On the Bimeromorphic Geometry of Compact Complex Contact Threefolds
Kristina Frantzen, Thomas Peternell
证明了与Kaehler流形双同态等价且非有理连通的紧切触三重是Kaehler曲面的射影化切丛。
We prove that a compact contact threefold which is bimeromorphically equivalent to a Kaehler manifold and not rationally connected is the projectivised tangent bundle of a Kaehler surface.
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We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have Jordan property.
We prove the Jordan property for groups of bimeromorphic selfmaps of three-dimensional compact Kähler varieties of non-negative Kodaira dimension and positive irregularity.
Let $X$ be a non-uniruled compact Kähler space of dimension 3. We show that the group of bimeromorphic automorphisms of $X$ is Jordan. More generally, the same result holds for any compact Kähler space admitting a quasi-minimal model.
The purpose of this paper is to study the bimeromorphic invariants of compact complex manifolds in terms of Bott-Chern cohomology. We prove a blow-up formula for Bott-Chern cohomology. As an application, we show that for compact complex threefolds the non-Kählerness degrees, introduced by Angella-Tomassini [Invent. Math. 192, (2013), 71-81], are bimeromorphic invariants. Consequently, the $\partial\bar{\partial}$-Lemma on threefolds admits the bimeromorphic invariance.
Let $X$ be a compact K{ä}hler manifold of dimension three. We prove that there exists a projective manifold $Y$ such that $π\_1(X)\simeq π\_1(Y)$. We also prove the bimeromorphic existence of algebraic approximations for compact K{ä}hler manifolds of algebraic dimension $\dim(X)-1$. Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact K{ä}hler threefolds.
Let X be a compact Kaehler threefold such that the base of the MRC-fibration has dimension two. We prove that X is bimeromorphic to a Mori fibre space. Together with our earlier result arXiv:1304.4013 this completes the MMP for compact Kaehler threefolds: let X be a non-projective compact Kaehler threefold. Then X has a minimal model or X is bimeromorphic to a Mori fibre space over a non-projective Kaehler surface.
Compact K\" ahler threefolds of $π_1$-general type
我们对这三个折叠进行分类，这三个折叠是这样的：它们的泛覆盖不是紧的，并且不被正维紧解析子集覆盖。我们表明，这三个褶皱有非负Kodaira尺寸，和Iitaka-Moishezon纤维化，经过适当的有限etale盖，双半纯淹没纤维复环面，和基地的一般类型和$π_1$ -一般类型。
We prove that every compact Kähler threefold $X$ of Kodaira dimension $κ= 0$ or $1$ has a $\mathbf{Q}$-factorial bimeromorphic model $X'$ with at worst terminal singularities such that for each curve $C \subset X'$, the pair $(X',C)$ admits a locally trivial algebraic approximation such that the restriction of the deformation of $X'$ to some neighborhood of $C$ is a trivial deformation. As an application, we prove that every compact Kähler threefold with $κ= 0$ or $1$ has an algebraic approximation.
In this note, we study the normal compact Kähler (possibly singular) threefold $X$ admitting the action of a free abelian group $G$ of maximal rank, all the non-trivial elements of which are of positive entropy. If such $X$ is further assumed to have only terminal singularities, then we prove that it is either a rationally connected projective threefold or bimeromorphic to a quasi-étale quotient of a complex $3$-torus.
Bimeromorphic invariance of $\partial\bar\partial$-property and blow-up formulae I: counterexamples and derived category approach
本文通过构造反例否定 $\partial\bar\partial$-引理的遗传性，并建立导出 blow-up 公式，引入相对性质与判据，揭示经典公式的局限。
We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic invariance of $(\bullet,0)$- and $(0,\bullet)$-Hodge numbers on a compact complex manifold, and obtain the equality for the numbers of the blow-ups and blow-downs in the weak factorization of the bimeromorphic map between two compact complex manifolds with equal $(1,1)$-Hodge number or equivalently second Betti number. Many examples of the latter one are listed. Inspired by these, we obtain the bimeromorphi…
Contact Moishezon threefolds with second Betti number one
证明了唯一具有第二贝蒂数等于1的三重Moishezon切触是射影空间.
Erratum and addendum to the paper: Abundance for Kähler threefolds
本文勘误并补遗 Kähler 三维簇丰饶性定理，指出证明漏洞，并明确除“无正维子簇且非环面商”这一待定情形外定理仍成立。
Existence of the equivariant minimal model program for compact Kähler threefolds with the action of an abelian group of maximal rank
本文针对容许极大秩、正熵自由交换群作用的 $\mathbb{Q}$-factorial 紧 Kähler klt 三维簇，建立 $G$-等变对数极小模型纲领，并据此给出其有理连通或双纯等价于 $Q$-复环面的分类结论。
Holomorphic 1-forms without zeros on Kähler threefolds
本文分类了所有容许以圆为底的 $C^\infty$-纤维丛结构的连通紧致光滑 Kähler 三流形，证明其存在有限平展覆盖双亚满射于特定曲面丛，从而证明了 Kotschick 在 3 维情形下的猜想。
Complex Contact Threefolds and Their Contact Curves
复接触三重是具有二维不可积全纯分布的三重。三重接触上的接触曲线是分布的可积曲线。这项工作的灵感来自两篇论文的布莱恩特，他在其中使用复杂的接触几何研究超极小曲面在四球和调查异国holonomies。本文系统地研究了接触三重及其上的接触曲线。我们推广了Bryant的一个结果并回答了他的一个问题.
In this paper we introduce the notion of a formal complex contact structure on an odd dimensional complex manifold. Our main result is that every formal complex contact structure on a Stein manifold $X$ is homotopic to a holomorphic contact structure on a Stein domain $Ω\subset X$ which is diffeotopic to $X$. We also prove a parametric h-principle in this setting, analogous to Gromov's h-principle for contact structures on smooth open manifolds. On Stein threefolds we obtain a complete homotopy classification of formal complex contact structures. Our methods also furnish a parametric h-princip…
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