Operatori differenziali su varieta complesse

In the thesis, we generalize the classical characterization of Bott-Chern and Aeppli harmonic forms, holding on compact Hermitian manifolds, to the non compact case, namely on special families of complete Stein manifolds and of complete Hermitian manifolds. It turns out that a suitable space of differentiable forms satisfying these characterizations is the space of smooth forms with bounded L2 norm and whose Chern covariant derivative has bounded L2 norm. Consequently, we study a Sobolev space of forms introduced by Andreotti and Vesentini, showing a weak Bott-Chern (also Aeppli and Dolbeault) orthogonal decomposition of this Hilbert space, which holds on every, possibly non compact, Kähler manifold. After that, we focus on the compact case, investigating the relation between Aeppli cohomology and a special family of Hermitian metrics called Gauduchon metrics, which always exist on compact complex manifolds. Then, we consider another class of special Hermitian metrics, namely Strong Kähler with torsion metrics, briefly SKT, and study small deformations of the complex structure on compact complex manifolds, showing a necessary condition, which involves the Bott-Chern cohomology, for the existence of a smooth curve of SKT metrics along a curve of deformations, originating from a SKT Hermitian manifold. This is motivated by the fact that SKT metrics are a generalization of Kähler metrics, and Kodaira and Spencer, proved that the Kähler condition is stable under small deformations of the complex structure. The property of admitting SKT metrics has been shown to be unstable under small deformations, therefore it is interesting to study when this property may be stable. Finally, we investigate the role of the Bott-Chern Laplacian and of Bott- Chern harmonic forms on compact almost Hermitian manifolds, analysing the differences and the similarities of their behaviours with respect to the case of compact Hermitian manifolds. This inserts in the very recent study of Dolbeault cohomology, Dolbeault harmonic forms, and Bott-Chern cohomology, on compact almost complex manifolds.