Given a compact Kähler manifold it is a classical question whether it admits a Kähler metric of constant scalar curvature (cscK metric for short). In this talk we prove that there always exist cscK metrics on compact Kähler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song and extends the seminal result of Aubin and Yau, giving a large class of new examples of cscK manifolds. As a byproduct we show that the connected component of the automorphism group of any compact Kähler manifold with nef canonical bundle is either trivial or a complex torus.

The tools used are from the variational approach in Kähler geometry, and some related results on Donaldson's J-equation are discussed along the way.

Based on arXiv:1907.01486 and arXiv:2004.02832.

Via Zoom, e-mail Cristiano Spotti (c.spotti@math.au.dk) for log-in details.