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Birational positivity in dimension 4. (Positivité birational en dimension 4.) (English. French summary) Zbl 1326.14093
The article under review studies the relationship between the Kodaira dimension $$\kappa (X)$$ and a more general notion $$\kappa^+ (X)$$ introduced by F. Campana [J. Algebr. Geom. 4, No. 3, 487–502 (1995; Zbl 0845.32027)].
Let $$X$$ be a projective manifold of dimension $$n$$ and let $$L$$ be a holomorphic line bundle on $$X$$. The Iitaka dimension of $$L$$ is defined as the largest number $$k\in\mathbb{N}$$ such that $\varlimsup_{m\rightarrow +\infty} \frac{h^0 (X, L^{\otimes m})}{m^k} > 0 .$ The Kodaira dimension $$\kappa (X) := \kappa (K_X)$$, and $$\kappa^+ (X)$$ is defined by $\kappa^+ (X) := \max \{ \kappa (\det \mathcal{F}) | \mathcal{F} \text{ is a coherent subsheaf of }\Omega_X ^p, \text{ for some }p\} .$ It was conjectured by Campana [loc. cit.] that $$\kappa (X) =\kappa^+ (X)$$ when $$\kappa (X) \geq 0$$. In the same article, Campana proved that, if the good minimal model conjecture holds for dimension $$n$$, then we have $$\kappa (X) =\kappa^+ (X)$$.
The main result of the article under review is concerned with replacing the abundance conjecture for dimension $$n$$ with the abundance conjecture in lower dimension. More precisely, the main theorem of the article is:
Let $$X$$ be a projective manifold of dimension $$n$$. Assume that the good minimal model conjecture holds for terminal varieties with zero Kodaira dimension up to dimension $$n-m$$. If $$\kappa (X) \geq m-1$$ then $$\kappa (X) =\kappa ^+ (X)$$.
One of the key ingredient of the proof is a result of F. Campana and T. Peternell [Bull. Soc. Math. Fr. 139, No. 1, 41–74 (2011; Zbl 1218.14030), with an appendix by Matei Toma] about the pseudo-effectivity of quotients of $$\Omega_X ^p$$ for non-uniruled projective manifolds.
As an application, we know that for $$4$$-dimensional projective manifolds with $$\kappa (X) \geq 0$$, we have $$\kappa (X) =\kappa ^+ (X)$$. As another interesting application, combining this result with a previous result of Campana, we know that (cf. Thm 1.7), for $$\kappa (X)=0$$ and $$\chi (X,\mathcal{O}_X) \neq 0$$, then $$\pi_1 (X)$$ is finite.
Reviewer: Junyan Cao (Paris)
##### MSC:
 14J35 $$4$$-folds 14E30 Minimal model program (Mori theory, extremal rays)
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##### References:
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