arXiv subjects

Yibo Ren

Publications and source records attributed to Yibo Ren.

3 recordsLinked to original sources

arXiv2026

Torsion algebras of Hermitian manifolds and rigidity

We regard the Chern torsion of a Hermitian manifold as a skew-symmetric complex-bilinear product on its holomorphic tangent bundle. For a compact connected Chern--K\"{a}hler-like Hermitian manifold, we prove that this product satisfies the Jacobi identity pointwise. If, in addition, the Chern holomorphic sectional curvature is strongly quasi-positive, we show that the resulting torsion Lie algebra is nilpotent and must be abelian. Consequently, the metric is K\"{a}hler. The underlying complex manifold is therefore projective and rationally connected. For general Hermitian manifolds, we construct a metric with positive real bisectional curvature on a Hopf surface which is neither simply connected nor rationally connected.

math.DG↗

arXiv2026

The $\chi_y$-genus, Chern number inequalities and signature

This article has two parts. In the first part we introduce two positivity conditions for the modified $\chi_y$-genus on almost-complex manifolds and show that each of them implies a family of optimal Chern number inequalities. It turns out that many important K\"{a}hler and symplectic manifolds satisfy either of the two positivity conditions, and hence these Chern number inequalities hold true on them. In the second part we focus on the signature, a special value of the $\chi_y$-genus, of symplectic manifolds equipped with symplectic circle actions and give applications. Our results in this part unify and generalize various related results in the existing literature.

math.DG↗

arXiv2025

Decomposition of Cliques into $k$-Star-Forests

A $k$-star-forest is a forest with at most $k$ connected components where each component is a star. Let $F_k(n)$ be the minimum integer such that the complete graph on $n$ vertices can be decomposed into $F_k(n)$ $k$-star-forests. Pach, Saghafian and Schnider showed that $F_2(n)=\lceil 3n/4 \rceil$. In this paper, we show that $F_3(n)=5n/9$ when $n$ is a multiple of 27. Further, for $k\ge 4$, we show that $F_k(n)=n/2+2$ when $n>2k$ and $n\equiv 4 \pmod{12}$. Our results disprove a conjecture of Pach, Saghafian and Schnider.

math.CO↗