**Houda Bellitir**^{[a]} and **Dan Popovici**^{[b]}

*Positivity cones under deformations of complex structures
*

**Received:** 19 March 2018

**Accepted:** 16 July 2018

**Mathematics Subject Classification (2010):** 32G05, 14F40, 14C30, 32J25, 32Q57.

**Keywords:** Frölicher spectral sequence; deformations of complex structures;
positivity; \(\partial\bar\partial\)-manifolds; strongly Gauduchon (sG) metrics; sGG manifolds.

**Authors address:**

[a]: Ibn Tofail University, Faculty of Sciences, Departement of Mathematics, PO 242 Kenitra, Morocco

[b]: Université Paul Sabatier, Institut de Mathématiques de Toulouse,
118, route de Narbonne, 31062, Toulouse Cedex 9, France

**Abstract:**
We investigate connections between the sGG property of compact complex manifolds, defined in earlier work by the
second author and L. Ugarte by the requirement that every Gauduchon metric be strongly Gauduchon, and a possible
degeneration of the Frölicher spectral sequence. In the first approach that we propose, we prove a partial
degeneration at \(E_2\) and we introduce a positivity cone in the \(E_2\)-cohomology of bidegree \((n-2,\,n)\) of
the manifold that we then prove to behave lower semicontinuously under deformations of the complex structure.
In the second approach that we propose, we introduce an analogue of the \(\partial\bar\partial\)-lemma property
of compact complex manifolds for any real non-zero constant \(h\) using the partial twisting \(d_h\), introduced
recently by the second author, of the standard Poincaré differential \(d\). We then show, among other things,
that this \(h\)-\(\partial\bar\partial\)-property is deformation open.

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