arXiv subjects

Shengyuan Zhao

Publications and source records attributed to Shengyuan Zhao.

7 recordsLinked to original sources

arXiv2026

Periods of Oka K3 surfaces are dense

We give the first known examples of Oka Calabi--Yau manifolds in any dimension: every smooth hypersurface of multidegree $(2,\ldots,2)$ in a product of projective lines is Oka. Building on this example, we prove that in every local Kuranishi family of K3 surfaces the points with Oka fibers form a dense $G_δ$ subset of the base, and the sublocus of Oka fibers with vanishing Néron--Severi group is likewise a dense $G_δ$ subset. We also obtain projective and nonprojective Oka Kummer surfaces and prove an analogous density statement in the Kummer period domain. We also prove the existence of Oka Enriques surfaces and Oka irreducible hyperkähler fourfolds.

math.CV↗

arXiv2026

Kobayashi Hyperbolicity of General Surfaces via the Poincaré Problem

We prove that a general surface in $\mathbb{P}^3$ of degree at least $18$ contains no rational or elliptic curves, strengthening the classical result of Clemens by replacing the original very general assumption by a genuine Zariski-open condition. Previously, nonexistence results in the ``general'' setting were known only in much higher degrees. Combining this with established algebraic degeneracy results for entire curves, we deduce the Kobayashi hyperbolicity of a general surface in $\mathbb{P}^3$ of degree at least $18$, thereby resolving a question asked by Demailly--El Goul. Our proof uses foliations induced by $2$-jet differentials. Two independent such differentials give rise to a multi-foliation tangent to all rational and elliptic curves. We establish a Poincaré-type bound for its algebraic leaves, yielding a mechanism to upgrade very general statements to general ones. Our method also applies to complements of plane curves. In particular, we prove that the complement of two general cubic curves in $\mathbb{P}^2$ is hyperbolically embedded.

math.AG↗

arXiv2026

Decay of weighted cusp counts for congruence subgroups of $SL_2$ over number fields

For congruence subgroups commensurable with $\operatorname{SL}_2$ over number fields, we study cusp counts with arithmetic multiplicities. We prove that the ratio of the total weighted cusp count to the group index is bounded by a negative power of the norm of the congruence level, with an exponent that can be chosen explicitly in terms of the degree of the number field. This is a generalization of a theorem of Cox--Parry over rational numbers. The main input is an explicit local orbit estimate for arbitrary subgroups of exact level in $\operatorname{SL}_2(\mathcal O_K/\mathfrak p^e)$, uniform over all number fields of fixed degree. This estimate is covered by a general result of Finis--Lapid. We give a new explicit proof in our setting based on an analysis reminiscent of additive combinatorics.

math.NT↗

arXiv2026

Equidistribution of currents under Anosov group actions

We study the behavior of currents on flag-manifolds under actions of Anosov subgroups of complex semisimple Lie groups G. Given a current T of bidimension (k, k) on the full flag-manifold F = G/B, we average it under the group action via a construction analogous to the construction of Patterson-Sullivan measures. We show that under certain genericity conditions (in the case of currents of integration over subvarieties), the limiting current is a Gibbs current, i.e. is given by integration (with respect to a Gibbs measure) over the flag-limit set of suitable multiples of currents of integration along Schubert varieties based at the limit points of the group. We prove that the same equidistribution result (but without any genericity assumptions) for currents defined by smooth forms on F. We also prove a form of Axiom A property for the suspension flow of the group action on F.

math.DS↗

arXiv2026

On cube and Cremona rigidity for higher-rank lattices

For irreducible lattices in semisimple Lie groups of real rank at least $2$, we prove a cohomological vanishing result implying that any action on a CAT(0) cube complex fixes a vertex whenever every hyperplane stabilizer is solvable. As an application, we prove regularizability for actions of all higher-rank lattices by birational transformations on projective surfaces. We first use superrigidity for actions on infinite-dimensional real hyperbolic spaces to reduce to the de Jonquières group, and then apply our fixed-point theorem to the Jonquières complex. Our proof bypasses the direct use of property FW.

math.DS↗

arXiv2024

Birational Kleinian groups

In this paper we study birational Kleinian groups, i.e.\ groups of birational transformations of complex projective varieties acting in a free, properly discontinuous and cocompact way on an open set of the variety with respect to the usual topology. We obtain a classification in dimension two.

math.DS↗

arXiv2023

Berkovich dynamics of twisted rational maps

A twisted rational map over a non-archimedean field $K$ is the composition of a rational function over $K$ and a continuous automorphism of $K$. We explore the dynamics of some twisted rational maps on the Berkovich projective line.

math.DS↗