On compact complex manifolds with nonnegative mean curvature

In this paper, we establish a structure theorem for projective manifolds with semipositive mean curvature. We prove that the rational dimension of a projective manifold with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -positive mean curvature is at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We give an estimate of Hodge number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Superscript p comma 0"> <mml:semantics> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">h^{p,0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for compact complex manifolds with mean curvature bounded below. As an application, we get a maximal Betti number rigidity for compact HN-nonnegative Kähler manifolds.