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Kefeng Liu

Publications and source records attributed to Kefeng Liu.

At least 19 recordsLinked to original sources
arXiv2026

Khovanov homology: pro-tangles, derived colimits and spectral sequences

This paper introduces \emph{pro-tangles}, a natural generalization of classical tangles, which are functors from the Boolean cube to Bar-Natan's cobordism category. By employing the simplicial Yoneda embedding, we construct the Khovanov simplicial presheaf of a pro-tangle as the total homotopy cofiber over the punctured Boolean cube, and prove that this simplicial presheaf is representable, with representing object the classical Khovanov simplicial object. We establish a fully faithful embedding showing that the weak equivalence class of this simplicial presheaf is determined by the chain homotopy type of the Khovanov complex. Furthermore, we utilize Boolean cube decompositions to construct an algebraic spectral sequence for pro-tangles. This spectral sequence converges to the total Khovanov homology, and its $E_1$ page is explicitly expressed in terms of the Khovanov homology of reduced tangles. This categorical setup yields a functorial interpretation of Reidemeister invariance in terms of morphisms of spectral sequences. By applying the tangle TQFT construction, we study this spectral sequence for Hopf clasps, the fundamental structural building blocks in tangle and link theory. We show that the spectral sequence collapses at the $E_3$ page, which further specializes to an $E_2$-collapse under the restriction to Hopf sums. Finally, we investigate connected sums of pro-tangles and pro-links. To address the module-action dependencies arising from tensor products in multi-connected sums, we introduce a state-dependent modified tensor operator and prove a structural decomposition theorem that generalizes the classical result at the chain complex level.

math.GT↗
arXiv2026

Degenerations and Stability of K\"ahler Structures on Calabi--Yau Manifolds

In this paper, we study the degeneration and stability of K\"ahler structures on Calabi--Yau manifolds, namely compact K\"ahler manifolds with trivial canonical bundles, from the viewpoint of deformation theory and Hodge theory. Using the global deformation theory of Calabi--Yau manifolds together with estimates relating the Weil--Petersson distance and Beltrami differentials, we prove that certain limits of Calabi--Yau manifolds remain K\"ahler. As applications, we give a new proof of Siu's theorem on the K\"ahlerness of K3 surfaces. We further prove that deformation limits of hyperk\"ahler manifolds with bounded periods remain K\"ahler, which gives a complete and stronger solution to the conjecture of Soldatenkov--Verbitsky. Finally, we prove that the moduli spaces of stable sheaves on K3 surfaces are hyperk\"ahler manifolds, which gives a complete solution to the conjecture of Perego.

math.AG↗
arXiv2026

Local Laplacian: theory and models for data analysis

While topological data analysis has emerged as a powerful paradigm for structural inference, its foundational tools, notably persistent homology and the persistent Laplacian, are frequently insensitive to localized structural fluctuations and suffer from prohibitive computational costs on large-scale datasets. To bridge this gap, we introduce the persistent local Laplacian formalism, which is designed to extract fine-grained local topological and geometric signatures while enabling a highly efficient, parallelizable computational workflow. On the theoretical front, we prove a generalized persistent Hodge isomorphism, establishing that the harmonic space of the persistent local Laplacian is isomorphic to the persistent local homology. Furthermore, we derive a unitary equivalence between the persistent local Laplacian and the persistent Laplacian of its corresponding link complex at a shifted dimension. This spectral conjugacy establishes the mathematical foundation for developing efficient computational schemes to resolve persistent local spectral invariants. We further extend this construction to point clouds and graph-structured data, characterizing their persistent local spectral properties through combinatorial filtrations. The resulting architecture is inherently decoupled, facilitating massive parallelization and rendering it uniquely scalable for large-scale network analysis and distributed computational environments.

math.AT↗
arXiv2026

Sections of Hodge bundles I: Global theory and applications to period maps

We study global sections of Hodge bundles arising from two complementary constructions: a deformation-theoretic construction, which yields global geometric consequences for period maps, and a construction from the matrix representation of the image of the period map, which provides an explicit Euclidean realization. Combining these perspectives, we prove that the image of the lifted period map on the universal cover is contained in a complex Euclidean subspace of the period domain, thereby giving a partial solution to a conjecture of Griffiths on the global behavior of period maps. As an application, we construct a global complex affine structure on the Teichm\"uller space of Calabi--Yau type manifolds.

math.AG↗
arXiv2026

Sections of Hodge bundles II: Deformation of $(p,p)$-classes and applications to K\"ahler geometry

Let $(X,\omega_0)$ be a compact K\"ahler manifold and $\mathcal X\to B$ its Kuranishi family, where $B$ may be singular and $\dim_{\C}B\ge1$. Using explicit sections of Hodge bundles, we define an intrinsic period map and a Hodge map parametrizing nearby $(p,p)$-classes. For deformations over irreducible analytic bases, we introduce two flat extensions of K\"ahler cones defined by the reference and moving Hodge connections. The extension associated with the reference connection admits explicit positive representatives and yields uniform upper semicontinuity, while that associated with the moving connection identifies the K\"ahler cones away from a countable union of proper analytic subsets and admits an explicit expression in terms of the period map and the Beltrami differential. These constructions provide a description of K\"ahler cones through analytic cycles and yield both local and large-scale K\"ahler stability without assuming unobstructedness. As further applications, we generalize Green's density criterion to strong algebraic approximation and to the approximation of real $(p,p)$-forms. We also obtain an intrinsic analytic description of Hodge loci, leading to a Beltrami-differential criterion for the variational Hodge conjecture.

