotients of ${\mathcal O}^{\oplus r}_X$ of degree $n$. If $g \geq 2$ and $n \leq 2g-2$, we prove that ${\mathcal Q}_X(r,n)$ does not admit a Kähler structure such that all the holomorphic bisectional curvatures are nonnegative.

math.DG↗
arXiv2013

On the Gromov hyperbolicity of convex domains in Cn

We give a necessary complex geometric condition for a bounded smooth convex domain in Cn, endowed with the Kobayashi distance, to be Gromov hyperbolic. More precisely, we prove that if a smooth bounded convex domain contains an analytic disk in its boundary, then the domain is not Gromov hyperbolic for the Kobayashi distance. We also provide examples of bounded smooth convex domains that are not strongly pseudoconvex but are Gromov hyperbolic.

math.CV↗
arXiv2013

Noncoercive Ricci flow invariant curvature cones

This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions greater than 4, if a Ricci flow invariant condition is weaker than "Einstein with nonnegative scalar curvature", then this condition has to be "nonnegative scalar curvature". As a corollary, we obtain that a Ricci flow invariant curvature condition which is stronger than "nonnegative scalar curvature" cannot be (strictly) satisfied by compact Einstein symmetric spaces such as S^2xS^2 or CP^2. We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.

math.DG↗
arXiv2012

A gap theorem for Ricci-flat 4-manifolds

Let $(M,g)$ be a compact Ricci-flat 4-manifold. For $p \in M$ let $K_{max}(p)$ (respectively $K_{min}(p)$) denote the maximum (respectively the minimum) of sectional curvatures at $p$. We prove that if $$K_{max} (p) \le \ -c K_{min}(p)$$ for all $p \in M$, for some constant $c$ with $0 \leq c < \frac{2+\sqrt 6}{4}$, then $(M,g)$ is flat. We prove a similar result for compact Ricci-flat Kähler surfaces. Let $(M,g)$ be such a surface and for $p \in M$ let $H_{max}(p)$ (respectively $H_{min}(p)$) denote the maximum (respectively the minimum) of holomorphic sectional curvatures at $p$. If $$H_{max} (p) \le -c H_{min}(p)$$ for all $p \in M$, for some constant $c$ with $0 \leq c < \frac {1+\sqrt 3}{2}$, then $(M,g)$ is flat.

math.DG↗
arXiv2012

Totally geodesic discs in strongly convex domains

We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let $n_1, n_2$ be positive integers and let $Ω_i \subset \C^{n_i}, \ i=1,2$, be bounded $C^3$ strongly convex domains. If $ϕ: (Ω_1, d^K_{Ω_1}) \rightarrow (Ω_2, d^K_{Ω_2})$ is an isometry, i.e. $ d^K_Ω_{n_2}(f(ζ),f(η)) = d^K_{n_1} (ζ,η)$ for all $ζ,η\in Ω_1,$ then $ϕ$ is either holomorphic or anti-holomorphic.

math.CV↗