Weakly smooth forms and Dolbeault cohomology of curves

Abstract: Gubler and I work out a theory of weakly smooth forms on non-Archimedean analytic spaces closely following the construction of Chambert-Loir and Ducros, but in which harmonic functions are forced to be smooth. We call such forms "weakly smooth". We compute the Dolbeault cohomology groups of rig-smooth, compact non-Archimedean curves with respect to this theory, and show that they have the expected dimensions and satisfy Poincaré duality. We carry out this computation by giving an alternative characterization of weakly smooth forms on curves as pullbacks of certain "smooth forms" on a skeleton of the curve. This yields an isomorphism between the Dolbeault cohomology of the skeleton, which can be computed using standard combinatorial methods, and the Dolbeault cohomology of the curve.

This work is joint with Walter Gubler.

algebraic geometrycombinatorics

Audience: researchers in the topic

(LAGARTOS) Latin American Real and Tropical Geometry Seminar

Series comments: Please email one of the organizers to receive login info and join our mail list.