Positivity cones under deformations of complex structures. (English) Zbl 1417.32016

Bellitir, Houda; Popovici, Dan

Riv. Mat. Univ. Parma (N.S.) 9, No. 1, 133-176 (2018).

Summary: We investigate connections between the sGG property of compact complex manifolds, defined in earlier work by the second author and L. Ugarte [Proc. Lond. Math. Soc. (3) 116, No. 5, 1161–1186 (2018; Zbl 1394.32013)] 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 \(\partial\overline\partial\)-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 \(h-\partial\overline\partial\)-property is deformation open.

MSC:

32G05	Deformations of complex structures
14F40	de Rham cohomology and algebraic geometry
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
32J25	Transcendental methods of algebraic geometry (complex-analytic aspects)
32Q57	Classification theorems for complex manifolds

Keywords:

Frölicher spectral sequence; deformations of complex structures; \(\partial \overline\partial\)-manifolds; strongly Gauduchon metrics

Citations:

Zbl 1394.32013