Jingcao Wu
Let $E$ be a holomorphic vector bundle endowed with a singular Hermitian metric $H$. In this paper, we develop the harmonic theory on $(E,H)$. Then we extend several canonical results of J. Kollár and K. Takegoshi to this situation. In the end, we generalise Nakano's vanishing theorem.
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In this paper we study numerical properties of quotients of holomorphic log-tensors.
$ω$-trace and Griffiths positivity for singular Hermitian metrics
本文研究了奇异 Hermite 度量的 Griffiths、$ω$-迹和 RC 等多种正性，证明了这些拟正性概念能诱导第 $0$ 阶上同调消没和有理连通性等性质。
In this note we study a positivity notion for the curvature of the Bismut connection; more precisely, we study the notion of \emph{Bismut-Griffiths-positivity} for complex Hermitian non-Kähler manifolds. Since the Kähler-Ricci flow preserves and regularizes the usual Griffiths positivity we investigate the behaviour of the Bismut-Griffiths-positivity under the action of the Hermitian curvature flows. In particular we study two concrete classes of examples, namely, linear Hopf manifolds and six-dimensional Calabi-Yau solvmanifolds with holomorphically-trivial canonical bundle. From these exampl…
Nakano Positivity of direct image sheaves of adjoint line bundles with mild singularities
本文将Berndtsson关于直接像层Nakano正性的经典结果推广到具有温和奇点的伴随线丛情形，拓展了该正性理论在奇异几何背景下的适用范围。
We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.
In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if $E$ is an ample vector bundle over a compact Kähler manifold $X$, $S^kE\ts \det E$ is both Nakano-positive and dual-Nakano-positive for any $k\geq 0$. Moreover, $H^{n,q}(X,S^kE\ts \det E)=H^{q,n}(X,S^kE\ts \det E)=0$ for any $q\geq 1$. In particular, if $(E,h)$ is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle $(S^k…
Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares, which was the first Hermitian analogue of Hilbert's seventeenth problem in the nondegenerate case. Later Catlin-D'Angelo generalized this positivstellensatz of Quillen to the case of Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds by proving the eventual positivity of an associated integral operator. The arguments of Catlin-D'Angelo, as well as that of a subsequent refinement by …
It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend to have positivity properties. From these considerations several issues arise: (i) In general for bundles that are semi-positive but not strictly positive; what further natural conditions lead to the existence of sections of its symmetric powers? (ii) In Hodge theory the Hodge met…
Uniform RC-positivity of tangent bundles
本文证明有理连通射影流形切丛存在一致RC正度量，回答Yang的问题并刻画有理连通性；同时证明该性质在爆破下保持，并在非Kähler情形下对Hopf和Kato曲面构造度量并给出分类。
This is a survey of results on positivity of vector bundles, inspired by the Brunn-Minkowski and Prékopa theorems. Applications to complex analysis, Kähler geometry and algebraic geometry are also discussed.
We prove that the $L^2$ metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers of an ample vector bundle tensorized by its determinant are Nakano positive, in particular Griffiths positive. This in turn gives vanishing theorems and an analytic characterization of ample vector bundles.
We prove that the invariant part, with respect to a compact group action satisfying certain condition, of the direct image of a Nakano positive Hermitian holomorphic vector bundle over a bounded pseudoconvex domain is Nakano positive. We also consider the action of the noncompact group $\mathbb{R}^m$ and get the same result for a family of tube domains, which leads to a new method to the matrix-valued Prekopa's theorem originally proved by Raufi. The two main ingredients in our method are Hörmander's $L^2$ theory of $\bar\partial$ and the recent work of Deng-Ning-Zhang-Zhou on characterization…
Given a vector bundle of arbitrary rank with ample determinant line bundle on a projective manifold, we propose a new elliptic system of differential equations of Hermitian-Yang-Mills type for the curvature tensor. The system is designed so that solutions provide Hermitian metrics with positive curvature in the sense of Griffiths-and even in the dual Nakano sense. As a consequence, if an existence result could be obtained for every ample vector bundle, the Griffiths conjecture on the equivalence between ampleness and positivity of vector bundles would be settled.
