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Xueyuan Wan

Publications and source records attributed to Xueyuan Wan.

At least 19 recordsLinked to original sources
arXiv2026

Topological components of surface group representations into the unitary group

We study representations of compact oriented surface groups into $\mathrm U(p)$ with elliptic-unipotent boundary holonomies, meaning that no boundary holonomy has eigenvalue $1$. We prove that the signature of the associated flat Hermitian bundle completely determines the connected component, and that every component is path-connected. For genus $g\geq1$ and $n\geq1$ boundary components, there are $np-1$ components; for $g=0$ and $n\geq2$, there are $(n-2)p+1$. The disk case is empty, while the closed-surface representation spaces are connected. The same component classification holds after taking the quotient by conjugation. Our proofs use the boundary rho invariant and explicit matrix deformations. For a three-holed sphere, the representation components have the homotopy types of complex Grassmannians, and their conjugation quotients are contractible.

math.GT↗
arXiv2026

Rigid and obstructed tangent bundles on Calabi--Yau threefolds

We address two questions of Huybrechts concerning rigidity and singular deformation spaces of tangent bundles on Calabi--Yau threefolds, without assuming simple connectedness. Our first example is a free $(\mathbb Z/2)^3$ quotient of a smooth intersection of four quadrics in $\mathbb P^7$. Its tangent bundle is stable and infinitesimally rigid, and it also provides a three-dimensional counterexample to the proposed classification in Peternell's Question~1.6. For a classical Igusa quotient of a product of three elliptic curves, we determine the analytic semiuniversal deformation germ of its polystable tangent bundle, keeping the underlying threefold fixed. This germ is isomorphic to the product of two copies of the union of the three coordinate axes in $\mathbb C^3$. It is therefore reduced and singular, with nine two-dimensional irreducible components.

math.AG↗
arXiv2026

On possible values of the signature of flat unitary bundles over compact surfaces

We determine all possible signatures of flat Hermitian bundles over compact, connected, oriented surfaces of positive genus with nonempty boundary. If $Σ_{g,n}$ has genus $g\geq 1$ and $n\geq 1$ boundary components, then the signatures arising from representations $π_1(Σ_{g,n})\to\mathrm{U}(p,q)$ are exactly the integers $m$ satisfying \[ |m| \leq (p+q)(2g+n-2) -(2g-2)|p-q| -\min\{2,n|p-q|\}. \] Every such integer, including the two extremal values, is realized by a block-diagonal representation whose image is contained, after possibly interchanging $p$ and $q$, in $\mathrm{U}(1,1)^{\times\min\{p,q\}}\times\mathrm{U}(|p-q|)$.

math.GT↗
arXiv2026

Non-strict negativity of holomorphic bisectional curvature for compact relative Kähler fibrations

Strict negativity of holomorphic bisectional curvature need not pass from the base and fibers of a compact holomorphic fibration to its total space, even when the Kodaira-Spencer map is everywhere injective. We construct a compact relative Kähler fibration over a genus-two curve, with a smooth projective threefold as total space and an everywhere injective Kodaira--Spencer map, whose induced fiber metrics have strictly negative holomorphic bisectional curvature, whereas the total space admits no Kähler metric with this curvature property. This gives a negative answer to an open problem posed by To and Yeung. We also construct compact, effectively parametrized Monge-Ampère fibrations from universal quaternionic abelian surfaces over Shimura curves. For products of these families, the generalized Weil-Petersson metric has nonpositive holomorphic bisectional curvature and strictly negative holomorphic sectional curvature, but its mixed bisectional curvatures vanish. Thus strict bisectional negativity can fail both for the existence of a metric on the total space and for the natural metric on the parameter space, despite effective variation at every point.

