Differential Geometry and Topology

course: Scalar-flat Kaehler metrics on compact complex surfaces

speaker: Michael Singer (University of Edinburgh)

abstract: shall start by recalling Calabi's notion of extremal metric and then turn to the special case of zero scalar curvature in complex dimension 2. The existence of a scalar-flat Kaehler (SFK) metric on a compact complex surface M imposes complex-geometric conditions on M: I will discuss these and give the simplest examples. Having set the scene, I shall discuss more exotic constructions of SFK metrics on M, culminating in my recent work with Yann Rollin. This work involves several ingredients, including:

(i) Elements of the geometry of ruled surfaces, including parabolic bundles;

(ii) Resolution of two-dimensional cyclic singularities (Hirzebruch-Jung strings);

(iii) Deformation theory of SFK metrics;

(iv) `Analytic resolution of singularities' (i.e. finding a SFK metric on the resolution, given a SFK metric on an orbifold).

It will not be possible to cover all these in detail: I will either make a random choice, or take suggestions from the audience.

timetable:
Mon 27 Sep, 17:00 - 18:15, Sala Conferenze Centro De Giorgi
Wed 29 Sep, 11:15 - 12:15, Sala Conferenze Centro De Giorgi
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