On the monodromy of moduli spaces of sheaves on K3 surfaces II
On the monodromy of moduli spaces of sheaves on K3 surfaces II
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K3 面上滑轮模空间的单调性 II
DOI：
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发表时间：
2003
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通讯作者：
E. Markman
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作者：
E. Markman
让S成为一个K3曲面。在本文的第一部分，我们构造了S的派生范畴的自等价的群Aut D(S)的一个表示，我们将这个无限维表示解释为Aut D(S)在S上稳定层(具有本原Mukai向量)的所有模空间上同调的自然作用。第一部分的主要结果是这种作用与曲面上点的Hilbert格式S^[n]的单调性的精确关系。在第一部分中，将上述结果的证明简化到S^[n]的两个单调算子的情形，这两个算子分别与选择S曲面上的线丛有关，次数分别为2n-4和2n。当n=1时，第一序列的单行算子专门用于通过一条-2曲线的反射。第二个序列的n=1情形与伽罗瓦对合有关，它是射影平面的一个沿六次曲线分支的双重覆盖。我们通过处理这两个序列的例子来完成证明。
Let S be a K3 surface. In part I of this paper, we constructed a representation of the group Aut D(S), of auto-equivalences of the derived category of S. We interpreted this infinite dimensional representation, as the natural action of Aut D(S) on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on S. The main result, of part I, is the precise relation of this action with the monodromy of the Hilbert schemes S^[n] of points on the surface. The proof of the above result was reduced, in part I, to the case of two monodromy operators of S^[n], associated with choices of line bundles on the surface S, of degree 2n-4 and 2n respectively. When n=1, the first sequence of monodromy operators specializes to the reflection by a -2 curve. The n=1 case of the second sequence is related to the Galois involution, of a double cover of the projective plane, branched along a sextic. We complete the proof by treating these two sequences of examples.