nLab
Hilbert scheme

Contents

Contents

Idea

Hilbert schemes are moduli spaces of subvarieties, hence configuration spaces in algebraic geometry.

For instance a scheme of 0-dimensional sub-schemes is called a Hilbert scheme of points, etc.

Specifically for quasi-projective variety with fixed Hilbert polynomial?, Hilbert schemes are well behave as moduli spaces go, in that they’re actually quasi-projective varieties themselves.

The existence and construction of Hilbert schemes is due to Grothendieck (FGA).

The Hilbert scheme of 2\mathbb{C}^2 is widely studied in combinatorics and geometric representation theory for its connections to Macdonald polynomials and Cherednik algebras.

Properties

Compact hyperkähler structure

The only known examples of compact hyperkähler manifolds are Hilbert schemes of points X [n+1]X^{[n+1]} (for nn \in \mathbb{N}) for XX either

  1. a K3-surface

  2. a 4-torus (in which case the compact hyperkähler manifolds is really the fiber of (𝕋 4) [n]𝕋 4(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4)

(Beauville 83) and two exceptional examples (O’Grady 99, O’Grady 03 ), see Sawon 04, Sec. 5.3.

References

General

See also

Hilbert schemes of points

  • Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999 (ams:ulect-18)

  • Hiraku Nakajima, More lectures on Hilbert schemes of points on surfaces, Advanced Studies in Pure Mathematics 69, 2016, Development of Moduli Theory – Kyoto 2013, 173-205 (arXiv:1401.6782)

  • J. Bertin, The punctual Hilbert scheme: An introduction (pdf)

  • Dori Bejleri, Hilbert schemes: Geometry, combionatorics, and representation theory (pdf)

  • Barbara Bolognese, Ivan Losev, A general introduction to the Hilbert scheme of points on the plane (pdf)

  • Joachim Jelisiejew, Pathologies on the Hilbert scheme of points (arXiv:1812.08531)

Discussion in relation to the Fulton-MacPherson compactifications of configuration spaces of points:

Discussion of Euler numbers of Hilbert schemes of points:

  • Hiraku Nakajima, Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras (arXiv:2001.03834)

As moduli spaces of instantons

Discussion in their role as moduli spaces of instantons:

Specifically in relation to Donaldson-Thomas theory:

  • Michele Cirafici, Annamaria Sinkovics, Richard Szabo, Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory, Nucl. Phys. B809: 452-518, 2009 (arXiv:0803.4188)

  • Artan Sheshmani, Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories, Notices of the International Congress of Chines Mathematics (2019) (j.mp:2U7qd01, pdf)

  • Artan Sheshmani, Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten theories (arxiv:1911.01796)

See also

  • Jian Zhou, K-Theory of Hilbert Schemes as a Formal Quantum Field Theory (arXiv:1803.06080)

Of ADE singularities

Discussion of the Hilbert schemes of points of ADE-singularities:

See also:

Of K3 surfaces

Discussion of the Hilbert schemes of points of K3-surfaces:

Discussion of configuration spaces of possibly coincident points on K3-surfaces XX, hence of symmetric products X n/Sym(n)X^n/Sym(n) as moduli spaces of D0-D4-brane bound states wrapped on K3-surfaces:

Suggestion that this is to be resolved by the Hilbert scheme of points:

Hilbert schemes on K3 as moduli space of stable vector bundles:

  • Laura Costa Farràs, K3 surfaces: moduli spaces and Hilbert schemes, Collectanea Mathematica, 1998, vol. 49, núm. 2-3, p. 273-282 (hdl:2445/16925

with an eye towards Rozansky-Witten theory (ground field-valued Rozansky-Witten weight systems):

Hilbert schemes and Higgs/Coulomb branches

Identification of Higgs branches/Coulomb branches in D=3 N=4 super Yang-Mills theory with Hilbert schemes of points of complex curves:

Discussion in the context of Witten indices and K-theoretic enumerative geometry:

Last revised on January 14, 2020 at 01:28:08. See the history of this page for a list of all contributions to it.