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Hanlong Fang

Publications and source records attributed to Hanlong Fang.

17 recordsLinked to original sources
arXiv2026

Classification of compact homogeneous strongly pseudoconvex hypersurfaces in $\mathbb C^n$

By constructing explicit entire maps, we prove that every exceptional Morimoto--Nagano model $M_t^3$, $t>1$, can be realized as a compact real-analytic hypersurface in $\mathbb C^3$. This extends Isaev's realization results from restricted parameter ranges to all $t>1$ and provides a complete solution to a long-standing open problem in the Morimoto--Nagano classification of compact, simply connected, real-analytic hypersurfaces in complex Euclidean spaces that are homogeneous under the action of a connected Lie group of CR automorphisms.

math.CV↗
arXiv2026

Moduli of Persson surfaces: The compactification via KSBA stable pairs and a generic global Torelli type theorem

We study a family of canonically polarized surfaces introduced by Persson, which arise as Galois $G=(\mathbb{Z}/2\mathbb{Z})^4$-covers of $\mathbf{P}^2$ branched along eight general lines. For this family, we construct the compactified moduli space and explicitly describe the stable degenerations in the sense of Koll\'ar, Shepherd-Barron, and Alexeev (KSBA) via stable pairs of weighted hyperplane arrangements. By computing the $\mathbb{Q}$-Gorenstein obstructions and using the KSBA wall crossings, we show that the resulting compactified moduli stack is smooth. Furthermore, we establish a generic global Torelli type result: up to two possibilities, a generic smooth Persson surface can be recovered from the Hodge structure on the anti-invariant part of the second cohomology of its \'etale double cover, together with the associated $\widetilde{G}=(\mathbb{Z}/2\mathbb{Z})^5$-action.

math.AG↗
arXiv2026

Compactifications of spaces of symmetric matrices and pointed Kontsevich spaces of isotropic Grassmannians

We study two closely related families of varieties arising from genus $0$ stable maps to the Lagrangian Grassmannian $\operatorname{LG}(n,2n)$. First, we construct the Kausz--type compactification $\mathcal {TL}_n$ of the space of symmetric matrices and give an explicit description of its birational geometry. Second, we realize $\mathcal {TL}_n$ as a general evaluation fiber in a Kontsevich space, and then exploit this modular interpretation to derive consequences for the birational geometry of the space of pointed conics $\overline{M}_{0,1}(\operatorname{LG}(n,2n),2)$. Analogous compactifications related to orthogonal Grassmannians are also presented.

math.AG↗
arXiv2026

Canonical blow-ups of Lagrangian and Orthogonal Grassmannians

Let $\mathbf{LG}(V\oplus V^*)$ and $\mathbf{OG}^+(V\oplus V^*)$ denote the Lagrangian and orthogonal Grassmannians endowed with the natural $\mathbb{G}_m$-actions, respectively. Thaddeus proved that over $\mathbb{C}$, the Hilbert quotients $\mathbf{LG}(V\oplus V^*)\!/\!/\mathbb{G}_m$ and $\mathbf{OG}^+(V\oplus V^*)\!/\!/\mathbb{G}_m$ are isomorphic to the wonderful compactifications of the spaces of symmetric and skew-symmetric matrices of maximal ranks, that is, the spaces of complete quadrics and complete skew-forms, respectively. In this paper, we construct the universal families of these Hilbert quotients by explicitly blowing up the corresponding isotropic Grassmannians, resulting in smooth toroidal compactifications of the spaces of symmetric and skew-symmetric matrices of maximal ranks (before projectivization), which have simple normal crossing boundary divisors and include the spaces of the complete bilinear forms among these divisors. Specifically, we prove that over any algebraically closed field, the universal families of these Hilbert quotients are smooth, and the Hilbert quotients themselves are isomorphic to the spaces of the complete bilinear forms. Over an algebraically closed field of characteristic zero, we prove that the universal families are weak Fano varieties with vanishing higher cohomology groups for their tangent bundles, and are therefore locally rigid. Furthermore, we show that these universal families naturally resolve the Landsberg-Manivel rational maps from projective spaces to isotropic Grassmannians.

math.AG↗
arXiv2026

The automorphism groups of generalized Kausz compactifications and spaces of complete collineations

