arXiv subjects

Binggang Qu

Publications and source records attributed to Binggang Qu.

3 recordsLinked to original sources

arXiv2026

Closing the gap around the essential minimum of height functions with linear programming

For many common height functions, it is notoriously hard to compute the essential minimum. Nevertheless there are two classical methods, one giving lower bounds and the other giving upper bounds. In this paper, we show that the two methods are actually dual to each other in the sense of linear programming. The main theorem is that they satisfy strong duality, which closes the gap around the essential minimum from both ends. As applications we prove that this essential minimum can be realized by a generic sequence of algebraic integers, and that if the associated Green function is computable then this essential minimum is a computable real number.

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arXiv2024

Arakelov geometry on flag varieties over function fields and related topics

Let $k$ be an algebraically closed field of characteristic zero. Let $G$ be a connected reductive group over $k$, $P \subseteq G$ be a parabolic subgroup and $λ: P \longrightarrow G$ be a strictly anti-dominant character. Let $C$ be a projective smooth curve over $k$ with function field $K=k(C)$ and $F$ be a principal $G$-bundle on $C$. Then $F/P \longrightarrow C$ is a flag bundle and $\mathcal{L}_λ=F \times_P k_λ$ on $F/P$ is a relatively ample line bundle. We compute the height filtration, successive minima, and the Boucksom-Chen concave transform of the height function $h_{\mathcal{L}_λ}: X(\overline{K}) \longrightarrow \mathbb{R}$ over the flag variety $X=(F/P)_K$. An interesting application is that the height of $X$ equals to a weighted average of successive minima, and one may view this as a refinement of Zhang's inequality of successive minima. Let $f \in N^1(F/P)$ be the numerical class of a vertical fiber. We compute the augmented base loci $\mathrm{B}_+(\mathcal{L}_λ-tf)$ for any $t \in \mathbb{R}$, and it turns out that they are almost the same as the height filtration. As a corollary, we compute the $k$-th movable cones of flag bundles over curves for all $k$.

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arXiv2023

Arithmetic Demailly Approximation Theorem

We generalize the Demailly approximation theorem from complex geometry to Arakelov geometry.As an application, let $X/\mathbb{Q}$ be an integral projective variety and $\overline N$ be an adelic line bundle on $X$, we prove that $\operatorname{ess}(\overline N) \geq 0$ $\Longrightarrow $ $\overline N$ pseudo-effective. This was proved in [Bal21], assuming $\overline{N}$ relatively semipositive. We show in the appendix that the above assertion is also true for adelic line bundles on quasi-projective varieties, under the framework of [YZ22].

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