Let (M,J) be a compact complex manifold of complex dimension n. A p-K\"ahler 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)-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. 

In this seminar, we begin by describing some interesting aspects of the positivity condition called transversality and the difference between other notions of positivity. Next, we address the Alessandrini-Bassanelli conjecture in the context of nilpotent Lie algebras equipped with a nilpotent complex structure. Finally, we describe obstruction to the exactness of the cohomology class of a $p$-K\"ahler structure.