Xueyuan Wan's research while affiliated with Chongqing University of Technology and other places
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Publications (42)
In this paper, we define two types of strongly decomposable positivity, which are the generalizations of (dual) Nakano positivity, and are stronger than the decomposable positivity introduced by S. Finski. We give the criteria of strongly decomposable positivity of type I and type II, and show that the Schur forms of a strongly decomposably positiv...
Let ( M, g) be a completely connected n-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies R g ≥ − n( n − 1), and let [Formula: see text] be an asymptotically hyperbolic end. We prove that the mass functional of the end [Formula: see text] is timelike future-directed or zero. Moreover, it vanishes if and...
In this paper, we consider a special relative Kähler fibration that satisfies a homogenous Monge–Ampère equation, which is called a Monge–Ampère fibration. There exist two canonical types of generalized Weil–Petersson metrics on the base complex manifold of the fibration. For the second generalized Weil–Petersson metric, we obtain an explicit curva...
For a holomorphic vector bundle $E$ over a Hermitian manifold $M$ there are two important notions of curvature positivity, the Griffiths positivity and Nakano positivity. We study the consequence of these positivities and the relevant estimates. If $E$ is Griffiths negative over K\"ahler manifold, then there is a K\"ahler metric on its total space...
In this paper, we first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by Hrmander’s $L^2$ estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new vanishing theorems for sheaves of logarithmic differential forms on compact Kähler manifolds with simple normal...
Let $(M,g)$ be a complete connected $n$-dimensional Riemannian spin manifold without boundary such that the scalar curvature satisfies $R_g\geq -n(n-1)$ and $\mathcal{E}\subset M$ be an asymptotically hyperbolic end, we prove that the mass functional of the end $\mathcal{E}$ is timelike future-directed or zero. Moreover, it vanishes if and only if...
We generalize a band width estimate of Gromov to CMC initial data sets. We give three independent proofs: via the stability of a hypersurface with prescribed null expansion, via a perturbation of the spacetime harmonic function and via the Dirac operator.
This paper deals with the representations of the fundamental groups of compact surfaces with boundary into classical simple Lie groups of Hermitian type. We relate work on the signature of the associated local systems, due to Meyer and Atiyah, to Burger-Iozzi-Wienhard's Toledo invariant. To measure the difference, we extend Atiyah-Patodi-Singer's r...
Let $\pi:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relative ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $\pi_*(L^k+K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$, for large $k$. As an application we prov...
In this paper, by using Atiyah-Patodi-Singer index theorem, we obtain a formula for the signature of a flat symplectic vector bundle over a surface with boundary, which is related to the Toledo invariant of a surface group representation in the real symplectic group and the Rho invariant on the boundary. As an application, we obtain a Milnor-Wood t...
By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira–Spencer’s local stability theorem of Kähler structures. We also obtai...
In this paper, we introduce the associated geodesic-Einstein flow for a relative ample line bundle L over the total space \(\mathcal {X}\) of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair \((\mathcal {X}, L)\) is nonlinear semistable if the associated Donaldson type functional is bounded fro...
In the paper, we consider the harmonic maps between surfaces $\Sigma$ and $S$ in the homotopy class of a (branched) covering map $u_0$. We prove the uniqueness of critical points of energy function and the injectivity of Hopf differential if $u_0$ is a covering map. On the other hand, if $u_0$ is a branched covering, we show that the uniqueness of...
We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier in [1] using Toeplitz operator...
We consider harmonic maps u(z):Xz→N in a fixed homotopy class from Riemann surfaces Xz of genus g≥2 varying in the Teichmüller space T to a Riemannian manifold N with non-positive Hermitian sectional curvature. The energy function E(z)=E(u(z)) can be viewed as a function on T and we study its first and the second variations. We prove that the recip...
For a fixed smooth map $u_0$ between two Riemann surfaces $\Sigma$ and $S$ with non-zero degree, we consider the energy function on Teichm\"uller space $\mc{T}$ of $\Sigma$ that assigns to a complex structure $t\in \mc{T}$ on $\Sigma$ the energy of the harmonic map $u_t:\Sigma_t:=(\Sigma,t) \to S$ homotopic to $u_0$. We prove that the energy functi...
Let X be a compact Kähler manifold and D be a simple normal crossing divisor. If D is the support of some effective q-ample divisor, we show Hi(X,ΩXj(logD))=0,fori+j>n+q.
We start from a finite dimensional Higgs bundle description of a result of Burns on negative curvature property of the space of complex structures, then we apply the corresponding infinite dimensional Higgs bundle picture and obtain a precise curvature formula of a Weil--Petersson type metric for general relative K\"ahler fibrations. In particular,...
