We prove that any class $VII$ surface with $b_2=1$ has curves. This implies the "Global Spherical Shell conjecture" in the case $b_2=1$: Any minimal class $VII$ surface with $b_2=1$ admits a global spherical shell, hence it is isomorphic to one of the surfaces in the known list. The main idea of the proof is to show that a certain moduli space of PU(2)-instantons on a surface $X$ with no curves (if such a surface existed) would contain a closed Riemann surface $Y$ whose general points correspond to non-filtrable holomorphic bundles on $X$. Then we pass from a family of bundles on $X$ parameterized by $Y$ to a family of bundles on $Y$ parameterized by $X$, and we use the algebraicity of $Y$ to obtain a contradiction. The proof uses essentially techniques from Donaldson theory: compactness theorems for moduli spaces of PU(2)-instantons and the Kobayashi-Hitchin correspondence on surfaces.
In this paper, we first prove a Donaldson-Uhlenbeck-Yau theorem over projective normal varieties smooth in codimension two. As a consequence we deduce the polystability of (dual) tensor products of stable reflexive sheaves, and we give a new proof of the Bogomolov-Gieseker inequality, along with a precise characterization of the case of equality. This also improves several previously known algebro-geometric results on normalized tautological classes. We study the limiting behavior of semistable bundles over a degenerating family of projective normal varieties. In the case of a family of stable…
We established a Yau--Tian--Donaldson type correspondence, expressed in terms of a single Delzant polytope, concerning the existence of extremal Kähler metrics on a large class of toric fibrations, introduced by Apostolov--Calderbank--Gauduchon--Tonnesen-Friedman and called semi-simple principal toric fibrations. We use that an extremal metric on the total space corresponds to a weighted constant scalar curvature Kähler metric (in the sense of Lahdili) on the corresponding toric fiber in order to obtain an equivalence between the existence of extremal Kähler metrics on the total space and a su…
Gluing formulae for Donaldson invariants for connected sums along surfaces
我们解决了Morgan和Szabo的一个猜想（嵌入亏格2曲面在四维流形，预印本）的关系的基本类的两个四维流形$X_i$的简单型与$b_1=0$，$b^+>1$，使得存在亏格为$g \geq 2$和自相交零点的嵌入黎曼曲面（并表示奇同调类）与基本类的流形X出现作为一个连接和沿表面（假设后者是简单的类型）。这也表示为X的基本类中的约束。这个结果与关于Seiberg-Witten不变量的结果（Morgan，Szabo and Taubes，A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture）是一致的.
For compact complex surfaces (M^4, J) of Kaehler type, it was previously shown that the sign of the Yamabe invariant Y(M) only depends on the Kodaira dimension Kod (M, J). In this paper, we prove that this pattern in fact extends to all compact complex surfaces except those of class VII. In the process, we give a simplified proof of a result that explains why the exclusion of class VII is essential here.
Extending Donaldson's HyperKähler Structure via Minimal Surfaces into $\mathrm{PSL}(2,\mathbb{C})$
University of Turin（都灵大学）；University of California Riverside（加州大学河滨分校）；Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)（数学科学研究所（西班牙高等科研理事会-马德里自治大学-马德里理工大学-马德里 Complutense 大学））；University of Pisa（比萨大学）