math.DG↗
arXiv2026

Quasi-convexity of energy functions along Teichmüller geodesics

Hyperbolic length functions are among the most fundamental ones on Teichmüller space, and they are quasi-convex along Teichmüller geodesics. In this paper, we investigate the same question for energy functions of harmonic maps in two natural settings, which may be viewed as nonlinear and higher-dimensional analogs of the length functions. For a fixed domain and a varying hyperbolic target, we prove the energy is quasi-convex along Teichmüller geodesics under a filling hypothesis. Furthermore, we generalize Masur's result on asymptotic growth of the length function along the Teichmüller geodesic determined by a Jenkins-Strebel differential to the energy functions. We also prove the quasi-convexity for covering maps between closed hyperbolic surfaces with fixed target and varying domains. We derive first and second variation formulas of energy functions along Teichmüller geodesics and explain why the natural global statement is quasi-convexity rather than genuine convexity.

math.DG↗
arXiv2026

Positivity of simplicial volume for closed nonpositively curved four-manifolds with nonzero Euler characteristic

In this paper, by employing the Gauss-Bonnet theorem for Riemannian simplices due to Allendoerfer and Weil, we show that if a closed nonpositively curved $4$-manifold has nonzero Euler characteristic, then its simplicial volume is necessarily positive. This result partially resolves conjectures posed by Connell-Ruan-Wang and Gromov concerning the relationship between the simplicial volume and the Euler characteristic for four-dimensional manifolds. As an application, we show that if a closed nonpositively curved $4$-manifold has negative Ricci curvature, then its simplicial volume is positive, thereby confirming in dimension four another conjecture of Gromov on the positivity of simplicial volume.

math.GT↗
arXiv2026

Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature

We prove the Chern version of the constant holomorphic sectional curvature conjecture for compact locally conformal Kähler manifolds. More precisely, let $(M^n,h)$, $n\geq2$, be a compact locally conformal Kähler manifold whose Chern holomorphic sectional curvature is a constant $c$. We show that $h$ is necessarily Kähler and therefore is a complex space form metric of holomorphic sectional curvature $c$. In particular, when $c=0$, the metric is Kähler flat. This removes the nonpositivity assumption from a theorem of Chen, Chen, and Nie. The proof derives a curvature identity on the universal Kähler cover and shows that the covering metric is Bochner--Kähler. The globally conformally Kähler case is then treated by compact Bochner--Kähler rigidity, while the strict LCK case is excluded by Kamishima's uniformization theorem and the automorphy of the conformal factor.

math.DG↗
arXiv2026

Llarull type theorems for bands in Three and Four dimensions

Llarull's theorem asserts that the scalar curvature and the metric on the $n$-sphere cannot be bounded below at the same time by those of the standard $n$-sphere. Using the warped $μ$-bubble method, we develop Llarull type theorems for three and four-dimensional bands with spectral scalar curvature bounds.

math.DG↗
arXiv2026

Positivity of the third Chern form for Griffiths positive vector bundles

In this paper, we prove the positivity of the double mixed discriminant associated with a positive linear map between spaces of third-order complex matrices, thereby settling the three-dimensional case of Finski's open problem. As an application, we obtain the weak positivity of the third Chern form for Griffiths positive vector bundles. Moreover, we show that all Schur forms are weakly positive for Griffiths positive vector bundles of rank three over complex threefolds. This yields a complete affirmative answer, in the case where both the rank and the dimension are three, to the question posed by Griffiths in 1969.

math.AG↗
arXiv2026

Kähler metrics of negative holomorphic (bi)sectional curvature on a compact relative Kähler fibration

For a compact relative Kähler fibration over a compact Kähler manifold with negative holomorphic sectional curvature, if the relative Kähler form on each fiber also exhibits negative holomorphic sectional curvature, we can construct Kähler metrics with negative holomorphic sectional curvature on the total space. Additionally, if this form induces a Griffiths negative Hermitian metric on the relative tangent bundle, and the base admits a Kähler metric with negative holomorphic bisectional curvature, we can also construct Kähler metrics with negative holomorphic bisectional curvature on the total space. As an application, for a non-trivial fibration where both the fibers and base have Kähler metrics with negative holomorphic bisectional curvature, and the fibers are one-dimensional, we can explicitly construct Kähler metrics of negative holomorphic bisectional curvature on the total space, thus resolving a question posed by To and Yeung for the case where the fibers have dimension one.

