Griffiths positivity does not imply positivity of the top Chern form

Yun-Heng Du

Home/Archive/arXiv

arXiv · 2610.05931

Griffiths positivity does not imply positivity of the top Chern form

Yun-Heng Du

View original source ↗Read source document ↗Export metadata ↓

Abstract

For every integer $r\geq9$, we construct a smooth Griffiths-positive Hermitian metric on $\mathcal O_{\mathbb P^r}(1)^{\oplus r}$ whose top Chern form is pointwise negative on a nonempty open set. This gives counterexamples on compact projective manifolds to Griffiths' conjecture on the positivity of Chern--Weil forms.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yun-Heng Du. 2026-10-05. Griffiths positivity does not imply positivity of the top Chern form. https://arxiv.org/abs/2610.05931

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Proof of the Koumandos--Ruscheweyh Conjecture

We prove the Koumandos--Ruscheweyh conjecture for every $0<ρ\leq1$. If $ν(ρ)$ is the unique root in $(0,1]$ of $\int_0^{(1+ρ)π}t^{μ-1}\sin(t-ρπ)\,\mathrm{d}t=0$, then $(1-z)^ρs_n^μ(z)\prec((1+z)/(1-z))^ρ$ for every $n\geq0$, $0<μ\leqν(ρ)$ and $z\in\mathbb{D}$, where $s_n^μ(z)=\sum_{k=0}^n(μ)_kz^k/k!$. The parameter $ν(ρ)$ is optimal. The proof combines an all-parameter gamma-coefficient comparison with an exact beta integral and a binomial variance estimate, reducing the infinitely many degrees to finitely many continuous interval inequalities. The remaining inequalities are certified by 256-bit ball arithmetic; rational covers, source code and alternative-formula verifiers are provided. The weak positive-real-part conjecture follows as a sharp corollary. We also derive sharp consequences for starlike functions and Gegenbauer polynomial sections: a full-parameter convolution subordination, the optimal starlike order for uniform partial-sum sectors, and the optimal Gegenbauer parameter and sector angle. The necessary bounds and angular sharpness are obtained from explicit kernels and interior scaling limits.

math.CV↗

arXiv2026

The Differential Hilbert Operator Between Weighted Bergman Spaces

In this paper, a complete characterization of the boundedness, compactness, norm and essential norm of the differential Hilbert operator $\mathcal{H}_2:A^2_α\to A^2_β$ is obtained. More precisely, $\mathcal H_2:A^2_α\to A^2_β$ is bounded if and only if $-1<α<0$ and $β\geqα+2$ and it is compact if and only if $-1<α<0$ and $β>α+2$. The norm and essential norm of $ \|\mathcal H_2\|_{A^2_α\to A^2_β}$ are also investigated. In particular, when $β=α+2$, \[ \|\mathcal H_2\|_{A^2_α\to A^2_{α+2}} =\|\mathcal H_2\|_{\mathrm e,A^2_α\to A^2_{α+2}} =\frac{π\sqrt{(α+2)(α+3)}} {\sin(π(α+2)/2)},\qquad -1<α<0. \] When $β>α+2$, $\|\mathcal H_2\|_{\mathrm e,A^2_α\to A^2_β}=0$. Furthermore, for every $1\leq p<\infty$, the operator $\mathcal H_2$ belongs to the Schatten class $\mathcal S_p$ if and only if it is compact.

math.CV↗

arXiv2026

Local-to-global maximality for plurisubharmonic functions with locally analytic singularities

We prove that a locally maximal plurisubharmonic function on a domain in $\bC^n$, $n\geq2$, is maximal if it has locally analytic singularities with a two-sided locally bounded remainder. McAdam's theorem gives a necessary bound on the analytic spread of the local defining ideals. Comparison on a normalized blow-up then controls the singularities of competing functions. As a consequence, maximality is characterized by the vanishing of the Bedford--Taylor Monge--Ampère measure off the pole set together with the analytic spread bound.

math.CV↗