arXiv subjects
Publications and source records attributed to Xiaokui Yang.
Let $X$ be a compact Kähler manifold. We prove that if $X$ admits a smooth Hermitian metric $ω$ with quasi-positive second Chern-Ricci curvature $\mathrm{Ric}^{(2)}(ω)$, then $X$ is projective and rationally connected. In particular, $X$ is simply connected.
Suppose that $X$ is a projective manifold whose tangent bundle $T_X$ contains a locally free strictly nef subsheaf. We prove that $X$ is isomorphic to a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group $π_1(X)$ is virtually abelian, then $X$ is isomorphic to a projective space.
Let $(M,g,J)$ be a Riemannian manifold with a compatible integrable complex structure $J\in\mathrm{End}(T_\mathbb{R} M)$ and $\mathcal{A}_{g,J}$ be the space of real connections on $T_\mathbb{R} M$ preserving both $g$ and $J$. In this paper, we investigate the relationship between the geometry of real connections in $\mathcal{A}_{g,J}$ and that of Hermitian connections on $T^{1,0}M$. In particular, we study the geometry of the real Chern connection $\nabla^{\mathrm{Ch,\mathrm{R}}}$ on $(M,g,J)$, and obtain Kähler-Einstein metrics by using real Chern-Einstein metrics.
In this paper, we study smooth complex projective varieties $X$ such that some exterior power $\bigwedge^r T_X$ of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If $T_X$ is strictly nef, then $X$ isomorphic to the projective space $\mathrm{P}^n$. If $\bigwedge^2 T_X$ is strictly nef and if $X$ has dimension at least $3$, then $X$ is either isomorphic to $\mathrm{P}^n$ or a quadric $\mathrm{Q}^n$.
Let $X$ be a smooth projective manifold with $\dim_\mathbb{C} X=n$. We show that if a line bundle $L$ is $(n-1)$-ample, then it is $(n-1)$-positive. This is a partial converse to the Andreotti-Grauert theorem. As an application, we show that a projective manifold $X$ is uniruled if and only if there exists a Hermitian metric $ω$ on $X$ such that its Ricci curvature $\mathrm{Ric}(ω)$ has at least one positive eigenvalue everywhere.
In this paper, we introduce a new energy density function $\mathscr Y$ on the projective bundle $\mathbb{P}(T_M)\>M$ for a smooth map $f:(M,h)\>(N,g)$ between Riemannian manifolds $$\mathscr Y=g_{ij}f^i_αf^j_β\frac{W^αW^β}{\sum h_{γδ} W^γW^δ}.$$ We get new Hessian estimates to this energy density and obtain various new Liouville type theorems for holomorphic maps, harmonic maps and pluri-harmonic maps. For instance, we show that there is no non-constant holomorphic map from a compact \emph{Hermitian manifold} with positive (resp. non-negative) holomorphic sectional curvature to a \emph{Hermitian manifold} with non-positive (resp. negative) holomorphic sectional curvature.
In this paper, we show that every harmonic map from a compact Kähler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact Kähler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.
Let $X$ be a ruled surface over a curve of genus $g$. We prove that $X$ has a scalar-flat Hermitian metric if and only if $g\geq 2$ and $m(X)>2-2g$ where $m(X)$ is an intrinsic number depends on the complex structure of $X$.
In this paper, we prove that if a compact Kähler manifold $X$ has a smooth Hermitian metric $ω$ such that $(T_X,ω)$ is uniformly RC-positive, then $X$ is projective and rationally connected. Conversely, we show that, if a projective manifold $X$ is rationally connected, then the tautological line bundle $\mathscr{O}_{T_X^*}(-1)$ is uniformly RC-positive (which is equivalent to the existence of some RC-positive complex Finlser metric on $X$). As an application, we prove that if $(X,ω)$ is a compact Kähler manifold with certain quasi-positive holomorphic sectional curvature, then $X$ is projective and rationally connected.
Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle $\mathscr{O}_{T_M^*}(1)$ is not pseudo-effective and $\mathscr{O}_{T_N^*}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.
Let $X$ be a compact Kähler manifold and $D$ be a simple normal crossing divisor. If $D$ is the support of some effective $k$-ample divisor, we show $$ H^q(X,Ω^p_X(\log D))=0,\quad \text{for}\quad p+q>n+k.$$
In this paper, we study the geometry of compact complex manifolds with Levi-Civita Ricci-flat metrics and prove that compact complex surfaces admitting Levi-Civita Ricci-flat metrics are Kahler Calabi-Yau surfaces or Hopf surfaces.
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 Kähler manifold $X$, if $Λ^p T_X$ 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 Kähler manifold $(X,ω)$ has positive holomorphic sectional curvature, then $Λ^p T_X$ 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]).
Let $X$ be a smooth projective variety over an algebraically closed field $\mathbb{K}$ with arbitrary characteristic. Suppose $L$ is an ample and globally generated line bundle. By Castelnuovo--Mumford regularity, we show that $K_X \otimes L^{\otimes \dim X} \otimes A$ is globally generated and $K_X \otimes L^{\otimes (\dim X+1)} \otimes A$ is very ample, provided the line bundle $A$ is nef but not numerically trivial. On complex projective varieties, by investigating Kawamata-Viehweg-Nadel type vanishing theorems for vector bundles, we also obtain the global generation for adjoint vector bundles. In particular, for a holomorphic submersion $f:X\longrightarrow Y$ with $L$ ample and globally generated, and $A$ nef but not numerically trivial, we prove the global generation of $ f_*(K_{X/Y})^{\otimes s}\otimes K_Y \otimes L^{\otimes \dim Y} \otimes A$ for any positive integer $s$.
Let $M$ be a compact Riemannian manifold, $π:\widetilde{M}\rightarrow M$ be the universal covering and $ω$ be a smooth $2$-form on $M$ with $π^*ω$ cohomologous to zero. Suppose the fundamental group $π_1(M)$ satisfies certain radial quadratic (resp. linear) isoperimetric inequality, we show that there exists a smooth $1$-form $η$ on $\widetilde M$ of linear (resp. bounded) growth such that $π^*ω=d η$. As applications, we prove that on a compact Kahler manifold $(M,ω)$ with $π^*ω$ cohomologous to zero, if $π_1(M)$ is $\mathrm{CAT}(0)$ or automatic (resp. hyperbolic), then $M$ is Kahler non-elliptic (resp. Kahler hyperbolic) and the Euler characteristic $(-1)^{\frac{\dim_\mathbb{R} M}{2}}χ(M)\geq 0$ (resp. $>0$).
In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety $X^n$ $(n>4)$ has strictly nef $Λ^2 TX$, then it is isomorphic to $\mathbb{P}^n$ or quadric $\mathbb{Q}^n$. We also prove that on elliptic curves, strictly nef vector bundles are ample, whereas there exist Hermitian flat and strictly nef vector bundles on any smooth curve with genus $g\geq 2$.
In this paper, we show that any compact K$ä$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$ä$hler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure $J$ compatible with the symplectic form, there is no non-constant $J$-holomorphic entire curve $f:C \rightarrow X$.
In this paper, we prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp. negative) scalar curvature if and only if $K_X$ (resp. $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\geq 2$, there exist smooth Hermitian metrics with positive total scalar curvature and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Székelyhidi, V. Tosatti and B. Weinkove.