math.DG↗
arXiv2025

Obstructions of deforming complex structures and cohomology contractions

The Kodaira principle asserts that suitable cohomological contraction maps annihilate obstructions to deforming complex structures. In this paper, we revisit these phenomena from a purely analytic point of view, developing a refined power series method for the deformation of $(p,q)$-forms and complex structures. Working with the Frölicher spectral sequence, we show that under natural partial vanishing conditions on its differentials, all obstruction classes lie in the kernel of the corresponding contraction maps. This yields a refined Kodaira principle that recovers and strictly extends the known results. As a main application, we obtain new unobstructedness criteria for compact complex manifolds with trivial canonical bundle.

math.CV↗
arXiv2025

Topological components of surface group representations and signature

We study the topological components of the surface group representations into $\mathrm{SL}(2,\mathbb{R})$ and $\mathrm{PSL}(2,\mathbb{R})$. Utilizing the signature formula established in [14], we determine the number of connected components of the representation spaces with boundary elliptic, hyperbolic, and parabolic holonomies.

math.GT↗
arXiv2025

A logarithmic $\bar{\partial}$-equation on a compact Kähler manifold associated to a smooth divisor

In this paper, we solve a logarithmic $\bar{\partial}$-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 $E_1$, and we also prove that the pair $(X,D)$ has unobstructed deformations for any smooth divisor $D\in|-2K_X|$.

math.AG↗
arXiv2025

Deformations of $(p,q)$-forms and degenerations of the Frölicher spectral sequence

It is well-known that Hodge numbers are invariant under deformations of complex structures if the Frölicher spectral sequence of the central fiber degenerates at the first page (i.e. $E_1=E_\infty$). As a result, the deformations of $(p,q)$-forms are unobstructed for all $(p,q)$ if $E_1=E_\infty$. We refine this classical result by showing that for any fixed $(p,q)$ the deformations of $(p,q)$-forms are unobstructed if the differentials $d_r^{p,q}$ in the Frölicher spectral sequence satisfy \[ \bigoplus_{r\geq 1}d_r^{p,q}=0\quad\text{and}\quad \bigoplus_{ r\geq i\geq 1 } d_r^{p-i,q+i}=0. \] Moreover, the deformation stability of the degeneration property for Frölicher spectral sequences in the first page and higher pages is also studied. In particular, we have found suitable conditions to ensure the deformation stability of $E_r^{p,q}=E_\infty^{p,q}$ ($r\geq1$) for fixed $(p,q)$.

math.DG↗
arXiv2024

Scalar curvature rigidity of parabolic convex polytopes in hyperbolic space

In odd dimensions, we prove a scalar curvature rigidity for parabolic convex polytopes in hyperbolic space enclosed by linear planes in the Poincare upper half-space model and convex with respect to the conformally related flat metric. Our method is based on spinor techniques and relies on the recent smoothing constructions of Brendle-Wang. We also prove a Llarull type rigidity for bounded smooth parabolic convex domains and a dihedral rigidity for polytopal initial data sets with dominant energy conditions.

math.DG↗
arXiv2024

Initial data set rigidity results for polyhedra

Using spinors, we show a dihedral type rigidity for polyhedral initial data sets. This rigidity connects spacetime positive mass theorem, dihedral rigidity and capillary marginally trapped surfaces. Our method is to extend the rigidity analysis of spacetime positive mass theorem due to Beig-Chrusciel to the settings of a twisted spinor bundle.

math.DG↗
arXiv2024

Spectral constant rigidity of warped product metrics

A theorem of Llarull says that if a smooth metric $g$ on the $n$-sphere $\mathbb{S}^n$ is bounded below by the standard round metric and the scalar curvature $R_g$ of $g$ is bounded below by $n (n - 1)$, then the metric $g$ must be the standard round metric. We prove a spectral Llarull theorem by replacing the bound $R_g \geq n (n - 1)$ by a lower bound on the first eigenvalue of an elliptic operator involving the Laplacian and the scalar curvature $R_g$. We utilize two methods: spinor and spacetime harmonic function.

math.DG↗