By use of a natural extension map and a power series method, we obtain a local stability theorem for $p$ -Kähler structures with the $(p,p+1)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma under small differentiable deformations.
Let $QF(S)$ be the quasifuchsian space of a closed surface $S$ of genus $g\geq 2$. We construct a new mapping class group invariant K\"ahler metric on $QF(S)$. It is an extension of the Weil-Petersson metric onthe Teichm\"uller space $\mathcal T(S)\subset QF(S)$. We also calculate its curvature and prove some negativity for the curvature along the...
We consider harmonic maps$u(z): \mathcal{X}_z\to N$ in a fixed homotopy class from Riemann surfaces $\mathcal{X}_z$ of genus $g\geq 2$ varying in the Teichm\"u{}ller space $\mathcal T$ to a Riemannian manifold $N$ with non-positive Hermitian sectional curvature. The energy function $E(z)=E(u(z))$ can be viewed as a function on $\mathcal T$ and we s...
We first solve an open problem of S. Kobayashi in 1975 by proving that a holomorphic vector bundle on compact complex manifold is ample if and only if it admits a strongly pseudoconvex complex Finsler metric with positive Kobayashi curvature. We derive several applications which include: (1). the proof of the famous Lefschetz type theorem for gener...
Let $\pi:\mc{X}\to \mc{T}$ be Teichm\"uller curve over Teichm\"uller space $\mc{T}$, such that the fiber $\mc{X}_z=\pi^{-1}(z)$ is exactly the Riemann surface given by the complex structure $z\in \mc{T}$. For a fixed Riemannian manifold $M$ and a continuous map $u_0: M\to \mc{X}_{z_0}$, let $E(z)$ denote the energy function of the harmonic map $u(z...
Let $\pi:\mathcal{X}\to M$ be a holomorphic fibration over a compact complex manifold $M$ with compact fibers, $L$ be a relative ample line bundle over $\mathcal{X}$. In this paper, we prove that the pair $(\mathcal{X}, L)$ is nonlinear semistable if the Donaldson type functional $\mathcal{L}(\cdot,\psi)$ is bounded from below and the long time exi...
Let $X$ be a compact K\"ahler 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,\Omega^p_X(\log D))=0,\quad \text{for}\quad p+q>n+k.
In this paper, we solve a logarithmic ¯ ∂-equation on a compact Kähler manifold associated to a smooth divisor by using the cyclic covering trick. As applications, we discuss the closedness of logarithmic forms, injectivity theorems and obtain a kind of degeneration of spectral sequence at E1, and we also prove that the pair (X, D) has unobstructed...
In this paper, we solve a logarithmic $\bar{\partial}$-equation on a compact K\"ahler manifold associated to a smooth divisor by using the cyclic covering trick. As applications, we discuss the closedness of logarithmic forms, injectivity theorems and obtain a kind of degeneration of spectral sequence at $E_1$, and we also prove that the pair $(X,D...
We consider the relative canonical line bundle $K_{\mathcal{X}/\mathcal{T}}$ and a relatively ample line bundle $(L, e^{-\phi})$ over the total space $ \mathcal{X}\to \mathcal{T}$ of fibration over the Teichm\"uller space by Riemann surfaces. We consider the case when the induced metric $\sqrt{-1}\partial\bar{\partial}\phi|_{\mathcal{X}_y}$ on $\ma...
In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and K\"ahler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eigenvector assumption. As appli...
By use of a natural extension map and a power series method, we obtain a local stability theorem for p-Kahler structures with the (p,p+1)-th mild ddbar-lemma under small differentiable deformations.
In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by
introducing a Donaldson type functional on the space $F^+(E)$ of strongly
pseudo-convex complex Finsler metrics on $E$ -- a holomorphic vector bundle
over a closed K\"ahler manifold $M$. This Donaldson type functional is a
generalization in the complex Finsler geometry setting of...
In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional to show...
In this short note, using Siu-Yau's method [14], we give a new proof that any n-dimensional compact Kähler manifold with positive orthogonal bisectional curvature must be biholomorphic to Pn.
In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we show a Schwarz Lemma from a compete Riemann manifold to a complex Finsler manifold. We also prove that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identically zero holomorphic sectional curv...
We present a new method to solve certain ∂ ¯ \bar \partial -equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a ∂ ∂ ¯ \partial \bar \partial -lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic...
In this paper, we first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by H\"{o}rmander's $L^2$-estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new vanishing theorems for sheaves of logarithmic differential forms on compact K\"ahler manifolds with simple...
By use of a natural map introduced recently by the first and third authors from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the small differentiable deformation of this manifold, we will give a power series proof for Kodaira-Spencer's local stability theorem of K\"ahler structures. We also obt...
