复曲面上层族的连续性定理
A continuity theorem for families of sheaves on complex surfaces
摘要：
我们证明了Gauduchon曲面上的任何秩为2的无挠层的平坦族$(\mathcal{ F}_u)_{u\in U}$在半稳定轨迹$U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$上定义了一个连续映射，其值在相应的瞬子模空间的Donaldson-Uhlenbeck紧化中。在一般（可能非Kählerian）情况下，Donaldson-Uhlenbeck紧化不是一个复空间，而集合$U^{\mathrm {ss}}$可以是底空间$U$的一个复杂子集，它在经典拓扑中既不是开的，也不是闭的，在Zebraki拓扑中也不是局部闭的。这一结果为任意Gauduchon曲面上的Donaldson-Uhlenbeck紧化的显式描述提供了一个有效的工具。
英文摘要：
We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$ with values in the Donaldson-Uhlenbeck compactification of the corresponding instanton moduli space. In the general (possibly non-Kählerian) case, the Donaldson-Uhlenbeck compactification is not a complex space, and the set $U^{\mathrm {ss}}$ can be a complicated subset of the base space $U$ that is neither open or closed in the classical topology, nor locally closed in the Zariski topology. This result provides an efficient tool for the explicit description of Donaldson-Uhlenbeck compactifications on arbitrary Gauduchon surfaces.
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