We study a singular Hermitian metric of a vector bundle. First, we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a Nadel-Nakano type vanishing theorem of a vector bundle with a singular Hermitian metric. We do not use an approximation technique of a singular Hermitian metric. We apply these theorems to a singular Hermitian metric induced by holomorphic sections and a big vector bundle, and we obtain a generaliz…
Universal vector bundles, push-forward formulae and positivity of characteristic forms
研究万能向量丛的推前公式及特征形式正性，通过计算Chern曲率验证推前公式的点wise成立，并提供微分形式层面的Jacobi-Trudi恒等式替代版本，部分验证Griffiths猜想。
Let $X$ be a complex smooth projective variety such that the exterior power of the tangent bundle $\bigwedge^{r} T_X$ is nef for some $1\leq r<\dim X$. We prove that, up to an étale cover, $X$ is a Fano fiber space over an Abelian variety. This gives generalizations of the structure theorem of varieties with nef tangent bundle by Demailly, Peternell and Schneider and that of varieties with nef $\bigwedge^{2} T_X$ by the author. Our result also gives an answer to a question raised by Li, Ou and Yang for varieties with strictly nef $\bigwedge^{r} T_X$ when $r < \dim X$.
Generalized Ueda Obstruction Classes and Applications to Non-semi-positivity of Line Bundles
引入沿复流形闭解析子簇的线束广义第一Ueda障碍类，允许奇异子簇和非平坦限制，通过Dolbeault分解和Chern曲率，类非零阻碍半正曲率光滑Hermite度量的存在，统一经典例子并给出新的nef非半正线束例子。
We develop some results on the positivity of direct image bundles in the particular case of a trivial fibration over a one-dimensional base. We also apply the results to study variations of Kahler metrics.
We study the positivity properties of Hermitian (or even Finsler) holomorphic vector bundles in terms of $L^p$-estimates of $\bar\partial$ and $L^p$-extensions of holomorphic objects. To this end, we introduce four conditions, called the optimal $L^p$-estimate condition, the multiple coarse $L^p$-estimate condition, the optimal $L^p$-extension condition, and the multiple coarse $L^p$-extension condition, for a Hermitian (or Finsler) vector bundle $(E,h)$. The main result of the present paper is to give a characterization of the Nakano positivity of $(E,h)$ via the optimal $L^2$-estimate condit…
We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.
In this note, we introduce a new type of positivity condition for the curvature of a Hermitian manifold, which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non-Kähler case. We derive a Bochner formula for closed $(1, 1)$-forms from which this condition appears naturally and prove that if a Hermitian manifold satisfy our positivity condition, then any class $α\in H^{1, 1}_{BC}(X)$ can be represented by a closed $(1, 1)$-form which is parallel with respect to the Bismut connection. Lastly, we show that such a curvature positivity condition holds on cert…
The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stabi…
We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.
The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold $(M,g,J)$ preserves many natural curvature positivity conditions. Following Wilking, for an $Ad\,{GL(T^{1,0}M)}$-invariant subset $S\subset End(T^{1,0}M)$ and a ncie function $F\colon End(T^{1,0}M)\to\mathbb R$ we construct a convex set of curvature operators $C(S,F)$, which is invariant under the HCF. Varying $S$ and $F$, we prove that the HCF preserves Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic orthogonal bisectional curvature, lower bounds on the second scalar …
We give a characterization of Nakano positivity of Riemannian flat vector bundles over bounded domains $D\subset\mathbb{R}^n$ in terms of solvability of the $d$ equation with certain good $L^2$ estimate condition. As an application, we give an alternative proof of the matrix-valued Prekopa's theorem that is originally proved by Raufi. Our methods are inspired by the recent works of Deng-Ning-Wang-Zhou on characterization of Nakano positivity of Hermitian holomorphic vector bundles and positivity of direct image sheaves associated to holomorphic fibrations.
