Monday

Revisiting a proof of formality of Kahler manifolds, with a view towards complex manifolds and special metrics (d-dc + 3)

  • Speaker: Scott Wilson
  • Time: 11:20-12:20
  • Abstract: From one such proof of formality we obtain a new long exact sequence, defined for all complex manifolds, that relates de Rham cohomology to two other complex analytic groups. These groups relate in a transparent way to other well-studied notions, such as the Bott-Chern and Aeppeli cohomologies, and pure Hodge structures, while yielding new numerical (in)equalities involving Betti numbers as well. A mild weakening of the so-called d-dc condition, that we call (d-dc + 3), is equivalent to the vanishing of the connecting homomorphism in this sequence. Moreover, it implies E1-degeneration and is stable under many blowups. I’ll discuss numerous examples: all compact complex surfaces, higher Hopf surfaces (but not all products of odd spheres), and complex manifolds which admit locally conformal Kahler metrics satisfying a certain parallel-ness condition (i.e., Vaisman metrics). This gives a new complex analytic obstruction to the existence of certain compatible metrics and highlights a metric condition that implies E1-degeneration and much more (namely, d-dc + 3). This is joint work with Jonas Stelzig.