In this paper, we present two kinds of total Chern forms $c(E,G)$ and
$\mathcal{C}(E,G)$ as well as a total Segre form $s(E,G)$ of a holomorphic
Finsler vector bundle $\pi:(E,G)\to M$ expressed by the Finsler metric $G$,
which answers a question of J. Faran (\cite{Faran}) to some extent. As some
applications, we show that the signed Segre forms $(-...
By using analytic method, we prove that there exist rational curves on
compact Hermitian manifolds with positive holomorphic bisectional curvature. It
confirms a question of S.-T. Yau. It is well-known that Mori proved in
\cite{Mori79} that every compact complex manifold $N$ with $c_1(N)>0$ contains
at least one rational curve. However, as a border...
Citations
... Recently, Cecchini and Zeidler [CZ21b] addressed Conjecture 1.1 by using Callias operators beyond Witten's spinorial method. Subsequently, the cases of asymptotically hyperbolic ends and asymptotically flat initial data ends were respectively extended by Chai-Wan [CW22] and Cecchini-Lesourd-Zeidler [CLZ23] using similar techniques. We also note that a proof of Conjecture 1.1 for n ≤ 7 based on the idea of density theorem and Lohkamp's compactification was found by Zhu [Zhu22]. ...
... [Dem93], [Nad89], [Wat22b]) involving multiplier ideal sheaves. Recently, Huang, Liu, Wan and Yang proved the following logarithmic vanishing theorem in [HLWY22]: ...
... Actually, the result we show is a discrete analogue of a result by Yamada; he proved that if the domain is a general closed Riemannian manifold M and f : M → (S, G) is a continuous map to a closed hyperbolic surface inducing a surjective homomorphism between fundamental groups, then the Dirichlet energy functional evaluated at a (unique) harmonic map homotopic to f is proper and strictly convex on the Teichmüller space with respect to the Weil-Petersson metric [Yam99, Theorem 3.2.1] (see also expositions [Yam14] and [Yam17], and the recent paper [KWZ18]). We mostly follow the strategy by Yamada to show the convexity and the properness of energy functional E C ; but we give an independent proof for the convexity that enables us to establish an explicit Hessian formula of E C (Theorem 3.9). ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/279438236373004-1443634675841_Q64/Inkang-Kim.jpg)
... For any ( p, q) ∈ Z 2 , consider the following maps [4,5,21,54,55], see [1] for a unified discussion. 7 We have been informed by the anonymous referee that by using the structure theory of double complexes one can show ∂ p,q A,∂(ker ∂) = 0 holds for all ( p, q) ∈ Z 2 is equivalent to the ∂∂-lemma on X , see [30,49,52,63] for more information. ...
Reference: Deformations of Dolbeault cohomology classes
![[object Object]](https://i1.rgstatic.net/ii/profile.image/368826068815876-1464946395739_Q64/Sheng-Rao-2.jpg)
... Toledo [13] extended Tromba's analysis to the case where N is a Riemannian manifold satisfying (1.1). Later, Kim et al. [4] recovered Toledo's results using the approach of fiber integration. The case where M t are Kähler-Einstein manifolds of negative Ricci curvature was treated in [14]. ...
... . Moreover, if the divisor is SNC, we can calculate each gr F p DR X (j * M) and deduce the logarithmic Akizuki-Nakano vanishing theorem for weakly ample divisor as in [LWY19]. ...
... Recently, extensive works have been done on the topics of blow-up formulae and the ∂∂ -lemma. We refer the readers to [7,8,16,24,25,[31][32][33][34][35][36]38,42,43,47,52,56,58] and the references therein for some recent results. ...
... In this section, we will recall some basic definitions and facts on geodesic-Einstein metrics of a relative ample line bundle over holomorphic fibration. For more details one may refer to [19,36]. ...
... In [27], B. Shen and Y. B. Shen obtained a Schwarz lemma for holomorphic mappings between two compact strongly pseudoconvex complex Finsler manifolds. In [33], X. Y. Wan obtained a Schwarz lemma for holomorphic mappings from a complete Riemann surface endowed with a conformal metric into a complex manifold endowed with a strongly pseudoconvex complex Finsler metric. In [24], J. Nie and C. P. Zhong obtained a Schwarz lemma for holomorphic mappings from a complete Kähler manifold into a strongly pseudoconvex complex Finsler manifold with some curvatures assumptions, and then in [25], J. Nie and C. P. Zhong obtained a Schwarz lemma on a complex manifold which admits a weakly Kähler-Finsler metric with some assumptions of the radial flag curvatures and holomorphic sectional curvatures. ...
... Other generalizations of Theorem 1.1 include generalizations to compact Hermitian manifolds ([Buc99; LY87]), and very recent work of Feng-Liu-Wan [FLW18], which expanded Theorem 1.1 to include the existence of Finsler-Einstein metrics. ...
