Ved Datar

Research

Research description

My work lies in geometric analysis, especially in complex differential geometry and Kähler geometry. I am broadly interested in the relation between curvature, canonical metrics, and nonlinear complex PDE, and in how these impose analytic, topological, and algebraic constraints on complex manifolds.

More specifically, I work on problems involving Kähler–Einstein and related canonical metrics, complex Monge–Ampère and Hessian equations, comparison and rigidity questions for positively curved Kähler manifolds, and geometric questions arising from degenerations, collapse, and gauge-theoretic phenomena.

Publications and preprints

  1. Uniformisation of complete Kähler surfaces with positive sectional curvature (with Vamsi Pingali and Harish Seshadri) arXiv preprint, 2026. arXiv Summary: Techniques from the previous paper are used to prove that a complete Kähler surface with positive sectional curvature is biholomorphic to C2, thereby establishing the surface and sectional curvature version of Yau’s uniformisation conjecture.
  2. The complex Monge–Ampère equation and an application to uniformisation of surfaces (with Vamsi Pingali and Harish Seshadri) arXiv preprint, 2025. arXiv Summary: Uses a complex Monge–Ampère approach to a uniformisation problem for complete noncompact Kähler surfaces.
  3. Minimal slopes and bubbling for complex Hessian equations (with Ramesh Mete and Jian Song) Advances in Mathematics 491 (2026), 110865. arXiv · journal Summary: Develops canonical singular solutions and bubbling phenomena for complex Hessian equations in unstable regimes.
  4. Diameter rigidity for Kähler manifolds with positive bisectional curvature (with Harish Seshadri) Mathematische Annalen 385 (2023), 471–479. arXiv · journal Summary: A rigidity theorem characterising the maximal-diameter case under positivity of bisectional curvature.
  5. Kähler–Einstein metric near an isolated log canonical singularity (with Xin Fu and Jian Song) Journal für die reine und angewandte Mathematik (Crelle’s Journal) 797 (2023), 79–116. arXiv · journal Summary: Studies complete negative Kähler–Einstein metrics near isolated log canonical singularities.
  6. Metric rigidity of Kähler manifolds with lower Ricci bounds and almost maximal volume (with Harish Seshadri and Jian Song) Proceedings of the American Mathematical Society 149 (2021), no. 8, 3569–3574. arXiv · journal Summary: An almost-rigidity result for Kähler manifolds under Ricci lower bounds and near-maximal volume.
  7. A numerical criterion for generalised Monge–Ampère equations on projective manifolds (with Vamsi Pingali) Geometric and Functional Analysis 31 (2021), no. 4, 767–814. arXiv · journal Summary: Gives a numerical solvability criterion for a class of inverse Hessian and generalized Monge–Ampère equations.
  8. On coupled constant scalar curvature Kähler metrics (with Vamsi Pingali) Journal of Symplectic Geometry 18 (2020), no. 4, 961–994. arXiv · journal Summary: Introduces and studies coupled cscK equations and their moment-map interpretation.
  9. Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces (with Adam Jacob and Yuguang Zhang) Journal of Differential Geometry 118 (2021), no. 2, 223–296. arXiv · journal Summary: Studies Yang–Mills connections in the collapsing limit of elliptically fibered K3 surfaces.
  10. Hermitian–Yang–Mills connections on collapsing elliptically fibered K3 surfaces (with Adam Jacob) The Journal of Geometric Analysis 32 (2022), article no. 69. arXiv · journal Summary: Analyzes Hermitian–Yang–Mills connections for collapsing Ricci-flat metrics on K3 surfaces.
  11. Expansions of solutions to extremal metric type equations on blow-ups of cscK surfaces Annals of Global Analysis and Geometry 55 (2019), no. 2, 215–241. arXiv · journal Summary: Obtains asymptotic expansions relevant to extremal metric type equations on blow-ups of cscK surfaces.
  12. Kähler–Einstein metrics along the smooth continuity method (with Gábor Székelyhidi) Geometric and Functional Analysis 26 (2016), no. 4, 975–1010. arXiv · journal Summary: Establishes Kähler–Einstein existence results through the smooth continuity method and equivariant K-stability.
  13. On convexity of the regular set of conical Kähler–Einstein metrics Mathematical Research Letters 23 (2016), no. 1, 105–126. arXiv · journal Summary: Shows convexity properties of the regular set for conical Kähler–Einstein metrics.
  14. A remark on Kähler metrics with conical singularities along a simple normal crossing divisor (with Jian Song) Bulletin of the London Mathematical Society 47 (2015), no. 6, 1010–1013. arXiv · journal Summary: A short note on conical Kähler metrics in the simple normal crossing setting.
  15. Connecting toric manifolds by conical Kähler–Einstein metrics (with Bin Guo, Jian Song and Xiaowei Wang) Advances in Mathematics 323 (2018), 38–83. arXiv · journal Summary: Studies conical Kähler–Einstein metrics and continuous paths between toric manifolds.

Surveys and expository articles

  • Comparison and Rigidity Theorems for the Volume and Diameter of Positively Curved Kähler Manifolds with Harish Seshadri · Acta Mathematica Sinica, English Series (published online 13 December 2024) journal
  • An overview of the work of Karen Uhlenbeck with Agnid Banerjee · The Mathematics Consortium Bulletin, Vol. 2, Issue 1 (July 2020) journal

Thesis

Public talks and lecture videos

Slides and notes