arXiv subjects
Publications and source records attributed to Wenhao Ou.
Assume that $X$ is a compact complex analytic variety which has quotient singularities in codimension 2, and that $\mathcal{F}$ is a reflexive sheaf on $X$. Using orbifold modifications, we can define first and second homological Chern classes for $\mathcal{F}$. If in addition $X$ has a Kähler form $ω$ and $\mathcal{F}$ is $ω$-stable, then we deduce Bogomolov-Gieseker inequality on the orbifold Chern classes of $\mathcal{F}$.
We prove that if $X$ is a compact complex analytic variety, which has quotient singularities in codimension 2, then there is a projective bimeromorphic morphism $f\colon Y\to X$, such that $Y$ has quotient singularities, and that the indeterminacy locus of $f^{-1}$ has codimension at least 3 in $X$. As an application, we deduce the Bogomolov-Gieseker inequality on orbifold Chern classes for stable reflexive coherent sheaves on compact Kähler varieties which have quotient singularities in codimension 2.
We show that a compact, complex analytic space $X$ has a bimeromorphic orbifold modification that is an isomorphism over the locally trivial orbifold locus of $X$.
In a previous paper, the orbifold Bogomolov-Gieseker inequality is proved for a stable reflexive sheaf on a compact Kähler variety with klt singularities. In this paper, we give a characterization on the stable reflexive sheaf when the Bogomolov-Gieseker equality holds.
In this article we show that if $(X, Δ)$ is a log canonical compact Kähler threefold pair such that $K_X+Δ$ is nef and the numerical dimension $ν(X, K_X+Δ)=2$, then $K_X+Δ$ is semi-ample. This result combined with our previous work in arXiv:2201.01202 shows that the log abundance holds for log canonical compact Kähler threefold pairs.
We adapt Bost's algebraicity characterization to the situation of a germ in a compact Kähler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-Păun and of Druel to foliations on compact Kähler manifolds. As an application, we prove that a compact Kähler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.
Let $\mathcal{F}$ be a foliation on a projective manifold $X$ with $-K_{\mathcal{F}}$ nef. Assume that either $\mathcal{F}$ is regular, or it has a compact leaf. We prove that there is a locally trivial fibration $f\colon X\to Y$, and a foliation $\mathcal{G}$ on $Y$ with $K_{\mathcal{G}} \equiv 0$, such that $\mathcal{F} = f^{-1}(\mathcal{G})$.
In this article we show that if $(X, Δ)$ is a log canonical compact Kähler $3$-fold such that $K_X+Δ$ is nef and the numerical dimension $ν(K_X+Δ)\neq 2$, then $K_X+Δ$ is semi-ample.
Let $X$ be a normal compact Kähler variety, and $\mathcal{F}$ a coherent reflexive sheaf on $X$. We investigate the existence of admissible Hermitian metrics on $\mathcal{F}$. If moreover $\mathcal{F}$ is slope stable, we also study the existence of admissible Hermitian-Yang-Mills metrics on it. The existence will hold if one can prove a uniform Sobolev inequality on singular spaces.
Let $(X,Δ)$ be a projective klt pair, and $f:X\to Y$ a fibration to a smooth projective variety $Y$ with strictly nef relative anti-log canonical divisor $-(K_{X/Y}+Δ)$. We prove that $f$ is a locally constant fibration with rationally connected fibres, and the base $Y$ is a canonically polarized hyperbolic projective manifold. In particular, when $Y$ is a single point, we establish that $X$ is rationally connected. Moreover, when $\dim X=3$ and $-(K_X+Δ)$ is strictly nef, we prove that $-(K_X+Δ)$ is ample, which confirms the singular version of a conjecture of Campana-Peternell for threefolds.
We show that the tangent bundle of a projective manifold with nef anticanonical class is generically nef. That is, its restriction to a curve cut out by general sufficiently ample divisors is a nef vector bundle. This confirms a conjecture of Peternell. As a consequence, the second Chern class of such a manifold has non-negative intersections with ample divisors. We also investigate under which conditions these positivities are strict, and answer a question of Yau.
In this note, we give a brief exposition on the differences and similarities between strictly nef and ample vector bundles, with particular focus on the circle of problems surrounding the geometry of projective manifolds with strictly nef bundles.
In this article, we give the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with klt singularities. Building on recent works of Spicer, Cascini - Spicer and Spicer - Svaldi, we then describe the birational geometry of rank two foliations with canonical singularities and canonical class of numerical dimension zero on complex projective threefolds.
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.
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 an $n$-dimensional smooth Fano complex variety of Picard number one. Assume that the VMRT at a general point of $X$ is smooth irreducible and non-degenerate (which holds if $X$ is covered by lines with index $ >(n+2)/2$). It is proven that $\dim \mathfrak{aut}(X) > n(n+1)/2$ if and only if $X$ is isomorphic to $\mathbb{P}^n, \mathbb{Q}^n$ or ${\rm Gr}(2,5)$. Furthermore, the equality $\dim \mathfrak{aut}(X) = n(n+1)/2$ holds only when $X$ is isomorphic to the 6-dimensional Lagrangian Grassmannian ${\rm Lag}(6)$ or a general hyperplane section of ${\rm Gr}(2,5)$.
We classify mildly singular Fano varieties $X$ such that $\mathrm{Nef}(X)=\mathrm{Psef}(X)$ and that the Picard number of $X$ is equal to the dimension of $X$ minus $1$.
We prove that there are at most two possibilities for the base of a Lagrangian fibration from a complex projective irreducible symplectic fourfold.