##
**Stability of tautological bundles on the Hilbert scheme of two points on a surface.**
*(English)*
Zbl 1319.14049

Let \(X\) be a \(K3\) surface, and consider a moduli space \(\mathcal M\) of semi-stable sheaves on \(X\). S. Mukai [Invent. Math. 77, 101–116 (1984; Zbl 0565.14002)] proved that, in many cases, \(\mathcal M\) is a holomorphic symplectic variety. The motivating question for the article under review is: What happens if we iterate this process? In other words: Are there moduli spaces of sheaves on \(\mathcal M\) (or, more generally, on any higher-dimensional holomorphic symplectic variety) that carry a holomorphic symplectic structure?

As a first step in the study of this question, one needs examples of stable sheaves on \(\mathcal M\). The author provides such examples in the case that \(\mathcal M=X^{[2]}\) is the Hilbert scheme of two points on the \(K3\) surface \(X\).

To every sheaf \(\mathcal F\) on \(X\), one can associate the sheaf \(\mathcal F^{[2]}:= a_*b^* \mathcal F\), called tautological sheaf, on \(X^{[2]}\). Here, \[ a: \Xi_2 \to X^{[2]}\quad \text{and}\quad b: \Xi_2\to X \] denote the projections from the universal family \(\Xi_2\subset X^{[2]}\times X\). Since \(a: \Xi_2\to X^{[2]}\) is flat and finite of degree 2, we have \(\text{rank} \mathcal F^{[2]}=2\cdot \text{rank}\mathcal F\). Furthermore, if \(\mathcal F\) is a vector bundle or torsion free, the same holds for \(\mathcal F^{[2]}\). In this article, it is shown that stable sheaves of low rank on \(X\) induce stable tautological sheaves on the moduli space \(X^{[2]}\).

More precisely, the results are as follows. Let \(X\) be a smooth projective surface with \(H^1(X,\mathcal O_X)=0\), and let \(H\) be a polarisation on \(X\). For \(N\in \mathbb N\) sufficiently large, we consider the polarisation \(H_N:=N\cdot H-\delta\) on \(X^{[2]}\) where we use the usual isomorphism \(\text{Pic}(X^{[2]})\cong \text{Pic}(X) \oplus \mathbb Z\cdot \delta\); see J. Fogarty [Am. J. Math. 95, 660–687 (1973; Zbl 0299.14020)]. The author of the paper under review proves that, for \(\mathcal F\) a torsion free sheaf of rank 1 or a \(\mu_H\)-stable vector bundle of rank 2 and \(c_1(\mathcal F)\neq 0\), the tautological sheaf \(\mathcal F^{[2]}\) is \(\mu_{H_N}\)-stable for \(N\gg 0\).

The special feature of \(X^{[2]}\) that is used in the proof is the identification of the universal family \(\Xi_2\) with the blow-up of \(X\times X\) along the diagonal. In Section 1, this fact is recalled and several results concerning the geometry of the varieties involved are collected. In Section 2, tautological sheaves are introduced and formulae for their first Chern class and their slope are derived. It is shown in Section 3 that, for \(\mathcal F\neq \mathcal O_X\) a \(\mu_H\)-stable vector bundle, the tautological bundle \(\mathcal F^{[2]}\) does not contain any \(\mu_{H_N}\)-destabilising line bundles if \(N\gg 0\). This is used in the proofs of the main results which are carried out in Section 4. The final Section 5 shows that the assumption on the non-triviality of the first Chern class of \(\mathcal F\) is really necessary. More concretely, \(\mathcal O_{X^{[2]}}\) is a \(\mu_{H_N}\)-destabilising line bundle in \(\mathcal O_X^{[2]}\) for every \(N\gg0\).

The paper under review generalises a result of [U. Schlickewei, Rend. Semin. Mat. Univ. Padova 124, 127–138 (2010; Zbl 1208.14036)]. The result concerning the stability of tautological bundles associated to line bundles was generalised to Hilbert schemes \(X^{[n]}\) of an arbitrary number of points \(n\) in the recent preprint [D. Stapleton, “Geometry and stability of tautological bundles on Hilbert schemes of points”, arXiv:1409.8229].

As a first step in the study of this question, one needs examples of stable sheaves on \(\mathcal M\). The author provides such examples in the case that \(\mathcal M=X^{[2]}\) is the Hilbert scheme of two points on the \(K3\) surface \(X\).

