Research

My research work deals with the study of the different notions of canonical Kähler metrics like Calabi's extremal Kähler metrics, cscK metrics and Kähler-Einstein metrics, and most importantly certain higher dimensional generalizations of these notions introduced recently by Pingali which are called as higher extremal Kähler metrics and higher cscK metrics respectively. These metrics often appear as the critical points or minimizers of some real-valued energy functionals defined on their respective Kähler classes (this is very much known to be the case for the three well-studied notions of canonical Kähler metrics), and hence they (or their associated Kähler forms) are precisely the solutions to some nice geometric PDEs on the underlying complex manifold.

In my work I deal with the problem of finding examples of these special metrics on some nice kinds of compact complex manifolds (viz. ruled complex surfaces like pseudo-Hirzebruch and Hirzebruch surfaces), and then asking questions like whether all Kähler classes contain these metrics, exactly which Kähler classes contain them and are these metrics unique in their Kähler classes in some way. For constructing explicit examples of these kinds of metrics in the setting known as Calabi symmetry, the momentum construction method of Hwang-Singer is applied, which by inducing a lot of symmetries on the metric to be constructed, boils down the problem of finding the metric in a given Kähler class to solving an ODE BVP on an interval of the real line, i.e. the PDE on the manifold to which the metric is a solution boils down to an ODE whose solution uniquely determines the resultant metric and is called as the momentum profile of the metric.

My work also deals with the study of some algebro-geometric obstructions to the existence of solutions to these PDEs on the manifold, i.e. to the existence of these metrics in a given Kähler class, and these obstructions are known as Futaki invariants in the case of cscK metrics and Bando-Futaki invariants in the case of higher cscK metrics.

Finally I have dealt with the construction of these metrics in the smooth setup as well as in the setup where the metrics have got conical singularities along some nice divisors of the manifold under consideration (viz. the zero and infinity divisors of the ruled surfaces). In the conical setup the question of global interpretation of these PDEs on the manifold is tackled with by using the theory of currents.

The conical analogues of the above mentioned algebro-geometric obstructions, which are called as log Futaki invariants in the case of conical cscK metrics and log Bando-Futaki invariants in the case of conical higher cscK metrics, are also discussed in my work.

List of my Research Publications and Preprints:

•  Existence of Higher Extremal Kähler Metrics on a Minimal Ruled Surface

 Journal: Bulletin des Sciences Mathématiques, Volume: 189 (December, 2023), Article No.: 103345

 Journal DOI: https://doi.org/10.1016/j.bulsci.2023.103345

 arXiv Preprint, Date: September 21, 2023, arXiv Identifier: arXiv:2302.07127

 arXiv DOI: https://doi.org/10.48550/arXiv.2302.07127

•  Existence of Conical Higher cscK Metrics on a Minimal Ruled Surface

 arXiv Preprint, Date: May 25, 2025, arXiv Identifier: arXiv:2505.19257

 arXiv DOI: https://doi.org/10.48550/arXiv.2505.19257