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Vamsi Pritham Pingali

Publications and source records attributed to Vamsi Pritham Pingali.

At least 19 recordsLinked to original sources
arXiv2026

Uniqueness for the Kähler-Yang-Mills equations

A formula for the $α$-K-energy functional for the Kähler-Yang-Mills (KYM) equations is provided in this paper. Using this formula and Chen's $ε$-geodesic equation on the space of Kähler potentials, we prove that if a solution exists to the KYM equations for a simple vector bundle on a Kähler manifold with discrete automorphism group, then it is unique and the $α$-K-energy is bounded from below. Inspired by the study of constant scalar curvature Kähler metrics, we introduce a coupled J-equation to study the $α$-K energy functional.

math.DG↗
arXiv2026

Uniformisation of complete Kähler surfaces with positive sectional curvature

We prove that any complete non-compact Kähler 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-Ampère 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 Bézout-type intersection and multiplicity estimates in considerable generality. In a different direction, we also prove a new obstruction to the existence of complete Kähler metrics with non-negative bisectional curvature on non-compact Kähler 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↗
arXiv2024

Non-abelian symmetric critical gravitating vortices on a sphere

We produce examples of solutions to the non-abelian gravitating vortex equations, which are a dimensional reduction of the Käher-Yang-Mills- Higgs equations. These are equations for a Kähler metric and a metric on a vector bundle. We consider a symmetric situation on a sphere with a relationship between the parameters involved (criticality), and perform a non-trivial reduction of the problem to a system of ordinary differential equations on the real line with complicated boundary conditions at infinity. This system involves a parameter whose dependence on the volume of the Kähler metric is non-explicit. We prove existence to this system using the method of continuity. We then prove that the parameter can be varied to make sure that all possible admissible volumes are attained.

math.DG↗
arXiv2023

Criteria for the ampleness of certain vector bundles

We prove that certain vector bundles over surfaces are ample if they are so when restricted to divisors, certain numerical criteria hold, and they are semistable (with respect to $\det(E)$). This result is a higher-rank version of a theorem of Schneider and Tancredi for vector bundles of rank two over surfaces. We also provide counterexamples indicating that our theorem is sharp.

math.AG↗
arXiv2022

A vector bundle version of the Monge-Ampere equation

We introduce a vector bundle version of the complex Monge-Ampere equation motivated by a desire to study stability conditions involving higher Chern forms. We then restrict ourselves to complex surfaces, provide a moment map interpretation of it, and define a positivity condition (MA positivity) which is necessary for the infinite-dimensional symplectic form to be Kahler. On rank-2 bundles on compact complex surfaces, we prove two consequences of the existence of a "positively curved" solution to this equation - Stability (involving the second Chern character) and a Kobayashi-Lubke-Bogomolov-Miyaoka-Yau type inequality. Finally, we prove a Kobayashi-Hitchin correspondence for a dimensional reduction of the aforementioned equation.

math.DG↗
arXiv2022

The deformed Hermitian Yang-Mills equation on three-folds

We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $\hatθ \in (\fracπ{2},\frac{3π}{2})$, on compact complex three-folds conditioned on a necessary subsolution condition. Our proof hinges on a delicate analysis of a new continuity path obtained by rewriting the equation as a generalised Monge-Ampère equation with mixed sign coefficients.

math.DG↗
arXiv2021

A numerical criterion for generalised Monge-Ampere equations on projective manifolds

We prove that generalised Monge-Ampère equations (a family of equations which includes the inverse Hessian equations like the $J$-equation, as well as the Monge-Ampère equation) on projective manifolds have smooth solutions if certain intersection numbers are positive. As corollaries of our work, we improve a result of Chen (albeit in the projective case) on the existence of solutions to the $J$-equation, and prove a conjecture of Székelyhidi in the projective case on the solvability of certain inverse Hessian equations. The key new ingredient in improving Chen's result is a degenerate concentration of mass result. We also prove an equivariant version of our results, albeit under the assumption of uniform positivity. In particular, we can recover existing results on manifolds with large symmetry such as projective toric manifolds.

math.DG↗
arXiv2021

A note on Demailly's approach towards a conjecture of Griffiths

We prove that a "cushioned" Hermitian-Einstein-type equation proposed by Demailly in an approach towards a conjecture of Griffiths on the existence of a Griffiths positively curved metric on a Hartshorne ample vector bundle, has an essentially unique solution when the bundle is stable. This result indicates that the proposed approach must be modified in order to attack the aforementioned conjecture of Griffiths.

math.DG↗
arXiv2021

Gravitating vortices with positive curvature

We give a complete solution to the existence problem for gravitating vortices with non-negative topological constant $c \geqslant 0$. Our first main result builds on previous results by Yang and establishes the existence of solutions to the Einstein-Bogomol'nyi equations, corresponding to $c=0$, in all admissible Kähler classes. Our second main result completely solves the existence problem for $c>0$. Both results are proved by the continuity method and require that a GIT stability condition for an effective divisor on the Riemann sphere is satisfied. For the former, the continuity path starts from a given solution with $c = 0$ and deforms the Kähler class. For the latter result we start from the established solution in any fixed admissible Kähler class and deform the coupling constant $α$ towards $0$. A salient feature of our argument is a new bound $S_g \geqslant c$ for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.

math.DG↗
arXiv2019

On coupled constant scalar curvature Kähler metrics

We provide a moment map interpretation for the coupled Kähler-Einstein equations introduced by Hultgren and Witt Nyström, and in the process introduce a more general system of equations, which we call coupled cscK equations. A differentio-geometric formulation of the corresponding Futaki invariant is obtained and a notion of K-polystability is defined for this new system. Finally, motivated by a result of Székelyhidi, we prove that if there is a solution to our equations, then small K-polystable perturbations of the underlying complex structure and polarizations also admit coupled cscK metrics.

math.DG↗
arXiv2018

Metric properties of parabolic ample bundles

We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favour for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type result holds, and that there are metrics on stable bundles over surfaces whose Schur forms are positive.

math.DG↗
arXiv2018

Gravitating vortices and the Einstein--Bogomol'nyi equations

In this work we consider the gravitating vortex equations. These equations couple a metric over a compact Riemann surface with a hermitian metric over a holomorphic line bundle equipped with a fixed global section --- the Higgs field ---, and have a symplectic interpretation as moment-map equations. As a particular case of the gravitating vortex equations on $\mathbb{P}^1$, we find the Einstein--Bogomol'nyi equations, previously studied in the theory of cosmic strings in physics. We prove two main results in this paper. Our first main result gives a converse to an existence theorem of Y. Yang for the Einstein--Bogomol'nyi equations, establishing in this way a correspondence with Geometric Invariant Theory for these equations. In particular, we prove a conjecture by Y. Yang about the non-existence of cosmic strings on $\mathbb{P}^1$ superimposed at a single point. Our second main result is an existence and uniqueness result for the gravitating vortex equations in genus greater than one.

math.DG↗
arXiv2018

A characterization of finite vector bundles on Gauduchon astheno-Kahler manifolds

A vector bundle E on a projective variety X is called finite if it satisfies a nontrivial polynomial equation with integral coefficients. A theorem of Nori implies that E is finite if and only if the pullback of E to some finite etale Galois covering of X is trivial. We prove the same statement when X is a compact complex manifold admitting a Gauduchon astheno-Kahler metric.

math.AG↗
arXiv2017

Quillen metrics and perturbed equations

We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the generalised Monge-Amp`ere equation is conditioned on a conjecture from algebraic geometry. In addition, we prove that for small values of the perturbation parameters, some of these equations have solutions.

math-ph↗