I am trying to understand a particular property of the first Chern class of a nef line bundle over a Kähler manifold.

We know in general, let $X$ be a complete complex projective variety, and $L$ a line bundle over it. Then $L$ is called nef if $$\int_Cc_1(L)\geq0$$ where $C$ is any irreducible complete curve on $X$.

Then is it true that $c_1(L)$ can is a limit of Kähler forms in $H^2(X,\mathbb{Q})$?

I appreciate any comments and answers. Thanks in advance!

  • $\begingroup$ Are you asking about general compact Kaehler manifolds, or projective varieties? In the latter case this is just Kleiman's criterion. $\endgroup$ – Nefertiti Jul 6 '16 at 7:44
  • $\begingroup$ Thanks for your comment. I'm asking about compact Kaehler manifold. $\endgroup$ – David Lucien Jul 6 '16 at 16:09
  • $\begingroup$ When you write "where $C$ is any irreducible complete curve" do you mean "for all irreducible curves $C$"? It is always better to make your quantifiers explicit. "Where" is not very good at that: it is only somewhat good to explain notation, and never useful to quantify. $\endgroup$ – Mariano Suárez-Álvarez Dec 24 '17 at 18:09

The answer to your question, at least with the definition that you suggest, is negative. The main trouble is that a compact Kähler manifold need not have any complete curve at all. This is the case for instance for a compact complex torus $\mathbb{C}^n/\Lambda$ when $\Lambda$ is a sufficiently general lattice, and $n > 1$.

However, a much better definition of nef in the Kähler case is precisely to ask that $c_1(L)$ is a limit of Kähler classes, because this has all the expected properties of nef classes on projective varieties.

Jean-Pierre Demailly has been considering such questions in his work since a long time. See especially on his webpage references [35], [40], [62], [65], [69].

  • J.P. Demailly. Singular Hermitian metrics on positive line bundles. Complex algebraic varieties (Bayreuth, 1990), 87–104, Lecture Notes in Math., 1507, Springer, Berlin, 1992.

  • J.P. Demailly, T. Peternell, and M. Schneider. Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom. 3, 295–345 (1994).

  • J.P. Demailly, T. Peternell, and M. Schneider. Pseudo-effective line bundles on compact Kähler manifolds. Internat. J. Math. 12, 689–741 (2001).

  • J.P. Demailly, T. Eckl, and T. Peternell. Line bundles on complex tori and a conjecture of Kodaira. Comment. Math. Helv. 80, 229–242 (2005).

  • S. Boucksom, J.P. Demailly, M. Păun, and T. Peternell. The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebraic Geom. 22, 201–248 (2013).

Even a definition of "nef" that works for general compact complex manifolds (or even spaces with singularities) is given. Since Hodge decomposition does not hold, one has to use Bott-Chern cohomology instead of the more usual de Rham cohomology. (As a matter of fact, all Chern classes are defined in these "richer" cohomology groups, which coincide with the Dolbeault groups in the Kähler case, but differ in general).

You will find most of the above stuff in his book "analmeth_book.pdf" (funniest file name ever, am I right) here.

  • J.P. Demailly. Analytic methods in algebraic geometry. Surveys of Modern Mathematics, 1. International Press, Somerville, MA; Higher Education Press, Beijing, 2012. viii+231 pp.

The "agbook" introduces more preliminary material, see here.

  • J.P. Demailly. Complex analytic and differential geometry. Monograph Grenoble, 1997.
  • $\begingroup$ Thank you so much for such a clear and beneficial answer! $\endgroup$ – David Lucien Aug 2 '16 at 19:18
  • $\begingroup$ Thanks for the reminder. It's done now. $\endgroup$ – David Lucien Aug 2 '16 at 20:00

Let $X$ be a compact complex manifold with a fixed hermitian metric $ω$. A line bundle $L$ over $X$ is nef if for every $ε > 0$ there exists a smooth hermitian metric $h_ε$ on $L$ such that the curvature satisfies $Θ_{h_ε} ≥ −εω$

A nef line bundle $L$ satisfies $L · C ≥ 0$ for all curves $C ⊂ X$, but the converse is not true. For projective algebraic $X$ both notions coincide.

A vector bundle $E$ is said to be strictly nef (resp. ample, nef) if the tautological line bundle $\mathcal O_E (1)$ of the projective bundle $P(E ) \to X $ is strictly nef (resp. ample, nef).The notion of strictly nefness is stronger than that of nefness, but weaker than that of ampleness. See this paper

When tangent bundle is nef see Theorem 1.1

If $(X, \omega)$ is compact Kähler, a class $\{\alpha\}$ is nef if and only if $\{\alpha + \epsilon\omega\}$ is a K\"ahler class for every $\epsilon > 0$ . Let $X$ be a compact Kähler manifold with numerically effective Ricci class, then, this means that this class contains closed $(1, 1)$-forms with arbitrary small negative part.

Let $X$ be a compact Kähler manifold with nef Ricci class, then we can translate it to existence of a sequence of Kähler metrics $(\omega_\epsilon)_{\epsilon>0}$ in the cohomology class of $\omega$, such that $Ric(\omega_\epsilon)\geq -\epsilon\omega_\epsilon$. Thus, this tells us that we have Kähler metrics which are closer and closer to having semi-positive Ricci curvature.

Demailly et al., by using the Aubin-Yau theorem, showed that the being nef on the Ricci class is equivalent to the existence of a sequence of Kähler metrics $(\omega_k)_{k>o}$ on $X$ such that

I) For each $k > 0$, the metric $\omega_k$ belongs to a fixed cohomology class, say $\{\omega\}$;

II) The Ricci curvature of $\omega_k$ is bounded from below by $-1/k$.

  • $\begingroup$ As an important property : For a projective variery $X$ , when $-K_{X}$ is nef, then for any surjective holomorphic map $f:X\to B$ , we must have $K_B\sim_{\mathbb Q} 0$ $\endgroup$ – user61135 Feb 13 '18 at 11:22

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