Upper semicontinuity of the dimensions of automorphism groups of domains in $C^n$

On the Wu metric in unbounded domains

We discuss the properties of the Wu pseudometric and present counterexamples for its upper semicontinuity that answers the question posed by Jarnicki and Pflug. We also give formulae for the Wu pseudometric in elementary Reinhardt domains.

代数维数的上半连续性

Upper semicontinuity of algebraic dimension

本文证明Moishezon流形在光滑形变下极限仍为Moishezon，并推广到满足∂∂̄-引理的浸没，通过Barlet循环空间与Rao-Tsai无挠方法及相位估计，确立代数维数的上半连续性。

On automorphisms of quasi-circular domains fixing the origin

It is known that automorphisms of quasi-circular domains fixing the origin are polynomial mappings. By introducing the so-called resonance order and quasi-resonance order, we provide a uniform upper bound for the degree of such polynomial automorphisms. As a particular consequence of our result, we obtain a generalization of the classical Cartan's theorem for circular domains to the quasi-circular case.

Zero entropy automorphisms of compact Kähler manifolds and dynamical filtrations

We study zero entropy automorphisms of a compact Kähler manifold $X$. Our goal is to bring to light some new structures of the action on the cohomology of $X$, in terms of the so-called dynamical filtrations on $H^{1,1}(X, {\mathbb R})$. Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations $(g^m)^* \, {\circlearrowleft} \, H^2(X, {\mathbb C})$ where $g$ is a zero entropy automorphism, in terms of ${\rm dim} \, X$ only. We also give an upper bound for the (essential) derived length $\ell_{\rm ess}(G, X)$ for every zero entropy subgroup $…

On the order of the automorphism group of foliations

Let $\mathcal F$ be a holomorphic foliation with ample canonical bundle on a smooth projective surface $X$. We obtain an upper bound on the order of its automorphism group which depends only on $K_{\mathcal F}^2$ and $K_{\mathcal F}K_{X}$, provided this group is finite. Here, $K_{\mathcal F}$ and $K_{X}$ are the canonical bundles of $\mathcal F$ and $X$, respectively.

Polynomial bounds for automorphisms groups of foliations

Let $(X, \mathcal{F})$ be a foliated surface and $G$ a finite group of automorphisms of $X$ that preserves $\mathcal{F}$. We investigate invariant loci for $G$ and obtain upper bounds for its order that depends polynomially on the Chern numbers of $X$ and $\mathcal{F}$. As a consequence, we estimate the order of the automorphism group of some foliations under mild restrictions. We obtain an optimal bound for foliations on the projective plane which is attained by the automorphism groups of the Jouanolou's foliations.

关于椭圆面的数值和上同调平凡自同构 II：χ(S)>0

On the numerically and cohomologically trivial automorphisms of elliptic surfaces II: $χ(S)>0$

本文研究了椭圆面的数值和上同调平凡自同构，揭示了Aut_{ℚ}(S)和Aut_{ℤ}(S)的结构及上界，解决了关于这些自同构群的若干问题。

关于椭圆曲面的上同调平凡自同构 I：χ(S)=0

On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$

本文研究了具有小平维数1且χ(O_S)=0的极小椭圆曲面的上同调平凡自同构群，证明了有限情形基数上界为4且仅可能为Z/2、Z/3或(Z/2)^2，无限情形连通分支数上界为2。

Several types of solvable groups as automorphism groups of compact Riemann surfaces

Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e.g. solvable, supersolvable, nilpotent, metabelian, metacyclic, abelian, cyclic. We refine these results by finding similar bounds for groups of odd order that are of these types. We also add more types of solvable groups to that long list by establishing the optimal bounds for, among others, groups of order $p^m q^n$. Moreover, we show that Zomorrodian's bound…