Numerically and cohomologically trivial Automorphisms of compact Kaehler manifolds and surfaces
Prof. Fabrizio Catanese
2025-06-05 16:00-17:00
MCM410
Speaker: Prof. Fabrizio Catanese (Universitat Bayreuth)
Time: 16:00-17:00 June 5, 2025 (Thursday)
Venue: MCM410
Title: Numerically and cohomologically trivial Automorphisms of compact Kaehler manifolds and surfaces
Abstract: An automorphism of a cKM X is said to be numerically trivial (in Aut_Q(X)) if it acts trivially on rational cohomology, and cohomologically trivial (in Aut_Z(X)) if it acts trivially on integral cohomology. The interest for these notions stems from the theory of period maps and from Teichmueller theory.
The quotients of these groups by Aut^0(S) were shown to be finite in the 70’s by Fujiki, and there has been ever since much work of several authors in the case of algebraic surfaces S, dedicated mostly to the group Aut_Q(S) of numerically trivial automorphisms.
After recalling the general status of the problematic of automorphisms with some topological triviality, and the earlier contributions which I gave with Wenfei Liu according to the classification of surfaces, I will concentrate on the difficult open case of properly elliptic surfaces, that is, minimal surfaces S of Kodaira dimension 1.
Here I shall report on joint results with Matthias Schuett and Wenfei Liu : for \chi >0 we show that all 2-generated abelian groups appear as Aut_Q(S), and that there are upper bounds for N : = |Aut_Q(S) |, depending on the bigenus P_2(S) and on the irregularity q(S). Our results are sharp in the isotrivial case.
As Noether said, curves were created by God, and surfaces by the devil, since it is very easy to make mistakes on subtle issues: several authors claimed that in this situation there are no numerically trivial automorphisms if \chi , p_g >0...
In the case where \chi=0, S is isogenous to an elliptic product, and if if Aut^0(S) is infinite (i.e., S is pseudo-elliptic), we showed that the index of Aut^0(S) inside Aut_Z(S) is at most 2, and classify exactly the cases where the index is 2.
If S is not pseudo elliptic, but with \chi=0, we show that Aut_Z(S) can be only Z/2, Z/3, (Z/2)^2, and that the first two cases do effectively occur. This is done in joint work also with Davide Frapporti and Christian Gleissner.
Also for \chi >0, we do not have examples where Aut_Z(S) has strictly more than 3 elements, but conjecturing the upper bound 3 would be too bold.
Together with Frapporti we found recently the record winning value 192 for N in the case where S is of general type (then N is bounded by a universal constant).
