Weiyi Zhang (University of Warwick) will speak in the geometry seminar on Tuesday 25 September, at 1.30pm in the Salle de Profs. Weiyi’s title is From smooth to almost complex and his abstract is below.
An almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. We will discuss differential topology of almost complex manifolds, explain how to use transversality statements for smooth manifolds to formulate and prove corresponding results for an arbitrary almost complex manifold. The examples include intersection of almost complex manifolds, structure of pseudoholomorphic maps and zero locus of certain harmonic forms.
One of the main technical tools is Taubes’ notion of “positive cohomology assignment”, which plays the role of local intersection number. I will begin with explaining its motivation through multiplicities of zeros of a smooth function.
Our results would lead to a notion of birational morphism between almost complex manifolds. Various birational invariants, including Kodaira dimension, for almost complex manifolds will be introduced and discussed (this part is joint with Haojie Chen). The invariants distinguish the standard almost complex structure from hypothetical complex structures on the six sphere.