Room P3.10, Mathematics Building — Online

Aleksandar Milivojevic

Aleksandar Milivojevic, Max Planck Institute for Mathematics - Bonn
Holomorphic notions of formality and Massey products

I will discuss joint work with Jonas Stelzig in which we consider the beginnings of a bigraded analogue of rational homotopy theory adapted to complex manifolds, in a somewhat different fashion than that of Neisendorfer-Taylor which appeared in the 1970’s soon after Sullivan’s Infinitesimal Computations in Topology. Taking cues from an additive decomposition theorem for double complexes, we define two natural notions of formality for our basic objects — commutative bigraded bidifferential algebras — which place both bigraded components of the de Rham differential on equal footing. These notions are related by the ddbar-lemma (the additive property used to show formality, in the usual sense, of compact complex manifolds admitting a Kähler metric). We consider obstructions to these notions of formality, taking in Bott-Chern cohomology classes and outputting classes in a chain complex of Demailly-Schweitzer, whose construction mimics those of classical Massey products and extends the triple products landing in Aeppli cohomology considered by Angella-Tomassini; we also touch upon their behavior under blow-ups and more generally positive-degree holomorphic maps.