Searcharxiv⌕ Search

arXiv subjects

Renjie Lyu

Publications and source records attributed to Renjie Lyu.

7 recordsLinked to original sources
arXiv2026

Limiting Hodge Structures for Cubic Hypersurfaces of Secant Type

We study limiting Hodge structures for deformations of cubic hypersurfaces degenerating to specific singular cubics of secant type. Collino, Hassett and Laza have investigated degenerations of cubic threefolds and fourfolds whose central fibers are respectively the secant varieties of the rational normal quartic and the Veronese surface. The limiting Hodge structure bridges the geometric degeneration of varieties with the Hodge-theoretic degeneration. The Veronese surface belongs to the class of Severi varieties, and the rational normal quartic is a hyperplane section of the Veronese surface. We previously investigated similar degeneration problems for the secant varieties of Severi varieties in higher dimensions. In the present paper, we continue to study the limiting Hodge structures for the hyperplane sections of Severi varieties.

math.AG↗
arXiv2026

Degenerations to secant cubic hypersurfaces and limiting Hodge structure

The secant variety of the Veronese surface is a singular cubic fourfold. Degenerations to this specific cubic fourfold and the associated limiting Hodge structures are key ingredients for Hassett and Laza in studying the moduli space of cubic fourfolds and the period mapping. We generalize some results to the cubic hypersurfaces of secant type. Specifically, we compute the limit mixed Hodge structure for families of smooth cubic hypersurfaces degenerating to the cubic hypersurface of secant type. Using Usui's partial compactification and the resulting limit mixed Hodge structure, we characterize a local extension of the period map associated with the degenerating family.

math.AG↗
arXiv2025

The Chevalley--Weil formula for finite group actions on higher dimensional compact complex manifolds

Building on the Atiyah--Singer holomorphic Lefschetz fixed-point theorem, we define ramification modules associated to the fixed loci of a finite group acting on a compact complex manifold. This allows us to generalize the Chevalley--Weil formula for compact Riemann surfaces to higher dimensions. More precisely, let $G$ be a finite group acting on a compact complex manifold $X$, and let $\mathcal{E}$ be a $G$-equivariant locally free sheaf on $X$. Then, in the representation ring $R(G)_\mathbb{Q}$, we have \[ χ_G(X, \mathcal{E}):=\sum_{i=0}^{\dim X}(-1)^i[H^i(X, \mathcal{E})]=\frac{1}{|G|}χ(X,\mathcal{E})[\mathbb{C}[G]] + \sum_ZΓ(\mathcal{E})_Z \] where $Z$ runs over all connected components of the fixed-point sets $X^g$ for $g\in G$, and each $Γ(\mathcal{E})_Z\in R(X)_\mathbb{Q}$, called the \emph{ramification module} at $Z$, depends only on the restriction $\mathcal{E}|_Z$ and the normal bundle $N_{Z/X}$ as $G_Z$-equivariant bundles. We illustrate the computation of $Γ(\mathcal{E})_Z$ in several special cases and provide a detailed example for faithful actions of $G\cong(\mathbb{Z}/2\mathbb{Z})^n$ on a compact complex surface.

math.AG↗
arXiv2025

Cylinder maps of algebraic cycles on cubic hypersurfaces

Let \(X\subset \mathbb{P}^{n+1}\) be a smooth cubic hypersurface, and let \(F(X)\) be the variety of lines on \(X\). We prove the surjectivity of the cylinder maps on the Chow groups of \(F(X)\) and \(X\) if \(X\) contains a one-cycle of degree \(1\). Mongardi and Ottem previously proved the integral Hodge conjecture for curve classes on hyperkähler manifolds. Using the cylinder maps, we provide an alternative proof for the \(F(X)\) of a smooth complex cubic fourfold \(X\), which is a special hyperkähler fourfold. In addition, we confirm the integral Tate conjecture for \(F(X)\) of a smooth cubic fourfold \(X\) over a finitely generated field.

math.AG↗
arXiv2021

Lines on the secant cubic hypersurfaces of Severi varieties

The secant varieties of Severi varieties provide special examples of (singular) cubic hypersurfaces. An interesting question asks when a given cubic hypersurface is projectively equivalent to a secant cubic hypersurface. Inspired by the ''geometric'' Torelli theorem for smooth cubic hypersurfaces due to F. Charles, we study the geometry of lines on secant cubics and describe the Fano variety of lines. Then we verify the ''geometric'' Torelli theorem for the case of secant cubics. Namely, a cubic hypersurface is isomorphic to a secant cubic if and only if their Fano varieties of lines are isomorphic.

math.AG↗
arXiv2021

Remarks on Automorphism and Cohomology of Cyclic Coverings

For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that the group of automorphisms acts faithfully on the cohomology except for a few cases. In characteristic zero, we study the equivariant deformation theory and automorphism groups for complex cyclic coverings. The proof uses the decomposition of the sheaf of differential forms due to Esnault and Viehweg. In positive characteristics, a lifting criterion of automorphisms reduces the faithfulness problem to characteristic zero. To apply this criterion, we prove the degeneration of the Hodge-de Rham spectral sequences for a family of smooth finite cyclic coverings, and the infinitesimal Torelli theorem for finite cyclic coverings defined over an arbitrary field.

math.AG↗