• almost complex manifolds
• Bott-Chern Laplacian
• complex manifolds
• deformation of the complex structure
• differential operators
• differential operators on complex manifolds
• Dolbeault Laplacian
• gauduchon metrics
• non compact manifolds
• skt metrics

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.