On the spectrum of asymptotic slopes

Asymptotic slopes and strong semistability on surfaces

In this article we study asymptotic slopes of strongly semistable vector bundles on a smooth projective surface. A connection between asymptotic slopes and strong restriction theorem of a strongly semistable vector bundle is shown. We also give an equivalent criterion of strong semistability of a vector bundle in terms of its asymptotic slopes under some assumptions on the surface and on the bundle.

Asymptotic slopes of the Aubin-Yau functional and calculation of the Donaldson-Futaki invariant

We derive an explicit formula for the asymptotic slope of the Aubin-Yau functional along a Bergman geodesic on a surface of complex dimension 2, extending the work of Phong-Sturm on Riemann surfaces. This is equivalent to an explicit calculation of the Donaldson-Futaki invariant of a test configuration. The slope is given as a rational linear combination of period integrals of rational functions that sum to a rational number. The result gives a way to check directly whether a two dimensional projective variety is K-stable.

Successive minima and asymptotic slopes in Arakelov Geometry

Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\overline{D}$ be an adelic Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $ζ_{\mathrm{ess}}(\overline{D})$ of $\overline{D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $ζ_{\mathrm{ess}}(\overline{D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom--Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion …

关于直接像的Harder-Narasimhan斜率

On Harder-Narasimhan slopes of direct images

本文研究复射影簇平坦族中直接像的Harder-Narasimhan斜率渐近分布，建立Mehta-Ramanathan型定理，并用于估计线束重数。

Maximal slope of tensor product of Hermitian vector bundles

We give an upper bound for the maximal slope of the tensor product of several non-zero Hermitian vector bundles on the spectrum of an algebraic integer ring. By Minkowski's theorem, we need to estimate the Arakelov degree of an arbitrary Hermitian line subbundle $\bar M$ of the tensor product. In the case where the generic fiber of $M$ is semistable in the sense of geometric invariant theory, the estimation is established by constructing, through the classical invariant theory, a special polynomial which does not vanish on the generic fibre of $M$. Otherwise we use an explicte version of a res…

渐近Samuel函数与奇点不变量

The asymptotic Samuel function and invariants of singularities

本文将渐近Samuel函数推广到任意环，引入诺特局部环的Samuel斜率并研究其性质，证明簇奇点处局部环的该斜率与奇点消解算法的部分不变量相关。