An L2-Poincaré–Dolbeault lemma of spaces with mixed cone-cusp singular metrics

The existence of Kähler Einstein metrics with mixed cone and cusp singularity has received considerable attentions in recent years. It is believed that such kind of metric would give rise to important geometric invariants. We computed their [Formula: see text]-Hodge–Frölicher spectral sequence under the Dirichlet and Neumann boundary conditions and examine the pure Hodge structures on them. It turns out that these cohomologies agree well with the de Rham cohomology of a good compactification.

Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

Kähler–Einstein metrics on symmetric general arrangement varieties

We calculate Chow quotients of some families of symmetric T-varieties. In complexity two we obtain new examples of Kähler–Einstein metrics by bounding the symmetric alpha invariant of their orbifold quotients. As an additional application we determine the homeomorphism class of the orbit space of the compact torus action.