Birational positivity in dimension 4
The article under review studies the relationship between the Kodaira dimension \(\kappa (X)\) and a more general notion \(\kappa^+ (X)\) introduced by \textit{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 \textit{F. Campana} and \textit{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.
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