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Fusheng Deng

Publications and source records attributed to Fusheng Deng.

At least 19 recordsLinked to original sources
arXiv2026

Period bounds for Bergman kernel variation and the Suita defect

The periods of holomorphic differentials give an explicit lower bound for the curvature of Bergman evaluation matrices. We prove this bound along the Green sublevels of a smooth plane domain. For scalar evaluation at the Green pole, equality at one regular level holds precisely when that sublevel is simply connected. For complete jets at the Green pole, the matrices have continuous first derivatives, including at critical levels, and their logarithmic determinant defects are integrals of nonnegative curvature densities. A further estimate bounds the accumulated period energy by the determinant defect, with a constant independent of the jet order. On a multiply connected domain, all these defects vanish up to the first critical Green value and become strictly positive above it.

math.CV↗
arXiv2026

Invariance of plurigenera for Kähler families with nef canonical bundles

Let $p:X\rightarrowΔ$ be a proper holomorphic submersion from a Kähler manifold $X$. We prove that, if $K_{X_t}$ is nef for every $t\inΔ$, then $\dim_{\mathbb C}H^0(X_t,mK_{X_t})$ is independent of $t$ for every integer $m\geq1$. This gives a partial answer to Siu's conjecture on the invariance of plurigenera for Kähler families.

math.CV↗
arXiv2026

An Exact Spherical-Solid Mean Formula and Vanishing Criteria for Lelong Numbers

We establish an exact additive formula expressing the Lelong number of a plurisubharmonic function in terms of the limiting difference between its spherical and solid means. Building on this identity, we derive several equivalent criteria for the vanishing of the Lelong number, formulated in terms of fixed-scale increments and the asymptotic agreement of radial maxima, spherical and solid means, and radial regularizations. We also give a pointwise $L^1$ mean-oscillation formulation of the VMO characterization for plurisubharmonic functions with zero Lelong number, together with a streamlined proof of the corresponding global result of Biard and Wu. Finally, we establish a nonlinear characterization in terms of the relative extremal depth and the Bedford-Taylor capacity of deep sublevel sets.

math.CV↗
arXiv2026

Dirichlet Green kernel estimates and Sobolev-type inequalities for twisted differential forms

We study the full-trace Dirichlet realization of the Dolbeault Laplacian on differential forms with values in a Hermitian holomorphic vector bundle over a relatively compact smooth domain in a Kähler manifold. We prove global Green kernel estimates that are uniform up to the boundary, including one- and two-boundary-factor bounds and estimates for the \(\bar\partial\)- and \(\bar\partial^*\)-derivatives. These estimates yield Sobolev-type inequalities with boundary terms. In real dimension two, the first-order bound has a logarithmic loss. In top antiholomorphic degree, we obtain quantitative \(L^r\)-to-\(L^k\) solvability for \(\bar\partial\) without pseudoconvexity. Analogous results hold for the twisted de Rham complex of a flat metric connection on a compact Riemannian manifold with smooth boundary.

math.AP↗
arXiv2026

Holomorphic Approximation for Real Diffeomorphism Groups

We show that every diffeomorphism of $\mathbb{R}^n$ for $n \ge 2$ can be approximated by an automorphism of $\mathbb{C}^n$ in the Whitney $C^k$-topology for any positive integer $k$ using the notion of density property. More precisely we find sufficient conditions for this holomorphic approximation of diffeomorphisms to hold and prove that the split real forms of most linear algebraic groups satisfy these conditions. In the same manner we also show holomorphic approximations of volume-preserving diffeomorphisms for the split real forms of linear algebraic groups equipped with the left-invariant volume form.

math.CV↗
arXiv2025

Estimates of heat kernels and Sobolev-type inequalities for twisted differential forms on compact Kähler manifolds

The main goal of this paper is to generalize the Sobolev-type inequalities given by Guo-Phong-Song-Sturm and Guedj-Tô from the case of functions to the framework of twisted differential forms. To this end, we establish certain estimates of heat kernels for differential forms with values in holomorphic vector bundles over compact Kähler manifolds. As applications of these estimates, we also prove a vanishing theorem and give certain $L^{q,p}$-estimates for the $\bar\partial$-operator on twisted differential forms.

math.CV↗
arXiv2025

$\bar\partial$ Sobolev-type inequality and an improved $L^2$-estimate of $\bar\partial$ on bounded strictly pseudoconvex domains

We prove several Sobolev-type inequalities related to the $\bar\partial$-operator on bounded domains in $\mathbb{C}^n$, which can be viewed as a $\bar\partial$-version of the classical Sobolev inequality and its various generalizations, and apply them to derive a generalization of the Sobolev Inequality with Trace in $\mathbb{R}^n$. As applications to complex analysis, we get an integral form of Maximum Modulus Principle for holomorphic functions, and an improvement of Hörmander's $L^2$-estimate for $\bar\partial$ on bounded strictly pseudoconvex domains.

math.CV↗
arXiv2025

Log truncated threshold and zero mass conjecture

For plurisubharmonic functions $φ$ and $ψ$ lying in the Cegrell class of $\mathbb{B}^n$ and $\mathbb{B}^m$ respectively such that the Lelong number of $φ$ at the origin vanishes, we show that the mass of the origin with respect to the measure $(dd^c\max\{φ(z), ψ(Az)\})^n$ on $\mathbb{C}^n$ is zero for $A\in \mbox{Hom}(\mathbb{C}^n,\mathbb{C}^m)=\mathbb{C}^{nm}$ outside a pluripolar set. For a plurisubharmonic function $φ$ near the origin in $\mathbb{C}^n$, we introduce a new concept coined the log truncated threshold of $φ$ at $0$ which reflects a singular property of $φ$ via a log function near the origin (denoted by $lt(φ,0)$) and derive an optimal estimate of the residual Monge-Ampère mass of $φ$ at $0$ in terms of its higher order Lelong numbers $ν_j(φ)$ at $0$ for $1\leq j\leq n-1$, in the case that $lt(φ,0)<\infty$. These results provide a new approach to the zero mass conjecture of Guedj and Rashkovskii, and unify and strengthen well-known results about this conjecture.

