##
**Threefolds with nef anticanonical bundles.**
*(English)*
Zbl 0980.14028

From the introduction: In this paper we study the global structure of projective threefolds \(X\) whose anticanonical bundle \(-K_X\) is nef. In differential geometric terms this means that we can find metrics on \(-K_X=\det T_X\) (where \(T_X\) denotes the tangent bundle of \(X)\) such that the negative part of the curvature is as small as we want. In algebraic terms nefness means that the intersection number \(-K_X\cdot C\geq 0\) for every irreducible curve \(C\subset X\). The notion of nefness is weaker than the requirement of a metric of semipositive curvature and is the appropriate notion in the context of algebraic geometry.

J.-P. Demailly, T. Peternell and M. Schneider [Compos. Math. 89, No. 2, 217-240 (1993; Zbl 0884.32023)] proved that the Albanese map \(\alpha:X \to\text{Alb}(X)\) is a surjective submersion if \(-K_X\) carries a metric of semi-positive curvature, or, equivalently, if \(X\) carries a Kähler metric with semipositive Ricci curvature. It was conjectured that the same holds if \(-K_X\) is only nef, but there are very serious difficulties with the old proof, because the metric of semi-positive curvature has to be substituted by a sequence of metrics whose negative parts in the curvature converge to 0. The conjecture splits naturally into two parts: surjectivity of \(\alpha\) and smoothness. Surjectivity was proved in dimension 3 already in the paper cited above and in general by Qi Zhang [J. Reine Angew. Math. 478, 57-60 (1996; Zbl 0855.14007)], using char \(p\). Our main result now proves smoothness in dimension 3:

Let \(X\) be a smooth projective threefold with \(-K_X\) nef. Then the Albanese map is a surjective submersion.

J.-P. Demailly, T. Peternell and M. Schneider [Compos. Math. 89, No. 2, 217-240 (1993; Zbl 0884.32023)] proved that the Albanese map \(\alpha:X \to\text{Alb}(X)\) is a surjective submersion if \(-K_X\) carries a metric of semi-positive curvature, or, equivalently, if \(X\) carries a Kähler metric with semipositive Ricci curvature. It was conjectured that the same holds if \(-K_X\) is only nef, but there are very serious difficulties with the old proof, because the metric of semi-positive curvature has to be substituted by a sequence of metrics whose negative parts in the curvature converge to 0. The conjecture splits naturally into two parts: surjectivity of \(\alpha\) and smoothness. Surjectivity was proved in dimension 3 already in the paper cited above and in general by Qi Zhang [J. Reine Angew. Math. 478, 57-60 (1996; Zbl 0855.14007)], using char \(p\). Our main result now proves smoothness in dimension 3:

Let \(X\) be a smooth projective threefold with \(-K_X\) nef. Then the Albanese map is a surjective submersion.