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Register a new userL^2 Castelnuovo-de Franchis, the cup product lemma, and filtered ends of Kaehler manifolds
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Simple approaches to the proofs of the L^2 Castelnuovo-de Franchis theorem and the cup product lemma which give n
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In this note, we propose an approach to the study of the analogue for unipotent harmonic bundles of Schmids Nilpotent Orbit Theorem. Using this approach, we construct harmonic metrics on unipotent bundles over quasi-compact Kahler manifolds with carefully controlled asymptotics near the compactifying divisor; such a metric is unique up to some isometry. Such an asymptotic behavior is canonical in some sense.
Non-existence of warped product semi-slant submanifolds of Kaehler manifolds was proved in [17], it is interesting to find their existence. In this paper, we prove the existence of warped product semi-slant submanifolds of nearly Kaehler manifolds by a characterization. To this end we obtain an inequality for the squared norm of second fundamental form in terms of the warping function and the slant angle. The equality case is also discussed.
We prove two results related to the Schwarz lemma in complex geometry. First, we show that if the inequality in the Schwarz lemmata of Yau, Royden and Tosatti becomes equality at one point, then the equality holds on the whole manifold. In particular, the holomorphic map is totally geodesic and has constant rank. In the second part, we study the holomorphic sectional curvature on an almost Hermitian manifold and establish a Schwarz lemma in terms of holomorphic sectional curvatures in almost Hermitian setting.
First we confirm a conjecture asserting that any compact Kahler manifold $N$ with $Ric^perp>0$ must be simply-connected by applying a new viscosity consideration to Whitneys comass of $(p, 0)$-forms. Secondly we prove the projectivity and the rational connectedness of a Kahler manifold of complex dimension $n$ under the condition $Ric_k>0$ (for some $kin {1, cdots, n}$, with $Ric_n$ being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollar-Miyaoka-Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of cite{Ni-Zheng2}. Thirdly, motivated by $Ric^perp$ and the classical work of Calabi-Vesentini cite{CV}, we propose two new curvature notions. The cohomology vanishing $H^q(N, TN)={0}$ for any $1le qle n$ and a deformation rigidity result are obtained under these new curvature conditions. In particular they are verified for all classical Kahler C-spaces with $b_2=1$. The new conditions provide viable candidates for a curvature characterization of homogenous Kahler manifolds related to a generalized Hartshone conjecture.
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