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Numerical characterization of the Kähler cone of a compact Kähler manifold. (English) Zbl 1064.32019
The Kähler cone of a compact Kähler manifold is the set of cohomology classes of smooth positive definite clossed $$(1,1)$$-forms. The authors show that this cone depends only on the intersection product of the cohomology ring, the Hodge structure and the homology classes of analytic cycles: if $$X$$ is a compact Kähler manifold, the Kähler cone $$\mathcal{K}$$ of $$X$$ is one of the connected components of the set $$\mathcal{P}$$ of real $$(1,1)$$-cohomology classes $$\{\alpha\}$$ which are numerically positive on the analytic cycles, i.e. such that $$\int_Y\alpha^p>0$$ for every irreducible analytic set in $$X, \;p=\dim Y$$. This result can be considered as a generalization of the Nakai-Moishezon criterion, which provide a necessary and sufficient criterion for a line bundle to be ample. If $$X$$ is projective then $$\mathcal{K}=\mathcal{P}$$. If $$X$$ is a compact Kähler manifold, the $$(1,1)$$-cohomology class $$\alpha$$ is nef (numerically effective free) if and only if there exists a Kähler metric $$\omega$$ on $$X$$ such that $$\int _Y\alpha^k\wedge \omega^{p-k}\geq 0$$ for all irreducible analytic sets $$Y$$ and all $$k=1,2,\dots, p=\dim Y$$. A $$(1,1)$$-cohomology class $$\{\alpha\}$$ on $$X$$ is nef if and only if for every irreducible analytic set $$Y$$ in $$X$$, $$p=\dim Y$$, and for every Kähler metric $$\omega$$ on $$X$$, one has $$\int_Y\alpha\wedge \omega^{p-1}\geq 0$$.
First, the authors obtain a sufficient condition for a nef class to contain a Kähler current. Then the main result is obtained by an induction on the dimension.
The obtained result has an important application to the deformation theory of compact Kähler manifolds: consider $${\mathcal{X}}\to S$$ a deformation of compact Kähler manifolds over an irreducible base $$S$$. There exists a countable union $$S^\prime =\bigcup S_\nu$$ of analytic subsets $$S_\nu \subset S$$, such that the Kähler cones $${\mathcal{K}}_t \subset H^{1,1}(X_t,\mathbb{C})$$ are invariant over $$S\setminus S^\prime$$ under parallel transport with respect to the $$(1,1)$$-projection $$\nabla^{1,1}$$ of the Gauss-Manin connection.

##### MSC:
 32Q15 Kähler manifolds 32Q25 Calabi-Yau theory (complex-analytic aspects) 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32J27 Compact Kähler manifolds: generalizations, classification
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