In this paper, we determine the automorphism groups of generalized Kausz compactifications $\mathcal T_{s,p,n}$. By establishing the (semi-)positivity of the anticanonical bundles of $\mathcal T_{s,p,n}$, we also determine the automorphism groups of generalized spaces of complete collineations $\mathcal M_{s,p,n}$. The results in this paper are partially taken from the author's earlier arxiv post (Canonical blow-ups of grassmann manifolds, arxiv:2007.06200).

math.CV↗
arXiv2025

Fineness and smoothness of a KSBA moduli of marked cubic surfaces

By work of Gallardo-Kerr-Schaffler, it is known that Naruki's compactification of the moduli space of marked cubic surfaces is isomorphic to the normalization of the Koll\'ar, Shepherd-Barron, and Alexeev compactification parametrizing pairs $\left(S,\left(\frac{1}{9}+\epsilon\right)D\right)$, with $D$ the sum of the $27$ marked lines on $S$, and their stable degenerations. In the current paper, we show that the normalization assumption is not necessary as we prove that this KSBA compactification is smooth. Additionally, we show it is a fine moduli space. This is done by studying the automorphisms and the $\mathbb{Q}$-Gorenstein obstructions of the stable pairs parametrized by it.

math.AG↗
arXiv2024

Canonical blow-ups of Grassmannians I: How canonical is a Kausz compactification?

In this paper, we develop a simple uniform picture incorporating the Kausz compactifications and the spaces of complete collineations by blowing up Grassmannians $G(p,n)$ according to a torus action $\mathbb G_m$. We show that each space of complete collineations is isomorphic to any maximal-dimensional connected component of the $\mathbb G_m$-fixed point scheme of a Kausz-type compactification. We prove that the Kausz-type compactification is the total family over the Hilbert quotient $G(p,n)/ \! \! / \mathbb G_m$ which is isomorphic to the space of complete collineations. In particular, the Kausz compactifications are toroidal embeddings of general linear groups in the sense of Brion-Kumar. We also show that the Kausz-type compactifications resolve the Landsberg-Manivel birational maps from projective spaces to Grassmannians, by comparing Kausz's construction with ours. As an application, by studying the foliation we derive resolutions of certain birational maps among projective bundles over Grassmannians. The results in this paper are partially taken from the first author's earlier arxiv post (Canonical blow-ups of grassmann manifolds, arxiv:2007.06200), which has been revised and expanded in collaboration with the second author.

math.AG↗
arXiv2024

Bounding smooth Levi-flat hypersurfaces in a Stein manifold

This paper is concerned with the problem of constructing a smooth Levi-flat hypersurface locally or globally attached to a real codimension two submanifold in $\mathbb C^{n+1}$, or more generally in a Stein manifold, with elliptic CR singularities, a research direction originated from a fundamental and classical paper of E. Bishop. Earlier works along these lines include those by many prominent mathematicians working both on complex analysis and geometry. We prove that a compact smooth (or, real analytic) real codimension two submanifold $M$, that is contained in the boundary of a smoothly bounded strongly pseudoconvex domain, with a natural and necessary condition called CR non-minimal condition at CR points and with two elliptic CR singular points bounds a smooth-up-to-boundary (real analytic-up-to-boundary, respectively) Levi-flat hypersurface $\widehat{M}$. This answers a well-known question left open from the work of Dolbeault-Tomassini-Zaitsev, or a generalized version of a problem already asked by Bishop in 1965. Our study here reveals an intricate interaction of several complex analysis with other fields such as symplectic geometry and foliation theory.

math.CV↗
arXiv2023

Canonical blow-ups of Grassmannians II

We give a linear algebraic construction of the Lafforgue spaces associated to the Grassmannians $G(2,n)$ by blowing up certain explicitly defined monomial ideals, which sharpens and generalizes a result of Faltings. As an application, we provide a family of homogeneous varieties with high complexity and with nice compactifications, which exhibits the notion of homeward compactification introduced in our previous work in a non-spherical setting.

math.AG↗
arXiv2023

Algebraic and analytic properties of invariant differential operators on a homogeneous space of complexity $1$

Denote by $SL_3(\mathbb R)$ the special linear group of degree 3 over the real numbers, $A$ the subgroup consisting of the diagonal matrices with positive entries. In this paper, we study the algebraic and analytic properties of the invariant differential operators on the homogeneous space $SL_3(\mathbb R)/A$. Firstly, we specify the noncommutative algebra of invariant differential operators in terms of generators and their relations. Secondly, we describe the center of this algebra and prove that all of its symmetric elements are essentially self-adjoint. Thirdly, for the first time on homogeneous spaces, we identify several essentially self-adjoint invariant differential operators which do not lie in the center of the algebra of invariant differential operators.