math.CV↗
arXiv2024

Uniform estimates of Green functions and Sobolev-type inequalities on real and complex manifolds

We prove certain $L^p$ Sobolev-type and Poincaré-type inequalities for functions on real and complex manifolds for the gradient operator $\nabla$, the Laplace operator $Δ$, and the operator $\bar\partial$. Integral representations for functions are key to get such inequalities. The proofs of the main results involves certain uniform estimates for the Green functions and their gradients on Riemannian manifolds, which are also established in the present work.

math.CV↗
arXiv2024

Uniqueness of irreducible desingularization of singularities associated to negative vector bundles

We prove that the irreducible desingularization of a singularity given by the Grauert blow down of a negative holomorphic vector bundle over a compact complex manifold is unique up to isomorphism, and as an application, we show that two negative line bundles over compact complex manifolds are isomorphic if and only if their Grauert blow downs have isomorphic germs near the singularities. We also show that there is a unique way to modify a submanifold of a complex manifold to a hypersurface, namely, the blow up of the ambient manifold along the submanifold.

math.AG↗
arXiv2024

Multiplicites and modifications, and singularities associated to blowing down negative vector bundles

We first present the mixed Hilbert-Samuel multiplicities of analytic local rings over \mathbb{C} as generalized Lelong numbers and further represent them as intersection numbers in the context of modifications. As applications, we give estimates or an exact formula for the multiplicities of isolated singularities that given by the Grauert blow-downs of negative holomorphic vector bundles.

math.CV↗
arXiv2023

Approximation and extension of Hermitian metrics on holomorphic vector bundles over Stein manifolds

We show that a singular Hermitian metric on a holomorphic vector bundle over a Stein manifold which is negative in the sense of Griffiths (resp. Nakano) can be approximated by a sequence of smooth Hermitian metrics with the same curvature negativity. We also show that a smooth Hermitian metric on a holomorphic vector bundle over a Stein manifold restricted to a submanifold which is negative in the sense of Griffiths (resp. Nakano) can be extended to the whole bundle with the same curvature negativity.

math.CV↗
arXiv2022

Curvature strict positivity of direct image bundles associated to pseudoconvex families of domains

We consider the curvature strict positivity of the direct image bundle associated to a pseudoconvex family of bounded domains. The main result is that the curvature of the direct image bundle associated to a strictly pseudoconvex family of bounded circular domains or Reinhardut domains are strictly positive in the sense of Nakano, even if the weight functions are not strictly plurisubharmonic. This result gives a new geometric insight about the property of strict pseudoconvexity, and has some applications in complex analysis and convex analysis. We investigate that the main result implies a remarkable result of Berndtsson which states that, for an ample vector bundle $E$ over a compact complex manifold $X$ and any $k\geq 0$, the bundle $S^kE\otimes\det E$ admits a Hermitian metric whose curvature is strictly positive in the sense of Nakano, where $S^kE$ is the $k$-th symmetric product of $E$. The two main ingredients in the argument of the main theorems are Berndtsson's estimate of the lower bound of curvature of direct image bundles and Deng-Ning-Wang-Zhou's characterization of the curvature Nakano positivity of Hermitian vector bundles in terms of $L^2$-estimate of $\bar\partial$.

math.CV↗
arXiv2022

Linear isometric invariants of bounded domains

We introduce two new conditions for bounded domains, namely $A^p$-completeness and boundary blow down type, and show that, for two bounded domains $D_1$ and $D_2$ that are $A^p$-complete and not of boundary blow down type, if there exists a linear isometry from $A^p(D_1)$ to $A^{p}(D_2)$ for some real number $p>0$ with $p\neq $ even integers, then $D_1$ and $D_2$ must be holomorphically equivalent, where for a domain $D$, $A^p(D)$ denotes the space of $L^p$ holomorphic functions on $D$.

math.CV↗
arXiv2021

Omnidirectional nonreciprocal absorber realized by the magneto-optical hypercrystal

Photonic bandgap design is one of the most basic ways to effectively control the interaction between light and matter. However, the traditional photonic bandgap is always dispersive (blueshift with the increase of the incident angle), which is disadvantageous to the construction of wide-angle optical devices. Hypercrystal, that the photonic crystal with layered hyperbolic metamaterials (HMMs), can strongly modify the bandgap properties based on the anomalous wavevector dispersion of the HMM. Here, based on phase variation compensation between HMM and isotropic dielectric layers, we propose for the first time to design nonreciprocal and flexible photonic bandgaps using magneto-optical HMMs in one-dimensional photonic crystals. Especially for the forward and backward incident light, the blueshift and dispersionless of the forward and backward cavity modes are designed respectively to realize the interesting omnidirectional nonreciprocal absorber. Our results show high (low) absorption about 0.99 (0.25) in an angle range of 20-75 degrees for the forward (backward) incident light at the wavelength of 367 nm. The nonreciprocal omnidirectional cavity mode not only facilitates the design of perfect unidirectional optical absorbers working in a wide-angle range, but also possesses significant applications for all-angle reflectors and filters.

physics.optics↗