**Speaker**: Behrouz Taji (Freiburg/Northwestern)

**Pre-Seminar: **Uniformization problems for higher dimensional complex projective varieties

The aim of this talk is to discuss how the theory of canonical metrics and its algebraic counterparts can be employed to address the uniformization problems for complex projective varieties. We will consider the case of vanishing first and second Chern classes in more depth, starting with a brief review of the solution to this problem in the smooth setting via Kähler-Einstein metrics.

**Seminar: **Stability and Chern class inequalities for log-minimal pairs

As a consequence of his solution to the Calabi's conjecture, Yau proved the the tangent sheaf of a canonically polarized manifold X verifies a remarkable Chern class inequality, the so-called Miyaoka-Yau inequality. In the case where equality (in the MY inequality) is attained, it was shown that X is a ball-quotient. When X is singular, the lack of control over the singularities of Kahler-Einstein metrics pose a difficulty in obtaining optimal Chern class inequalities and uniformization results. In a joint work with Greb, Kebekus and Paternell, we overcome this difficulty by resorting to Simpson's work on HYM theory and complex variation of Hodge structures and we extend Yau's result to the case of minimal varieties of general type. I will also briefly discuss my recent work with Guenancia, where we use recent developments in the theory of conical KE metrics to establish MY inequality for all minimal pairs. This result in particular provides an alternative path for proving the Abundance theorem for threefolds that is independent of generic semipositivity results for tangent sheaves and the related characteristic p-methods.

Host: Sandor Kovacs