math.RT↗
arXiv2021

A vanishing theorem for the canonical blow-ups of Grassmann manifolds

Let $\mathcal T_{s,p,n}$ be the canonical blow-up of the Grassmann manifold $G(p,n)$ constructed by blowing up the Pl\"ucker coordinate subspaces associated with the parameter $s$. We prove that the higher cohomology groups of the tangent bundle of $\mathcal T_{s,p,n}$ vanish. As an application, $\mathcal T_{s,p,n}$ is locally rigid in the sense of Kodaira-Spencer.

math.AG↗
arXiv2020

Canonical blow-ups of Grassmann manifolds

We introduce certain canonical blow-ups $\mathcal T_{s,p,n}$, as well as their distinct submanifolds $\mathcal M_{s,p,n}$, of Grassmann manifolds $G(p,n)$ by partitioning the Pl\"ucker coordinates with respect to a parameter $s$. Various geometric aspects of $\mathcal T_{s,p,n}$ and $\mathcal M_{s,p,n}$ are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of K\"ahler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which $\mathcal T_{s,p,n}$ are examples, as a generalization of the wonderful compactification. Lastly, a generalization of $\mathcal T_{s,p,n}$ according to vector-valued parameters $\overline s$ is given, and open questions are raised.

math.AG↗
arXiv2018

Construct holomorphic invariants in \v{C}ech cohomology by a combinatorial formula

In this paper, we give a combinatorial formula for the \v{C}ech cocycles representing the power sums of the Chern roots of a holomorphic vector bundle over a complex manifold. By an observation motivation by author's previous paper, we also construct some new holomorphic invariants refining the Chern classes. Firstly, we define the refined first $T$ invariants for all holomorphic vector bundles (or $\mathcal Q$-flat classes in the line bundle case) and give a criterion for determining whether a manifold has a line bundle whose $\mathcal Q$-flat class is strictly finer than its first Chern class in the Dolbeault cohomology. Then, we define the refined higher $T$ invariants for holomorphic vector bundles with a full flag structure. At last, we generalize the notion of the $T$ invariants (or equivalently the Chern classes) and the refined $T$ invariants for the locally free sheaves of schemes over general fields.

math.CV↗
arXiv2018

On the construction of a complete Kahler-Einstein metric with negative scalar curvature near an isolated log-canonical singularity

In this short note we are concerned with the Kahler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a complete Kahler-Einstein metric with negative scalar curvature in a neighborhood of the cone over a Calabi-Yau manifold, which provides a local model for the future study of the global Kahler-Einstein metrics on singular varieties. In the first approach, we show that the singularity is uniformized by a complex ball and hence the induced metric from the Bergman metric of the ball is a desired one. In the second approach, we obtain a complete Kahler-Einstein metric with negative curvature by using Calabi Ansatz. At last, we show that these two metrics are indeed the same.

math.DG↗
arXiv2018

A geometric criterion for prescribing residues and some applications

An old theorem of Weil and Kodaira says that for a compact K\"ahler manifold $X$ there is a closed logarithmic $1$-form with residue divisor $D$ if and only if $D$ is homologous to zero in $H_{2n-2}(X,\mathbb C)$. In the first part of this paper, we generalize the above theorem to general compact complex manifolds by showing that the necessary and sufficient condition in general is described by a holomorphic invariant called the $\mathcal Q$-flat class. Next, we prove that the holomorphic criterion is reduced to the topological one when $X$ has Property $(H)$. Since all K\"ahler manifolds have Property $(H)$, this gives an alternative proof of Weil and Kodaira's original theorem. Then, we prove some decomposition theorems for closed meromorphic $1$-forms by applying the above general theorem. In the second part of the paper, we turn to the study of pluriharmonic functions on projective manifolds and classify all the pluriharmonic functions with mild singularity.

math.CV↗
arXiv2017

Flattening a non-degenerate CR singular point of real codimension two

This paper continues the previous studies in two papers of Huang-Yin [HY3-4] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in ${\mathbb C}^{n+1}$ with $n+1\ge 3$, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY3] and a formal theory approach used in [HY4], we are able to provide a very general flattening theorem for a non-degenerate CR singular point. As an application, we provide a solution to the local complex Plateau problem and obtain the analyticity of the local hull of holomorphy near a real analytic definite CR singular point in a general setting.

math.CV↗