Numerical characterization of the Kähler cone of a compact Kähler manifold. (English) Zbl 1064.32019

Demailly, Jean-Pierre; Paun, Mihai

Ann. Math. (2) 159, No. 3, 1247-1274 (2004).

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.

Reviewer: Vasile Oproiu (Iaşi)

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

Keywords:

Kähler manifolds; Kähler cone; nef cohomology classes; Kähler currents