Special K-stability and positivity of CM line bundles
该研究证明了参数化极大变异特殊K-稳定簇的真族上CM线丛的丰沛性，并应用该结论推导出曲线上一致绝热K-稳定klt-平凡纤维化粗模空间的任意真子空间具有射影性。
Positivity of Singular Hermitian Metrics for Holomorphic Vector Bundles
本文为全纯向量丛的奇异 Hermite 度量引入 Nakano 和 Demailly 正性，支持消没定理并证明尖锐的 Ohsawa-Takegoshi 型 $L^2$ 延拓定理及 Berndtsson 型正性定理。
Positivity of the third Chern form for Griffiths positive vector bundles
本文证明了格里斯次正向量丛的第三陈形式弱正性，解决了芬西开放问题的三维情况。
Singular Nakano positivity of direct image sheaves of adjoint bundles
本文研究伴随丛直像层的正性问题，证明装备Narasimhan-Simha度量的奇异埃尔米特线丛直像层具有奇异中野半正性，且证明无需依赖Griffiths正性理论。
Uniform RC-positivity of direct image bundles
本文研究了直接像丛的均匀RC-正性，证明了弱RC-正性条件下某些向量丛的RC-正性，并探讨了相关问题的解决方法。
We consider a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a line bundle $\mathcal{L}\to \mathcal{X}$. Given that $\mathcal{L}^{-1}$ carries a singular hermitian metric that has Poincaré type singularities along a relative snc divisor $\mathcal{D}$, the direct image $f_*(K_{\mathcal{X}/S}\otimes \mathcal{D} \otimes \mathcal{L})$ carries a smooth hermitian metric. In case $\mathcal{L}$ is relatively positive, we give an explicit formula for its curvature. The result applies to families of log-canonically polarized pairs. Moreover we show that it improves the general…
Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,H)$ on $X\times M$ (the action of $G$ on $X$ is trivial), such that the corresponding Quillen determinant line bundle $({\mathcal Q}, h_Q)$, which is a $G$--equivariant holomorphic Hermitian line bundle on $M$, is isomorphic to the given $G$--equivariant holomorphic Hermitian lin…
On a construction of hermitian metrics on holomorphic vector bundles
本文针对Griffiths关于全纯向量丛正曲率埃尔米特度量的猜想，证明无法通过函子性纤维构造从关联线丛O_{PE}(1)的正曲率度量得到该向量丛的正曲率度量，为该猜想的相关研究提供了关键结论。
This paper is a sequel to \cite{Berndtsson}. In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth Kähler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Nakano). Here we adress the question if the curvature is strictly positive when the Kodaira-Spencer class does not vanish. We prove that this is so provided the twisting line bundle is stricty positive along fibers, but not in general.
For a holomorphic vector bundle over a compact Kähler orbifold, the slope stability of the bundle is shown to be equivalent to the existence of a Hermitian-Einstein metric or to the properness of a certain functional introduced by Donaldson.
Quillen proved that, if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventual positive-definiteness of an associated integral operator. Their arguments involve asymptotic expansions of the Bergman kernel. The goal of this article is to give an elementary proof of the positive-definiteness of this integral operator.
$m$-Positivity and Regularisation
本研究基于$m$-多重次调和函数概念，研究复埃尔米特流形上的$m$-（半）正$(1,1)$-流与埃尔米特全纯线丛，证明了$\bar\bar\beta$方程的消灭定理、$L^2$估计及$m$-半正$(1,1)$-流的正则化定理。
Positivity of the canonical bundle and Hermitian metrics with quasi-negative holomorphic sectional curvature
本文证明紧致 Kähler 流形若存在关于 Chern 联络的拟负全纯截面曲率的 Hermitian 度量，则其典范丛丰沛，且该度量无需为 Kähler 度量，从而验证了 Yang-Zheng 拟负猜想的 Kähler 情形。
In the present paper, we study the properties of singular Nakano positivity of singular hermitian metrics on holomorphic vector bundles, and establish an optimal $L^2$ extension theorem for holomorphic vector bundles with singular hermitian metrics on weakly pseudoconvex Kähler manifolds. As applications, we give a necessary condition for the holding of the equality in optimal $L^2$ extension theorem, and present singular hermitian holomorphic vector bundle versions of some $L^2$ extension theorems with optimal estimate.
We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is described in a systematic way.
Positivity properties of the vector bundle Monge-Ampère equation
本文研究向量丛蒙日-安培方程的MA-正性保持性质，证明复曲面上秩二丛的半正解必为正，并指出高秩情形下该性质不总成立，同时为复三维流形上对称秩二丛构造连续路径并证明受限的正性保持足以保证开性。
The Horrocks-Mumford bundle $E$ is a famous stable complex vector bundle of rank 2 on 4-dimensional complex projective space. By construction, $E$ has a natural Hermitian metric $h_1$. On the other hand, stability implies the existence of a Hermitian-Einstein metric in $E$ which is unique up to a positive scalar. Now the obvious question is if $h_1$ is in fact the Hermitian-Einstein metric. In this note we indicate how to show by computation that this is not the case.
