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Compact Complex Manifolds with Small Gauduchon Cone. (arXiv:1407.5070v2 [math.DG] UPDATED)
来源于:arXiv
This paper is intended as the first step of a programme aiming to prove in
the long run the long-conjectured closedness under holomorphic deformations of
compact complex manifolds that are bimeromorphically equivalent to compact
K\"ahler manifolds, known as Fujiki {\it class} ${\cal C}$ manifolds. Our main
idea is to explore the link between the {\it class} ${\cal C}$ property and the
closed positive currents of bidegree $(1,\,1)$ that the manifold supports, a
fact leading to the study of semi-continuity properties under deformations of
the complex structure of the dual cones of cohomology classes of such currents
and of Gauduchon metrics. Our main finding is a new class of compact complex,
possibly non-K\"ahler, manifolds defined by the condition that every Gauduchon
metric be strongly Gauduchon (sG), or equivalently that the Gauduchon cone be
small in a certain sense. We term them sGG manifolds and find numerical
characterisations of them in terms of certain relations between various 查看全文>>