Kähler manifolds with numerically effective Ricci class and maximal first Betti number are tori
Compact Kähler manifolds with semipositive anticanonical class are a classical object of study and it is known that their Albanese maps are smooth fibrations, in particular they are surjective. By a theorem of \textit{J.-P. Demailly, T. Peternell} and \textit{M. Schneider} [J. Algebr. Geom. 3, 295--345 (1994; Zbl 0827.14027)], the same holds (after finite étale cover) for compact Kähler manifolds with nef tangent bundle. Compact Kähler manifolds with nef anticanonical class appear as a natural generalization of these two classes of manifolds and it has been conjectured in \textit{J.-P. Demailly, T. Peternell} and \textit{M. Schneider} [Compos. Math. 89, No. 2, 217--240 (1993; Zbl 0884.32023)] that their Albanese maps are surjective. In the projective case this has been shown by \textit{Q. Zhang} [J. Reine Angew. Math. 478, 57--60 (1996; Zbl 0855.14007)], but the compact Kähler case remains open. \textit{M. Paun} [Bull. Sci. Math. 122, 83--92 (1998; Zbl 0946.53037)] has shown that the first Betti number of such a manifold \(M\) is bounded by \(2 \dim(M)\), thus providing some evidence for this conjecture. In the note under review, the author shows that if \(b_1(M)= 2 \dim(M)\), the manifold \(M\) is isomorphic to a complex torus. This generalizes corollary 3.1. of \textit{M. Paun} [Commun. Anal. Geom. 9, 35--60 (2001; Zbl 0980.53091)], where this had been shown with an additional technical assumption. In the case \(b_1(M)= 2 \dim(M)-2\), the author shows the surjectivity of the Albanese map modulo an assumption on the fundamental group of \(M\). The proofs use deep results from Riemannian geometry.
scientific article