We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kähler manifold $W_F$ which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction, and prove that the automorphism group of $Q$ satisfies the Jordan property.
Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.
In this survey we discuss holomorphic $\mathbb{P}^1$-bundles $p: X \to Y$ over a non-uniruled complex compact Kähler manifold $Y$, paying a special attention to the case when $Y$ is a complex torus. We discuss so called Jordan properties of the groups $Aut(X)$ and $Bim(X)$ of its biholomorphic and bimeromorphic automorphisms, respectively.
In topological dynamics, the Gromov--Yomdin theorem states that the topological entropy of a holomorphic automorphism $f$ of a smooth projective variety is equal to the logarithm of the spectral radius of the induced map $f^*$. In order to establish a categorical analogue of the Gromov--Yomdin theorem, one first needs to find a categorical analogue of a holomorphic automorphism. In this paper, we propose a categorical analogue of a holomorphic automorphism and prove that the Gromov--Yomdin type theorem holds for them.
We classify nonsymplectic automorphisms of prime order on irreducible holomorphic symplectic manifolds of O'Grady's 6-dimensional defamation type. More precisely, we give a classification of the invariant and coinvariant sublattices of the second integral cohomology group.
One central problem in real algebraic geometry is to classify the real structures of a given complex manifold. We address this problem for compact hyperkähler manifolds by showing that any such manifold admits only finitely many real structures up to equivalence. We actually prove more generally that there are only finitely many, up to conjugacy, faithful finite group actions by holomorphic or anti-holomorphic automorphisms (the so-called Klein actions). In other words, the automorphism group and the Klein automorphism group of a compact hyperkähler manifold contain only finitely many conjugac…
We characterize conjugacy classes of isometries of odd prime order in unimodular Z-lattices. This is applied to give a complete classification of odd prime order non-symplectic automorphisms of irreducible holomorphic symplectic manifolds up to deformation and birational conjugacy.
An automorphism of order $n$ of a K3 surface is called purely non-symplectic if it multiplies the holomorphic symplectic form by a primitive $n$-th root of unity. We give the classification of purely non-symplectic automorphisms with $φ(n)\geq 12$ where $φ$ denotes the Euler totient function.
On a theorem of Peters on automorphisms of Kahler surfaces
对于不存在非零全纯向量场的Kahler曲面，我们考虑了诱导第二有理上同调恒等式的全纯自同构群。假设正则线性系是没有基点和固定的组件，C. A.M.彼得斯表明，这个群体是平凡的，除非当卡勒曲面是一般类型和$c_1^2=2c_2$或$c_1^2=3c_2$举行。而且，这个群在前一种情况下是2-群，在后一种情况下是3-群。本说明的目的是提供关于这一群体的进一步资料。特别地，我们证明了$c_1^2$可被群的阶整除。我们的论证是基于C. H. Taubes在辛4流形，这是适用于这里在一个等变设置。
Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite collections of translation surfaces supported on the same algebraic curve. The underlying curves are hyperelliptic curves with many automorphisms. For each curve, the automorphism of maximal order acts on the space of holomorphic 1-forms. We present a translation surface corresponding to each of the eigenforms of this action.
