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Jixiang Fu

Publications and source records attributed to Jixiang Fu.

At least 19 recordsLinked to original sources
arXiv2026

Deformations of K\"{a}hler and Balanced Hyperbolicity

We study the deformation stability of K\"ahler and balanced hyperbolicity. Balanced hyperbolicity is not open in general: in every complex dimension $N\geq5$ we construct a one-parameter family with balanced hyperbolic central fibre and non-balanced nearby fibres. For positive results, we develop three complementary mechanisms. A finite-dimensional moving-intersection framework tracks $\widetilde d$-bounded de Rham classes through moving pure-type loci; its Aeppli and Dolbeault realizations yield continuation, transversality, and positivity criteria for balanced and K\"ahler hyperbolicity. A topological mechanism combines the graded-ideal property of hyperbolic cohomology with the hard Lefschetz theorem to obtain saturation and higher-power propagation results. Finally, on the universal cover, we reduce the passage from a bounded $(\partial+\bar\partial)$-potential to a bounded $d$-primitive to a single bounded top-row $\partial$-equation, and package the dependence on the potential into a canonical quotient obstruction. Together, these viewpoints yield a range of deformation stability results.

math.DG↗
arXiv2026

Limiting Behavior of a Class of Hermitian Yang--Mills Metrics, II: Exponential Approximation

This paper is a sequel to [9], where the first-named author constructed a family of approximate Hermitian Yang--Mills metrics $H_{0,\epsilon}$ on stable rank-two holomorphic vector bundles arising from double spectral covers over the product of two one-dimensional complex tori. We prove that these approximate metrics give an all-order, exponentially accurate asymptotic description of the exact Hermitian Yang--Mills metrics in the large K\"ahler limit. More precisely, the mean curvature of $H_{0,\epsilon}$ decays exponentially in every $C^k$-norm. Moreover, if $H_{1,\epsilon}$ denotes the exact Hermitian Yang--Mills metric and \[ H_\epsilon=H_{0,\epsilon}^{-1}H_{1,\epsilon}, \] then, after normalization, for every nonnegative integer $k$, there exist positive constants $C_k$ and $c_k$ such that \[ \|H_\epsilon- Id\|_{C^k}\leq C_k e^{-\frac{c_{k}}{\epsilon}}. \] The main analytic difficulty lies in the global $C^0$-comparison. Obtaining $C^0$-estimates for the coupled nonlinear Hermitian Yang--Mills system is intrinsically difficult; moreover, the equation controls only the contraction of the curvature, and hence only certain combinations of second derivatives, whereas one needs global control of the full matrix-valued metric.

math.DG↗
arXiv2026

Boundary Cases of the $J$-Equation: Divisorial Rigidity and a Global $C^0$ Estimate

We study two boundary cases of the stability condition for the $J$-equation. First, under the $J$-semistable condition, we show that the destabilizing prime divisors form an exceptional family in the sense of Boucksom, with a uniform numerical gap away from them. The related modified nef estimate can remove the $J$-big assumption in Liu's work~\cite{Liu2026Boundary}. Second, under the smooth boundary cone condition, we obtain a uniform global $C^0$ estimate for solutions of the approximating twisted $J$-equations. As a consequence, we construct a bounded-potential Bedford--Taylor solution of the $J$-equation, which is smooth outside the destabilizing prime divisors.

math.DG↗
arXiv2026

A Numerical Criterion for the 2-Hessian Equation on Compact K\"ahler Manifolds

We show that a Nakai--Moishezon-type criterion associated with the complex $2$-Hessian equation produces a Gauduchon class. In complex dimension three, this numerical criterion is equivalent to the existence of a smooth $2$-admissible representative and hence to the solvability of the $2$-Hessian equation. As consequences of these results, we prove the corresponding conjectures of Murakami for the complex Hessian equation and of Sz\'ekelyhidi for the Hessian quotient equation in dimension three. We also establish a boundary version of the above results.

math.DG↗
arXiv2026

Limiting behavior of a class of Hermitian Yang-Mills metrics, II: exponential decay

In this note, the geometric set-up, the rank two bundle, the local HYM ansatz, and the global gluing construction are the same as in the preceding work \cite{Fu}. The new point is an exponential estimate for the radial ordinary differential equation obtained near each branch point. If $u_\epsilon$ denotes the local radial solution and $\frac12\ln r$ the singular limiting solution, then for every integer $k\ge0$, there exist positive constants $C_k$ and $c_k$ such that \[ \big\| u_\epsilon - \frac12\ln r \big\|_{C^k([r_0,2r_0])} \le C_k e^{-c_k/\epsilon}. \] Consequently, all results of the preceding paper can be refined.

math.DG↗
arXiv2025

The Critical LYZ Equation in K\"ahler Geometry

We establish the existence of smooth solutions for the LYZ equation at the critical phase $\theta =(n-2)\frac{\pi}{2}$, thereby solving the critical case of a problem posed by Collins-Jacob-Yau and Li concerning the solvability for phase $\theta \leq (n-2)\frac{\pi}{2}$. As applications, we solve the 3D Hessian equation $\sigma_2 = 1$ and the 4D Hessian quotient equation $\sigma_3 = \sigma_1$ under weaker assumptions than previously required.

