Tata Institute of Fundamental Research 

I am a Reader at the School of Mathematics, TIFR, Mumbai. My research interest is in algebraic geometry. More specifically, I am interested in the birational/bimeromorphic classification of varieties through the Minimal Model Program (MMP) or Mori Program and its applications to the Moduli Spaces of higher dimensional varieties.  

During my PhD and Postdoc years, I worked on problems related to the Minimal Model Program for projective varieties defined over the fields of prime characteristics. More recently, I have focused my effort on developing the Minimal Model Program for compact Kähler manifolds.  There are several technical challenges here, the two basic ones are the following:

Question 1: If the canonical bundle K_X of a compact Kähler manifold X is not nef, then does it contain a (K_X-negative) rational curve?

Question 2: If K_X is not nef, does there exist a contraction of a K_X-negative extremal ray of the Mori cone of curves? 

When X is a projective variety defined over the complex field, Question 1 is answered positively by Mori's famous Bend-Break technique, which involves reducing the variety to prime characteristics. This method is clearly not available for non-algebraic varieties because they are not defined by polynomial equations. 

Question 2 also has a positive answer for projective varieties over the complex field due to a fundamental theorem of Kawamata and Shokurov, called the  Base-Point Free Theorem (BFT). However, the BFT is not available for compact Kähler manifolds in general, indeed, if a compact Kähler manifold satisfies the hypothesis of the BFT, then it poses a big line bundle, and hence X is a Moishezon manifold. Then an old result of Moishezon says that X must be a projective manifold.  (An analogous result for singular compact Kähler space with rational singularities also holds due to Namikawa.)

Observe that, Question 2 is at the crux of the Minimal Model Program, as MMP aims to improve the positivity of the canonical bundle K_X by contracting the K_X-negative curves.  Moreover, notice that a positive answer to Question 2 gives a positive answer to Question 1 via Kawamata's clever use of the Bend-Break technique on the fibers of K_X-negative extremal contractions. For analytic varieties, Kawamata's proof is verified in one of our papers (jointly with Wenhao Ou), see [Theorem 1.23  DO24]

In conclusion, we say that the current trend of Kähler MMP revolves around finding an answer to Question 2. (Of course, there are many other important questions we don't know the answer to, but for now, this seems to be the most pressing one in higher dimensions ≥4.)

Here is a link to my CV.