RC-positivity, rational connectedness and Yau's conjecture
By Xiaokui Yang. 2017
84 citations recorded by the source corpus.
In this paper, we introduce a concept of RC-positivity for Hermitian holomorphic vector bundles and prove that, if $E$ is an RC-positive vector bundle over a compact complex manifold $X$, then for any vector bundle $A$, there exists a positive integer $c_A=c(A,E)$ such that $$H^0(X,\mathrm{Sym}^{\otimes \ell}E^*\otimes A^{\otimes k})=0$$ for $\ell\geq c_A(k+1)$ and $k\geq 0$. Moreover, we obtain that, on a compact Kahler manifold $X$, if $\Lambda^p T_X$ is RC-positive for every $1\leq p\leq \dim X$, then $X$ is projective and rationally connected. As applications, we show that if a compact Kahler manifold $(X,\omega)$ has positive holomorphic sectional curvature, then $\Lambda^p T_X$ is RC-positive and $H_{\bar\partial}^{p,0}(X)=0$ for every $1\leq p\leq \dim X$, and in particular, we establish that $X$ is a projective and rationally connected manifold, which confirms a conjecture of Yau([57, Problem 47]).
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/c1083b386a8a377c98d712d3394b87e6db81062a
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Comparison and vanishing theorems for Kähler manifolds (2018, methodology): “…dimension m ≥ 2) are not, perhaps this metric criterion via Ric ⊥ (which is the trace of orthogonal bisectional curvature in some sense as explained a little later) perhaps reflects the projective embedding property more accurately than the positivity of the holomorphic sectional curvature [38].”