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Gauge Theory and Complex Geometry (1582)
Théorie de Jauge et géométrie complexe

Dates : 29 June – 10 July 2015 at CIRM (Marseille Luminy, France)  

The first topic of this Research in Pairs related to Teleman’s approach to prove the existence of curves on class VII surfaces,  using Donaldson theory (see [Te1]-[Te3]) and the Kobayashi-Hitchin correspondence (which identifies moduli spaces of instantons with moduli spaces of stable bundles). The collaboration of the four participants in this project took into account the newest developments on the classification of non-Kählerian surfaces obtained using this approach, and also considered new related problems, for instance: comparing the holomorphic deformations of a given class VII surface with the the holomorphic deformations of the associated moduli space, describing explicitly moduli stacks of class VII surfaces.

The second topic was concerned with moduli theory for holomorphic bundles on higher dimensional compact complex manifolds with emphasis on compactification problems. The starting point was the article [GT], in which the authors construct a modular compactification of the moduli space of vector bundles which are slope-stable with respect to an ample divisor, which generalizes the algebro-geometric construction of the Donaldson-Uhlenbeck compactification for complex surfaces.

We had in mind and discussed interesting interactions between the two topics, for instance: Can one extend Greb-Toma’s results to Kählerian non-algebraic and non-Kählerian manifolds? Can one use such compactified moduli spaces to prove the existence of proper analytic cycles on certain higher dimensional non-algebraic manifolds?