Event description
Abstract: Let (M,J) be a compact complex manifold of complex dimension n. A p-Kähler structure on (M,J) is a transverse, d-closed (p,p)-form. For 1 < p < n-1, p-Kähler structures do not have a metric meaning. However, the Alessandrini-Bassanelli conjecture states that on a compact complex manifold, the existence of a p-Kähler structure implies the existence of a (p+1)-Kähler structure. Since transverse (n-1,n-1)-forms are the (n-1)-th power of the fundamental form of a Hermitian metric, then (n-1)-Kähler structures coincide with balanced metrics. Hence, a direct consequence of the Alessandrini-Bassanelli conjecture is that p-Kähler geometry is a special case of balanced geometry.
We first deal with the positivity notions known as transversality, and we show that the power of a transverse form is not necessarily transverse. We then address the Alessandrini-Bassanelli conjecture on some classes of compact complex manifolds. Finally, we discuss the existence of (n-2)-Kähler structures on complex semisimple Lie algebras of complex dimension n. The talk is based on a joint work with Asia Mainenti.