Talk page
Title:
Kodaira Dimension and the Yamabe Problem, Revisited
Speaker:
Abstract:
Dimension four provides a surprisingly idiosyncratic setting for the interplay between scalar curvature and differential topology. This peculiarity becomes especially pronounced when discussing the Yamabe invariant (or “sigma constant”) of a smooth compact manifold; and Seiberg-Witten theory makes this especially apparent for those 4-manifolds that arise as compact complex surfaces. For compact complex surfaces of Kaehler type, I showed in the late 1990s that the sign of the Yamabe invariant is always determined by the Kodaira dimension, and moreover calculated the Yamabe invariant exactly in all cases where it is non-positive. In this talk, I will describe recent joint work with Michael Albanese that generalizes these results to all complex surfaces of non-Kaehler type. However, the complex surfaces of class VII actually violate many of the expected patterns, and navigating around this hazard represents a key aspect of our story.
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