Geodesic
On Projective Kähler Manifolds of Partially Positive Curvature and Rational Connectedness
Documenta mathematica, Tome 25 (2020), pp. 219-238.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we investigate the intermediate notion of $k$-positive Ricci curvature and prove that for a projective $n$-dimensional Kähler manifold of $k$-positive Ricci curvature the MRC fibration has generic fibers of dimension at least $n-k+1$. We also establish an analogous result for projective Kähler manifolds of semi-positive holomorphic sectional curvature based on an invariant which records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. In particular, the latter result confirms a conjecture of S.-T. Yau in the projective case.
Classification : 32Q10, 14J10, 14J32, 14M22
Keywords: complex projective manifolds, Kähler metrics, positive holomorphic sectional curvature, \(k\)-positive Ricci curvature, rational curves, uniruledness, rational connectedness 
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     author = {Heier, Gordon and Wong, Bun},
     title = {On {Projective} {K\"ahler} {Manifolds} of {Partially} {Positive} {Curvature} and {Rational} {Connectedness}},
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Heier, Gordon; Wong, Bun. On Projective Kähler Manifolds of Partially Positive Curvature and Rational Connectedness. Documenta mathematica, Tome 25 (2020), pp. 219-238. http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a62/