Let $X$ be a smooth projective variety over $\mathbb C$. We prove that a twisted Higgs vector bundle $(\calE\, ,θ)$ on $X$ admits an Einstein--Hermitian connection if and only if $(\calE\, ,θ)$ is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved by S. Wang in \cite{Wa}. Our approach is simpler.
In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations which can be used to test various K-stability notions of Fano varieties.
Let E_G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.
Given a smooth complex projective variety X and a smooth divisor D on X, we prove the existence of Hermitian-Einstein connections, with respect to a Poincaré-type metric on X - D, on polystable parabolic principal Higgs bundles with parabolic structure over D, satisfying certain conditions on its restriction to D.
Positivity of Schur forms for strongly decomposably positive vector bundles
本文定义两类强于 Finski 可分解正性的强可分解正性，给出判别准则并证明相应 Schur 形式的弱正性与正性，从而肯定回答 Griffiths 问题并代数重证 Nakano 情形。
We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space.
In this paper, we study the dimension of cohomology of semipositive line bundles over Hermitian manifolds, and obtain an asymptotic estimate for the dimension of the space of harmonic $(0,q)$-forms with values in high tensor powers of a semipositive line bundle when the fundamental estimate holds. As applications, we estimate the dimension of cohomology of semipositive line bundles on $q$-convex manifolds, pseudo-convex domains, weakly $1$-complete manifolds and complete manifolds. We also obtain the estimate of cohomology on compact manifolds with semipositive line bundles endowed with a Herm…
Let X be a smooth complex projective curve and S a finite subset of X. We show that an orthogonal or symplectic parabolic Higgs bundle on X with parabolic structure over S admits a Hermitian-Einstein connection if and only if it is polystable.
We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be written in terms of the mean curvature of the Hitchin-Simpson connection. We also study some properties of the evolution equation associated to that functional. Next, we study the problem of the existence of approximate Hermitian-Yang-Mills structure and its relation with the algeb…
Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal $G$-bundle $E_G$ over $M$ admits a Hermitian-Einstein structure if and only if $E_G$ is polystable. A polystable flat principal $G$--bundle over $M$ admits a unique Hermitian-Einstein connection. We also prove the existence and uniqueness of a Harder-Narasimhan filtration for …
We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern numbers of such a holomorphic vector bundle are nonnegative and can be bounded below and above respectively by two special Chern numbers. As applications, we obtain a family of new results on compact connected complex manifolds which are homogeneous or can be holomorphically imm…
On positivity of CM line bundles on the moduli space of klt good minimal models with $κ=1$
研究了具有κ=1的klt良好极小模型模空间上CM线包的正性，证明了其正则化后为 ample，并构造了数值等价类的模空间以克服先前方法的局限性。
We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that K_X \otimes [D] is ample. It turns out that the degree of a torsion-free coherent sheaf on X with respect to the polarization K_X \otimes [D] coincides with the degree with respect to the complete Kähler-Einstein metric g_{X \setminus D} on X \setminus D. For stable holomorphic vector bundles, we prove the existence of a…
In this paper, we show that, for every Hermitian vector bundle over a compact Kaehler Einstein manifold, if the projection is biharmonic, then it is harmonic.
On characteristic forms of positive vector bundles, mixed discriminants and pushforward identities
本文证明Nakano和对偶Nakano正向量丛的Schur多项式在微分形式下为正，并通过混合判别式的正性建立Griffiths正向量丛的Schur多项式弱正性，提供Fulton-Lazarsfeld不等式的微分几何版本。
Hermitian Yang-Mills connections on pullback bundles
研究全纯浸没全空间上拉回丛的埃尔米特-杨-米尔斯联络，给出严格半稳定向量丛拉回承载该联络的充要判据，结合Donaldson-Uhlenbeck-Yau对应证明稳定/不稳定丛拉回保持原性质并解决半稳定情形。
In this paper, using Donaldson's heat flow, we show that the semi-stability of a Higgs bundle over a compact Kähler manifold implies the existence of approximate Hermitian-Einstein structure on the Higgs bundle.
Positively curved Finsler metrics on vector bundles III
本文证明了小林正向量丛的商丛和张量积仍为小林正，从而加强了丰富性、格里菲斯正性与小林正性之间的猜想等价关系。
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