Grant: Discovery Grants Program - Individual

The proposal is at the intersection of two seemingly distinct mathematical domains: Riemannian geometry and the theory of complex projective varieties. The former goes back to the foundational ideas of B. Riemann in the 19th century, who proposed to study the shape of surfaces by allowing the euclidian structure to vary from point to point.  The latter arises in algebra, in the study of zeros of multivariate polynomials over the complex numbers. An emblematic example where these branches of mathematics intersect is the famous Poincaré uniformization theorem for 1-dimensional projective varieties,  which says that any compact complex curve is either the 2-sphere or is covered by the complex plane or the unit disc: this follows from the fact that each compact complex curve admits a Riemannian metric of constant scalar curvature. The extension of Poincaré's result to higher dimensions is known as the Calabi program of special metrics in Kähler Geometry. It has been at the forefront of most active research during the past 70 years or so, with landmark contributions by Yau (Harvard), Tian (Princeton and Beijing), and Donaldson (Imperial College) who set the directions that subsequently many followed and developed into a flourishing field. The central conjecture, called the Yau-Tian-Donaldson (YTD) conjecture, predicts that there exists a Kähler metric with constant scalar curvature in a given integer deRham class if and only if the corresponding polarized variety is stable in a suitable algebraic sense. During the past 15 years the subject witnessed a groundbreaking progress in both its algebraic and geometric aspects: The existence of Kähler metrics of constant scalar curvature is now solved on Fano and on toric varieties (of any dimension), by establishing sharp versions of the YTD conjecture in these cases. At the same time, other applications of the theory emerged and subjects such as the theory of special Riemannian metrics on pseudo-convex CR manifolds, known as Sasaki geometry, or special hermitian metrics on non-Kähler complex varieties became of great importance. Geometric structures coming from physics, for instance generalized Kähler geometry, have been put in a similar framework. All these recent events start to converge to a unified view of the field, in which the guiding thread is to obtain a correspondence between the existence of a special geometric structure at one hand, and an algebraic stability conditions at the other hand. The stability notion will have direct ramifications to questions about moduli (i.e. to study the set of all special geometric structures). We propose a program in which we study a number of such problems, including weighted Kähler geometry introduced by the applicant and his collaborators, as well as canonical generalized Kähler structures and its link with holomorphic Poisson geometry. We expect our program to open up new horizons and bridges between complex algebriac geometry and Riemannian geometry.