The uniformization problem for complex projective varieties and a singular analogue of the Donaldson-Uhlenbeck-Yau theorem


Shin-Yi Lu

15:45:00 - 17:00:00
103 , Mathematics Research Center Building (ori. New Math. Bldg.)

The uniformization problem even for that of a compact Riemann Surface is one of the most classical and difficult problem in the history of mathematics and was vital in the modern rigorous development of pure and applied mathematics. In the seventies, S.T. Yau obtained using his solution to the Calabi conjecture on the existence of Kahler-Einstein metrics that an n-dimensional compact Kahler manifold $X$ with positive (respectively trivial) canonical class and vanishing of the integral of $(n c_1^2 - 2(n+1) c_2)w^{n-2}$, where $c_1$ and $c_2$ are the Chern classes and $w$ the Kahler class of $X$, is uniformized by the complex n-dimensional ball with the standard hyperbolic metric (respectively by the n-dimensional complex vector space with the standard Euclidean metric). By the same token, one also has the same statement for uniformization by the n-dimensinoalcomplex projective space in the case of positive anti-canonical class given the existence of a Kahler-Einstein metric. All of these generalize directly from the original solutions to the case of compact Kahler orbifolds (i.e. Kahler varieties with at worst quotient singularities or V-manifolds) if one replaces the Chern classes by their orbifold counterpart, and hence in particular to the case of projective surfaces with at worst klt singularities, a natural class of surfaces of interest in birational geometry.

In 1994, a remarkable paper of Shepherd-Barron and Wilson shows that complex projective threefolds with at most canonical singularities with numerically trivial first and second orbifold-Chern classes (properly defined) are uniformized by Abelian threefolds giving the first instance of a uniformization result in higher dimensions that goes beyond quotient singularities. In this series of talks, I will describe from this approach at least a couple of ways to obtain the same result for varieties with at most klt-singularities and in any dimension. The first is from the known polystability result of the tangent bundle for such varieties and the second via the semistability of tangent bundle of such a variety (from the celebrated result of Y. Miyaoka on the generic semi-positivity of the cotangent sheaf of a non-uniruled variety). The first method requires a generalization of the classical theorem of Donaldson-Uhlenbeck-Yau to the orbifold setting while the second that of Simpson's correspondance between semi-stable reflexive sheaves and holomorphically flat vector bundles. Both generalize to the ball-quotient case.

(1) The joint work with Behrouz Taji, preprint: arXiv:1410.0063
(2) An important precursor to this work in the non-orbifold setting is given by the preprint arXiv:1307.5718 of Greb-Kebekus-Peternell whose relevant results we will review.