math.AG↗
arXiv2025

Several new Witten rigidity theorems for spin$^c$ manifolds

Using Liu's modular invariance method and its odd-dimensional extension by Han and Yu, we establish new Witten rigidity theorems for the generalized Witten genus of twisted Dirac operators on even-dimensional spin$^c$ manifolds and twisted Toeplitz operators on odd-dimensional spin$^c$ manifolds with circle actions.

math.DG↗
arXiv2025

Higher Weil-Petersson volumes of the moduli space of super Riemann surfaces

Inspired by the theory of JT supergravity, Stanford-Witten derived a remarkable recursion formula of Weil-Petersson volumes of moduli space of super Riemann surfaces. It is the super version of the celebrated Mirzakhani's recursion formula. In this paper, we generalize Stanford-Witten's formula to include high degree kappa classes.

math.AG↗
arXiv2024

Several new Witten rigidity theorems for elliptic genus

Using the Liu's method, we prove a new Witten rigidity theorem of elliptic genus of twisted Dirac operators in even dimensional spin manifolds under the circle action. Combined with the Han-Yu's method, we prove the Witten rigidity theorems of elliptic genus of twisted Toplitz operators of odd-dimensional spin manifolds under the circle action. Moreover, we have obtained several similar Witten rigidity theorems of elliptic genus.

math.DG↗
arXiv2024

Penrose transformation on flag domains

Building on our recent work, we construct the Penrose transformations of the cohomology groups of homogeneous line bundles on flag domains $D = G_\R / T$, where $G_\R$ is of Hermitian type. We provide sufficient conditions for the injectivity of the Penrose transformation and identify conditions under which the Penrose transformation of the automorphic cohomology groups on compact quotients of flag domains is an isomorphism. Finally, we prove that the higher automorphic cohomology groups of certain homogeneous line bundles are isomorphic to the groups of automorphic forms on the Hermitian symmetric domain, and we apply this result to the cup products of the automorphic cohomology groups.

math.AG↗
arXiv2024

The noncommutative residue and sub-Riemannian limits for the twisted BCV spaces

In this paper, we derive the sub-Riemannian version of the Kastler-Kalau-Walze type theorem and the Dabrowski-Sitarz-Zalecki type theorem for the twisted BCV spaces. We also compute the Connes conformal invariants for the twisted product, as well as the sub-Riemannian limits of the Connes conformal invariants for the twisted BCV spaces.

math.DG↗
arXiv2024

Vanishing Theorems and Complex Structures on Non-Classical Flag Domains

We prove that every nontrivial line bundle on a compact quotient of a non-classical flag domain has no nonzero global sections. The proof first establishes the Green--Griffiths--Kerr conjecture by showing that the curvature of every nontrivial locally homogeneous line bundle has a negative direction, and then extends this property to arbitrary line bundles by decomposing their curvature into a homogeneous part and a seminegative correction term. We also establish several equivalent geometric and root-theoretic characterizations of non-classical flag domains. As consequences, their compact quotients are not in Fujiki class $\mathcal C$, contain no nonzero effective divisors, admit no nonconstant meromorphic functions, and have algebraic dimension zero. When $D=G_\R/V$ is non-classical and $G_\R$ is of Hermitian type, we construct another natural $G_\R$-invariant complex structure on the underlying differentiable manifold of $D$. The resulting classical flag domain has projective compact quotients. Thus the same differentiable manifold admits two invariant complex structures with opposite algebro-geometric behavior: one gives a projective manifold, whereas the other gives a non-classical quotient with the vanishing and non-algebraicity properties above.

math.AG↗
arXiv2021

Affine geometry and Frobenius algebra

The associativity of the multiplication on a Frobenius manifold is equivalent to the WDVV equation of a symmetric cubic form in flat coordinates. Frobenius manifold could be regarded a very special type of statistical manifold. There is a natural commutative product on each tangent space of a statistical manifold. We show that it is associative, hence making it into a manifold with Frobenius algebra structure, if and only if the sectional $K$-curvature vanishes. In other words, WDVV equation is equivalent to zero sectional $K$-curvature. This gives a curvature interpretation for WDVV equation.

math.DG↗
arXiv2020

Cubic forms, anomaly cancellation and modularity

Motivated by the cubic forms and anomaly cancellation formulas of Witten-Freed-Hopkins, we give some new cubic forms on spin, spin$^c$, spin$^{w_2}$ and orientable 12-manifolds respectively. We relate them to $\eta$-invariants when the manifolds are with boundary, and mod 2 indices on 10 dimensional characteristic submanifolds when the manifolds are spin$^c$ or spin$^{w_2}$. Our method of producing these cubic forms is a combination of (generalized) Witten classes and the character of the basic representation of affine $E_8$.

math.DG↗
arXiv2019

Complex Finsler vector bundles with positive Kobayashi curvature

In this short note, we prove that a complex Finsler vector bundle with positive Kobayashi curvature must be ample, which partially solves a problem of S. Kobayashi posed in 1975. As applications, a strongly pseudoconvex complex Finsler manifold with positive Kobayashi curvature must be biholomorphic to the complex projective space; we also show that all Schur polynomials are numerically positive for complex Finsler vector bundles with positive Kobayashi curvature.

math.DG↗
arXiv2019

Harmonic 2-forms and positively curved 4-manifolds

We prove that if a compact Riemannian 4-manifold with positive sectional curvature satisfies a Kato type inequality, then it is definite. We also discuss some new insights for compact Riemannian 4-manifolds of positive sectional curvature.

math.DG↗