Classification of holomorphic Pfaff systems on Hopf manifolds

Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds

We construct covariant $q$-deformed holomorphic structures for all finitely-generated relative Hopf modules over the irreducible quantum flag manifolds endowed with their Heckenberger--Kolb calculi. In the classical limit these reduce to modules of sections of holomorphic homogeneous vector bundles over irreducible flag manifolds. For the case of simple relative Hopf modules, we show that this covariant holomorphic structure is unique. This generalises earlier work of Majid, Khalkhali, Landi, and van Suijlekom for line modules of the Podleś sphere, and subsequent work of Khalkhali and Moatadel…

Holomorphic bundles on diagonal Hopf manifolds

Let $A$ be a diagonal linear operator on $\C^n$, with all eigenvalues satisfying $0<|α_i|<1$, and $M = (\C^n\backslash 0)/<A>$ the corresponding Hopf manifold. We show that any stable holomorphic bundle on $M$ can be lifted to a $G$-equivariant coherent sheaf on $\C^n$, where $G=(\C^*)^l$ is a Lie group acting on $\C^n$ and containing $A$. This is used to show that all stable bundles on $M$ are filtrable, that is, admit a filtration by a sequence $F_i$ of coherent sheaves, with all subquotients $F_i/F_{i-1}$ of rank 1.

指标与留数：从庞加莱-霍普夫（Poincaré-Hopf）到鲍姆-博特（Baum-Bott），以及马尔科·布鲁内拉（Marco Brunella）

Indices and residues: from Poincaré-Hopf to Baum-Bott, and Marco Brunella

该综述性文章研究向量场与全纯叶状结构的不变量，旨在关联复解析奇异簇与复流形上奇异全纯叶状结构这两个有诸多共通点的理论领域。

Holomorphic injectivity and the Hopf map

We give sharp conditions on a local biholomorphism $F:X \to \mathbb C^{n}$ which ensure global injectivity. For $n \geq 2$, such a map is injective if for each complex line $l \subset \mathbb C^{n}$, the pre-image $F^{-1}(l)$ embeds holomorphically as a connected domain into $\mathbb C \mathbb P^{1}$, the embedding being unique up to Möbius transformation. In particular, $F$ is injective if the pre-image of every complex line is connected and conformal to $\mathbb C$. The proof uses the topological fact that the natural map $\mathbb R \mathbb P^{2n-1} \to \mathbb C \mathbb P^{n-1}$ associated …

Ricci-flat Deformations of Holomorphic Vector Bundles

In this paper we give a criterion for a deformation of a hermitian vector bundle to be Ricci-flat. As an application we show that on a Kähler manifold, every deformation of a vector bundle can be made Ricci-flat whereas on some Hopf manifolds, the non-existence of a Ricci-flat deformation is related to non-trivial vector bundles on the universal cover ${\mathbb C}^n\setminus\{0\}$. On a surface with $b_1(X)\not=0$ filtrable Ricci-rigid vector bundles prove to be very special. We apply this to Inoue and Hopf surfaces.

Vector Bundle Moduli Superpotentials in Heterotic Superstrings and M-Theory

The non-perturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator on this curve. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. By explicitly computing the determinant of this map, one can deduce whether or not the dimension of the space of zero modes vanishes. It is shown that this information is sufficient to completely determine the Pfaffian and, hence, …

Hopf surfaces in locally conformally Kahler manifolds with potential

An LCK manifold with potential is a compact quotient M of a Kahler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on $X$ by holomorphic homotheties and maps f to a function proportional to f. It is known that M admits an LCK potential if and only if it can be holomorphically embedded to a Hopf manifold. We prove that any non-Vaisman LCK manifold with potential contains a complex surface with normalization biholomorphic to a Hopf surface H. Moreover, H can be chosen non-diagonal, hence, also not admitting a Vaisman structure.

Hopf流形中子簇上的凝聚层

Coherent sheaves on subvarieties in Hopf manifolds

本文证明了对于具有中心为x的可逆全纯收缩γ的正规复解析簇X，GAGA定理的一个版本，并研究了X的仿射簇结构以及γ-等变凝聚层的代数化，进而通过商簇X₀/γ嵌入Hopf流形，给出了Hopf流形中正规子簇的结构定理和凝聚层的滤过性。

Automorphisms of locally conformally Kahler manifolds

A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kahler cover of M is not isometric. We show that the cover admits an automorphic Kahler potential, and hence can be embedded to a Hopf manifold.

Rational curves on Hermitian manifolds

By using analytic method, we prove that there exist rational curves on compact Hermitian manifolds with positive holomorphic bisectional curvature. It confirms a question of S.-T. Yau. It is well-known that Mori proved in \cite{Mori79} that every compact complex manifold $N$ with $c_1(N)>0$ contains at least one rational curve. However, as a borderline example, we show that the standard Hopf surface $S^1\times S^3$ has a Hermitian metric with non-negative holomorphic bisectional curvature (in particular, $c_1(S^1\times S^3)\geq 0$), but it contains no rational curve.

Cox构造的等变自同构及其应用

Equivariant automorphisms of the Cox construction and applications

本文刻画单扇Cox构造关于大环面中足够大的连通复李子群的等变全纯自同构群，并用于确定有理矩角流形及完备单扇环面簇的相应自同构群。

具有大基本群的簇上的反常层

Perverse sheaves on varieties with large fundamental groups

本文猜想紧致无球面Kähler流形上的反常层具有非负Euler示性数以推广Singer-Hopf猜想，并在流形具非正全纯双截面曲率或射影且含忠实半单刚性局部系统时，分别利用特征圈相交数与周期映射曲率性质完成证明。