On the geography of complex surfaces of general type with an arbitrary fundamental group

Abstract: Surfaces of general type are lovely unclassifiable objects in algebraic geometry. Geography refers to the problem of construction of such surfaces for a given set of invariants, classically the Chern numbers \(c_1^2\) (self-intersection of canonical class) and \(c_2\) (topological Euler characteristic). In this talk, we treat the question: What can be said about the distribution of Chern slopes \(c_1^2/c_2\) of surfaces of general type when we fix the fundamental group? It turns out that there are various well-known constraints, which will be pointed out during the talk, but at least we can prove the following theorem (joint with Sergio Troncoso): "Let \(G\) be the fundamental group of some nonsingular complex projective variety. Then Chern slopes of surfaces of general type with fundamental group isomorphic to \(G\) are dense in the interval \([1,3]\).". Remember that for complex surfaces of general type we have that \(c_1^2/c_2\) is a rational number in \([1/5,3]\), and so most open questions now refer to slopes in \([1/5,1]\). On the other hand, it is known that every finite group is the fundamental group of some nonsingular projective variety, and so a lot is going on for high slopes.

algebraic geometrydifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic

Free Mathematics Seminar

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