math.DG↗
arXiv2023

The Monge-Amp\`{e}re equation for $(n-1)$-quaternionic PSH functions on a hyperK\"{a}hler manifold

We prove the existence of unique smooth solutions to the quaternionic Monge-Amp\`{e}re equation for $(n-1)$-quaternionic plurisubharmonic functions on a hyperK\"{a}hler manifold and thus obtain solutions for the quaternionic form type equation. We derive $C^0$ estimate by establishing a Cherrier-type inequality as in Tosatti and Weinkove [22]. By adopting the approach of Dinew and Sroka [9] to our context, we obtain $C^1$ and $C^2$ estimates without assuming the flatness of underlying hyperK\"{a}hler metric comparing to previous results [14].

math.DG↗
arXiv2022

On the global generation of higher direct images of pluricanonical bundles

Given a fibration $f$ between two projective manifolds $X$ and $Y$, we discuss the effective generation of the higher direct images $R^{i}f_{\ast}(K^{m}_{X})$, where $K^{m}_{X}$ is the $m$-th tensor power of the canonical bundle of $X$. In particular, we answer two questions posed by Popa--Schnell in [PS14].

math.AG↗
arXiv2021

A new flow solving the LYZ equation in K\"ahler geometry

We introduced a new flow to the LYZ equation on a compact K\"ahler manifold. We first show the existence of the longtime solution of the flow. We then show that under the Collins-Jacob-Yau's condition on the subsolution, the longtime solution converges to the solution of the LYZ equation, which was solved by Collins-Jacob-Yau [5] by the continuity method. Moreover, as an application of the flow, we show that on a compact K\"ahler surface, if there exists a semi-subsolution of the LYZ equation, then our flow converges smoothly to a singular solution to the LYZ equation away from a finite number of curves of negative self-intersection. Such a solution can be viewed as a boundary point of the moduli space of the LYZ solutions for a given K\"ahler metric.

math.DG↗
arXiv2019

Extension formulae on almost complex manifolds

We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae. As applications, we study $(n,0)$-forms, the $(n,0)$-Dolbeault cohomology group and $(n,q)$-forms on almost complex manifolds.

math.DG↗
arXiv2019

Scalar curvatures in almost Hermitian geometry and some applications

On an almost Hermitian manifold, we have two Hermitian scalar curvatures with respect to any canonical Hermitian connection defined by P. Gauduchon. Explicit formulas of these two Hermitian scalar curvatures are obtained in terms of Riemannian scalar curvature, norms of decompositions of covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we get some inequalities of various total scalar curvatures and some characterization results of the K\"{a}hler metric, balanced metric, locally conformally K\"{a}hler metric and the k-Gauduchon metric. As corollaries, we show some results related to a problem given by Lejmi-Upmeier \cite{LeU} and a conjecture given by Angella-Otal-Ugarte-Villacampa \cite{AOUV}.

math.DG↗
arXiv2018

Complex Balanced Spaces

In this paper, the concept of balanced manifolds is generalized to reduced complex spaces: the class B and balanced spaces. Compared with the case of Kahlerian, the class B is similar to the Fujiki class C and the balanced space is similar to the Kahler space. Some properties about these complex spaces are obtained, and the relations between the balanced spaces and the class B are studied.

math.CV↗
arXiv2018

Twistor geometry of Hermitian surfaces induced by canonical connections

In this paper, following the constructions of N. R. O'Brian, J. H. Rawnsley and I. Vaisman, we define four almost Hermitian structures (up to conjugation) on the twistor space of a Hermitian surface by using canonical connections, including the Lichnerowicz connection and the Chern connection. We also study the relations between the natural geometry of the twistor spaces and the geometry of Hermitian surfaces.

math.DG↗
arXiv2016

On complete constant scalar curvature K\"ahler metrics with Poincar\'e-Mok-Yau asymptotic property

Let $X$ be a compact K\"ahler manifold and $S$ a subvariety of $X$ with higher co-dimension. The aim is to study complete constant scalar curvature K\"ahler metrics on non-compact K\"ahler manifold $X-S$ with Poincar\'e--Mok--Yau asymptotic property (see Definition \ref{def}). In this paper, the methods of Calabi's ansatz and the moment construction are used to provide some special examples of such metrics.

math.DG↗
arXiv2014

Teissier's problem on proportionality of nef and big classes over a compact Kähler manifold

We solve Teissier's proportionality problem for transcendental nef classes over a compact Kähler manifold which says that the equality in the Khovanskii-Teissier inequalities hold for two nef and big classes if and only if the two classes are proportional. This result recovers the previous one of Boucksom-Favre-Jonsson for the case of nef and big line bundles over a (complex) projective algebraic manifold.

math.AG↗