Complex Surfaces with Cat(0) Metrics

Abstract

We study complex surfaces with locally CAT(0) polyhedral Kähler metrics and construct such metrics on CP 2 with various orbifold structures. In particular, in relation to questions of Gromov and DavisMoussong we construct such metrics on a compact quotient of the two-dimensional unit complex ball. In the course of the proof of these results we give criteria for Sasakian 3-manifolds to be globally CAT(1). We show further that for certain Kummer coverings of CP 2 of sufficiently high degree their desingularizations are of type K(π, 1).

On rigid compact complex surfaces and manifolds

“…Panov [Pan11] asserts (without giving full details, hence without specifying explicitly the meaning of n >> 0) a positive answer to II) for the surfaces HK CQ (n) and other examples by Hirzebruch: his method consists in finding polyhedral metrics of negative curvature. So the following question is not yet settled:…”

Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

“…In this case, the existence of the desired metric is only known for the case where 5 divides n: this was done by Fangyang Zheng [Zhe99], extending a technique introduced by Mostow and Siu [MS80] for ramified coverings of ball quotients. In general, Panov [Pan11] showed the existence of a non smooth negative metric (a polyhedral metric) for n > n 0 , but where n 0 is unspecified.…”

Kodaira fibrations and beyond: methods for moduli theory

“…These surfaces are rigid, in particular they are not the Kodaira examples of Kodaira fibrations. (8) More examples are gotten from coverings of the plane branchedover configurations of lines, as we shall discuss in a later section (see[Zheng99] , and[Pan09],[Pan11] ).…”