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)-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 talk, we discuss the positivity condition known as transversality and compare it with other notions of positivity. We then address the Alessandrini–Bassanelli conjecture for different classes of compact complex manifolds and highlight some cohomological aspects related to compact p-Kähler manifolds.