Formality and non-zero degree maps
Organizers
Speaker
Aleksandar Milivojevic
Time
Thursday, March 13, 2025 2:30 PM - 3:30 PM
Venue
A3-2-303
Online
Zoom 712 322 9571
(BIMSA)
Abstract
In the mid 70’s, Deligne-Griffiths-Morgan-Sullivan demonstrated a strong topological condition a closed manifold would have to satisfy if it were to carry a Kähler complex structure. Namely, the manifold would have to be formal, in the sense of its de Rham algebra of forms being weakly equivalent to its cohomology. In particular, there can be no non-trivial Massey products on a compact Kähler manifold. The salient underlying property of compact Kähler manifolds which implies formality is preserved under surjective holomorphic maps (non-zero degree maps in the holomorphic setting). It turns out, formality itself is preserved under non-zero degree continuous maps of spaces satisfying Poincaré duality on their rational cohomology. I will explain the key components of this argument and show how one can then apply it to various situations. Using this result we can recover and extend several seemingly disparate results in this area: the formality of singular complex projective varieties satisfying rational Poincaré duality, the formality of closed manifolds with sufficiently large first Betti number and a non-negative Ricci curvature metric, descent of formality from field extensions, and some more. This is joint work with Jonas Stelzig and Leopold Zoller.