On the symplectic structures on moduli space of stable sheaves over a K3 or abelian surface and on Hilbert scheme of points

Abstract Let $f \colon X \to Y$ be the blow-up of a smooth projective variety $Y$ along its codimension two smooth closed subvariety. In this paper, we show that the moduli space of stable sheaves on $X$ and $Y$ are connected by a sequence of flip-like diagrams. The result is a higher dimensional generalization of the result of Nakajima and Yoshioka, which is the case of $\dim Y=2$. As an application of our general result, we study the birational geometry of the Hilbert scheme of two points.

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Stability of some vector bundles on Hilbert schemes of points on K3 surfaces

Mathematische Zeitschrift ◽

10.1007/s00209-021-02920-6 ◽

2021 ◽

Author(s):

Fabian Reede ◽

Ziyu Zhang

Keyword(s):

Moduli Space ◽

Hilbert Scheme ◽

Vector Bundles ◽

K3 Surfaces ◽

Hilbert Schemes ◽

Connected Component ◽

Stable Vector Bundles ◽

Universal Family ◽

Hilbert Scheme Of Points ◽

Flat Family

AbstractLet X be a projective K3 surfaces. In two examples where there exists a fine moduli space M of stable vector bundles on X, isomorphic to a Hilbert scheme of points, we prove that the universal family $${\mathcal {E}}$$ E on $$X\times M$$ X × M can be understood as a complete flat family of stable vector bundles on M parametrized by X, which identifies X with a smooth connected component of some moduli space of stable sheaves on M.

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Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points

Complex Manifolds ◽

10.1515/coma-2017-0003 ◽

2017 ◽

Vol 4 (1) ◽

pp. 16-36 ◽

Cited By ~ 4

Author(s):

Roger Bielawski

Keyword(s):

Moduli Space ◽

Hilbert Scheme ◽

Double Cover ◽

Hilbert Schemes ◽

Hilbert Scheme Of Points ◽

Adjoint Orbits ◽

Hyperkähler Metrics

Abstract We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of GL(n, ℂ) is isomorphic to the deformation of the D2-singularity if n = 2, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if n = 3, and to the Atiyah-Hitchin manifold itself if n = 4. For higher n, such slices to the sum of two orbits, each having only two distinct eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert scheme of points on one of the above surfaces. In particular, these open subsets of Hilbert schemes of points carry complete hyperkähler metrics. In the case of the double cover of the Atiyah-Hitchin manifold this metric turns out to be the natural L2-metric on a hyperkähler submanifold of the monopole moduli space.

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Stability and the Fourier–Mukai transform II

Compositio Mathematica ◽

10.1112/s0010437x08003758 ◽

2009 ◽

Vol 145 (1) ◽

pp. 112-142 ◽

Cited By ~ 15

Author(s):

Kōta Yoshioka

Keyword(s):

Moduli Space ◽

Abelian Surface ◽

Stable Sheaves ◽

The Stability

AbstractWe consider the problem of preservation of stability under the Fourier–Mukai transform ℱℰ:D(X)→D(Y) on an abelian surface and aK3 surface. IfYis the moduli space ofμ-stable sheaves onXwith respect to a polarizationH, we have a canonical polarization$\widehat {H}$onYand we have a correspondence between (X,H) and$(Y,\widehat {H})$. We show that the stability with respect to these polarizations is preserved under ℱℰ, if the degree of stable sheaves onXis sufficiently large.

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MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

Modern Physics Letters A ◽

10.1142/s0217732398002904 ◽

1998 ◽

Vol 13 (34) ◽

pp. 2731-2742 ◽

Cited By ~ 1

Author(s):

YUTAKA MATSUO

Keyword(s):

Hilbert Scheme ◽

Matrix Theory ◽

Fock Space ◽

Homology Class ◽

Orthogonal Basis ◽

Inner Product ◽

Virasoro Symmetry ◽

Hilbert Scheme Of Points ◽

The Matrix ◽

Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

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The integral cohomology of the Hilbert scheme of points on a surface

Forum of Mathematics Sigma ◽

10.1017/fms.2020.35 ◽

2020 ◽

Vol 8 ◽

Author(s):

Burt Totaro

Keyword(s):

Natural Number ◽

Hilbert Scheme ◽

Obstruction Theory ◽

Projective Surface ◽

Hilbert Schemes ◽

Complex Projective Surface ◽

Smooth Complex ◽

Hilbert Scheme Of Points ◽

Complex Projective ◽

Torsion Free

Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

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On the Singular Sheaves in the Fine Simpson Moduli Spaces of 1-dimensional Sheaves

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-059-7 ◽

2017 ◽

Vol 60 (3) ◽

pp. 522-535 ◽

Cited By ~ 1

Author(s):

Oleksandr Iena ◽

Alain Leytem

Keyword(s):

Projective Plane ◽

Moduli Space ◽

Moduli Spaces ◽

Blow Up ◽

Line Bundles ◽

Hilbert Polynomial ◽

Stable Sheaves ◽

Closed Subvariety

AbstractIn the Simpson moduli space M of semi-stable sheaves with Hilbert polynomial dm − 1 on a projective plane we study the closed subvariety M' of sheaves that are not locally free on their support. We show that for d ≥4 , it is a singular subvariety of codimension 2 in M. The blow up of M along M' is interpreted as a (partial) modification of M \ M' by line bundles (on support).

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