We prove existence and unicity of slope-stable vector bundles on a general polarized hyperkähler (HK) variety of type $K3^{[n]}$ with certain discrete invariants, provided the rank and the first two Chern classes of the vector bundle satisfy certain equalities. The latter hypotheses at first glance appear to be quite restrictive, but, in fact, we might have listed almost all slope-stable rigid projectively hyperholomorphic vector bundles on polarized HK varieties of type $K3^{[n]}$ with $20$ moduli.

中文翻译：

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HyperKähler 类型品种的刚性稳定向量束

我们证明了一般极化 hyperkähler (HK) 类型上斜率稳定矢量丛的存在性和唯一性 $K3^{[n]}$ 具有某些离散不变量，前提是向量束的秩和前两个 Chern 类满足某些等式。后者的假设乍一看似乎相当具有限制性，但事实上，我们可能已经列出了极化 HK 类型上几乎所有斜率稳定的刚性射影超全纯矢量丛 $K3^{[n]}$ 和 $20$ 模数。