To every sheaf \(\mathcal F\) on \(X\), one can associate the sheaf \(\mathcal F^{[2]}:= a_*b^* \mathcal F\), called tautological sheaf, on \(X^{[2]}\). Here, \[ a: \Xi_2 \to X^{[2]}\quad \text{and}\quad b: \Xi_2\to X \] denote the projections from the universal family \(\Xi_2\subset X^{[2]}\times X\). Since \(a: \Xi_2\to X^{[2]}\) is flat and finite of degree 2, we have \(\text{rank} \mathcal F^{[2]}=2\cdot \text{rank}\mathcal F\). Furthermore, if \(\mathcal F\) is a vector bundle or torsion free, the same holds for \(\mathcal F^{[2]}\). In this article, it is shown that stable sheaves of low rank on \(X\) induce stable tautological sheaves on the moduli space \(X^{[2]}\).

More precisely, the results are as follows. Let \(X\) be a smooth projective surface with \(H^1(X,\mathcal O_X)=0\), and let \(H\) be a polarisation on \(X\). For \(N\in \mathbb N\) sufficiently large, we consider the polarisation \(H_N:=N\cdot H-\delta\) on \(X^{[2]}\) where we use the usual isomorphism \(\text{Pic}(X^{[2]})\cong \text{Pic}(X) \oplus \mathbb Z\cdot \delta\); see J. Fogarty [Am. J. Math. 95, 660–687 (1973; Zbl 0299.14020)]. The author of the paper under review proves that, for \(\mathcal F\) a torsion free sheaf of rank 1 or a \(\mu_H\)-stable vector bundle of rank 2 and \(c_1(\mathcal F)\neq 0\), the tautological sheaf \(\mathcal F^{[2]}\) is \(\mu_{H_N}\)-stable for \(N\gg 0\).

The special feature of \(X^{[2]}\) that is used in the proof is the identification of the universal family \(\Xi_2\) with the blow-up of \(X\times X\) along the diagonal. In Section 1, this fact is recalled and several results concerning the geometry of the varieties involved are collected. In Section 2, tautological sheaves are introduced and formulae for their first Chern class and their slope are derived. It is shown in Section 3 that, for \(\mathcal F\neq \mathcal O_X\) a \(\mu_H\)-stable vector bundle, the tautological bundle \(\mathcal F^{[2]}\) does not contain any \(\mu_{H_N}\)-destabilising line bundles if \(N\gg 0\). This is used in the proofs of the main results which are carried out in Section 4. The final Section 5 shows that the assumption on the non-triviality of the first Chern class of \(\mathcal F\) is really necessary. More concretely, \(\mathcal O_{X^{[2]}}\) is a \(\mu_{H_N}\)-destabilising line bundle in \(\mathcal O_X^{[2]}\) for every \(N\gg0\).

The paper under review generalises a result of [U. Schlickewei, Rend. Semin. Mat. Univ. Padova 124, 127–138 (2010; Zbl 1208.14036)]. The result concerning the stability of tautological bundles associated to line bundles was generalised to Hilbert schemes \(X^{[n]}\) of an arbitrary number of points \(n\) in the recent preprint [D. Stapleton, “Geometry and stability of tautological bundles on Hilbert schemes of points”, arXiv:1409.8229].

Reviewer: Andreas Krug (Marburg)

### MSC:

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

14D20 | Algebraic moduli problems, moduli of vector bundles |

14J28 | \(K3\) surfaces and Enriques surfaces |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

### Keywords:

Hilbert squares of \(K3\) surfaces; tautological sheaves; stable sheaves on higher-dimensional holomorphic symplectic varieties### References:

[1] | G. Danila, Sur la cohomologie d’un fibré tautologique sur le schéma de Hilbert d’une surface , J. Algebraic Geom. 10 (2001), 247-280. · Zbl 0988.14011 |

[2] | G. Ellingsrud, L. Göttsche, and M. Lehn, On the cobordism class of the Hilbert scheme of a surface , J. Algebraic Geom. 10 (2001), 81-100. · Zbl 0976.14002 |

[3] | W. Fulton, Intersection Theory , Ergeb. Math. Grenzgeb. (3) 2 , Springer, Berlin, 1984. · Zbl 0541.14005 |

[4] | R. Hartshorne, Algebraic Geometry , Grad. Texts in Math. 52 , Springer, New York, 1977. · Zbl 0367.14001 |

[5] | S. Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface , Invent. Math. 77 (1984), 101-116. · Zbl 0565.14002 |

[6] | L. Scala, Some remarks on tautological sheaves on Hilbert schemes of points on a surface , Geom. Dedicata 139 (2009), 313-329. · Zbl 1169.14004 |

[7] | U. Schlickewei, Stability of tautological vector bundles on Hilbert squares of surfaces , Rend. Semin. Mat. Univ. Padova 124 (2010), 127-138. · Zbl 1208.14036 |

[8] | M. Wandel, Tautological sheaves: Stability, moduli spaces and restrictions to generalised Kummer varieties , preprint, [math.AG]. 1308.4304v1 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.