Stability of tautological vector bundles on Hilbert squares of surfaces
Stability of tautological vector bundles on Hilbert squares of surfaces
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希尔伯特平方曲面上同义反复向量丛的稳定性
DOI：
10.4171/rsmup/124-7
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
Ulrich Schlickewei
中科院分区：
文献类型：
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作者：
Ulrich Schlickewei
证明了曲面的Hilbert平方上秩为2的重言式丛的稳定性（在弱正性条件下），并计算了它们的Chern类。设S是光滑的复射影曲面，Hilbert（S）是S的长度为2的子概型的参数化的Hilbert概型。它是已知的一个经典定理Fogarty [Fo]，Hilb（S）是一个光滑的，射影各种尺寸4。设Z → S × Hilb（S）是泛子概型，p：Z → S和q：Z → Hilb（S）表示投影.给定S上的一个线丛L，层L[2]：= q <$pL是Hilb（S）上的一个秩为2的向量丛，称为与L相关的重言式向量丛。本文证明了下列定理。假设h 0（S，L）≥ 2。则对N 0，向量丛L[2]在Hilb（S）上是μHN -稳定的.在这里，HN是Sym（NH）− E形式的极化，其中H是S上的充分因子，E Hilb（S）表示希尔伯特-周态射的例外因子。定理的证明依赖于重言向量丛关于S ×S爆破的基本短正合列和米斯特雷塔[Mi]关于曲线的相应结果。最初，我们对这个结果的兴趣来自于希望在具有有趣度量和有趣陈类的K3曲面的Hilbert方案上产生向量丛。为此，我们给出了L[2]的Chern类的一个公式，该公式是由c1（L），[E]和Hilb（S）的特征类的对称积表示的。K3曲面上的稳定层的模空间在文献中已经被广泛研究（参见。例如[Mu]、[OG]、[HL]、[Ma]）。这些空间特别有趣，因为它们是紧致超卡勒流形的少数几个例子之一（参见图1）。Huybrechts的章节[GHJ]）。类似地，研究高维Hyperkahler流形上稳定层的模空间似乎是有希望的。本文给出了四维紧致Hyperkahler流形的两个原型之一的K3曲面的第二Hilbert方案上的稳定层的例子。在第一节中引入一些符号之后，我们在第二节中证明了定理。最后在第三节中我们计算了L[2]的Chern类。这项工作得到了DFG（德国研究基金会）的SFB/TR 45“周期，模空间和代数变种的算术”和波恩国际数学研究生院（BIGS）的支持。1 2 ULRICH SCHLICKEWEI鸣谢。这篇论文是我博士论文的一部分。在波恩大学撰写的论文。我非常高兴地感谢我的顾问丹尼尔·许布莱希茨的不断支持。我也感谢卢卡斯卡拉有益的讨论陈字符重言向量丛和埃内斯托米斯特雷塔向我解释他的结果稳定向量丛对称产品的曲线。1.一些记法设<$：→ S × S为对角线。用σ：S × S → S × S表示S × S在ε中的爆破。对称群S2在S ×S上的自然作用扩展为在S × S上的全纯作用，并且Hilb（S）= S × S/S2。设<$D：D → S × S是σ的例外因子。在下面的图表中，我们总结了这种情况，同时给各种自然地图命名。
We prove stability of rank two tautological bundles on the Hilbert square of a surface (under a mild positivity condition) and compute their Chern classes. Let S be a smooth, complex projective surface, let Hilb(S) be the Hilbert scheme parametrizing subschemes of S of length 2. It is known by a classical theorem of Fogarty [Fo] that Hilb(S) is a smooth, projective variety of dimension 4. Let Z ⊂ S × Hilb(S) be the universal subscheme, denote by p : Z → S and by q : Z → Hilb(S) the projections. Given a line bundle L on S, the sheaf L[2] := q∗pL is a rank two vector bundle on Hilb(S), called the tautological vector bundle associated with L. In this note we prove the following Theorem. Assume that h0(S,L) ≥ 2. Then for N 0, the vector bundle L[2] is μHN -stable on Hilb(S). Here, HN is a polarization of the form Sym(NH) − E where H is an ample divisor on S and E ⊂ Hilb(S) denotes the exceptional divisor of the Hilbert–Chow morphism. The proof of the theorem relies upon the fundamental short exact sequence for tautological vector bundles on the blowup of S ×S and upon the corresponding result for curves which was proved by Mistretta [Mi]. Originally, our interest in this result came from the desire to produce vector bundles on Hilbert schemes of K3 surfaces with interesting metrics and with interesting Chern classes. For this reason we give a formula for the Chern classes of L[2] in terms of the symmetric product of c1(L), of [E] and of the characteristic classes of Hilb(S). Moduli spaces of stable sheaves on K3 surfaces have been studied extensively in the literature (cf. e.g. [Mu], [OG], [HL], [Ma]). These spaces are particularly interesting because they are among the few examples of compact Hyperkahler manifolds (cf. Huybrechts’ chapter in [GHJ]). In analogy it seems to be promising to study moduli spaces of stable sheaves on higher-dimensional Hyperkahler manifolds. In this note we present examples of stable sheaves on the second Hilbert scheme of a K3 surface which is one of the two prototypes of four-dimensional compact Hyperkahler manifolds. After introducing some notation in Section 1 we prove the theorem in Section 2. Finally we calculate the Chern classes of L[2] in Section 3. This work was supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG (German Research Foundation) and by the Bonn International Graduate School in Mathematics (BIGS). 1 2 ULRICH SCHLICKEWEI Acknowledgements. This paper is a part of my Ph.D. thesis prepared at the University of Bonn. It is a great pleasure to thank my advisor Daniel Huybrechts for his constant support. I am also grateful to Luca Scala for helpful discussions on Chern characters of tautological vector bundles and to Ernesto Mistretta for explaining to me his results about stable vector bundles on symmetric products of curves. 1. Some notation Let ι∆ : ∆ ↪→ S × S be the diagonal. Denote by σ : S × S → S × S the blowup of S × S in ∆. The natural action of the symmetric group S2 on S ×S extends to a holomorphic action on S × S and Hilb(S) = S × S/S2. Let ιD : D ↪→ S × S be the exceptional divisor of σ. In the following diagram we summarize the situation and, at the same time, give names to the various natural maps.