Moduli Spaces of PU ( 2 ) -Instantons on Minimal Class VII Surfaces with b 2 = 1

Konrad Schöbel[1]

  • [1] Université de Provence Centre de Mathématiques et Informatique Laboratoire d’Analyse, Topologie et Probabilités 39 rue F. Joliot Curie 13453 Marseille cedex 13 (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 5, page 1691-1722
  • ISSN: 0373-0956

Abstract

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We describe explicitly the moduli spaces ℳ g pst ( S , E ) of polystable holomorphic structures ℰ with det ℰ ≅ 𝒦 on a rank two vector bundle E with c 1 ( E ) = c 1 ( K ) and c 2 ( E ) = 0 for all minimal class VII surfaces S with b 2 ( S ) = 1 and with respect to all possible Gauduchon metrics g . These surfaces S are non-elliptic and non-Kähler complex surfaces and have recently been completely classified. When S is a half or parabolic Inoue surface, ℳ g pst ( S , E ) is always a compact one-dimensional complex disc. When S is an Enoki surface, one obtains a complex disc with finitely many transverse self-intersections whose number becomes arbitrarily large when g varies in the space of Gauduchon metrics. ℳ g pst ( S , E ) can be identified with a moduli space of PU ( 2 ) -instantons. The moduli spaces of simple bundles of the above type lead to interesting examples of non-Hausdorff singular one-dimensional complex spaces.

How to cite

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Schöbel, Konrad. "Moduli Spaces of ${\rm PU}(2)$-Instantons on Minimal Class VII Surfaces with $b_2=1$." Annales de l’institut Fourier 58.5 (2008): 1691-1722. <http://eudml.org/doc/10359>.

@article{Schöbel2008,
abstract = {We describe explicitly the moduli spaces $\mathscr\{M\}^\{\rm pst\}_g(S,E)$ of polystable holomorphic structures $\mathcal\{E\}$ with $\det \mathcal\{E\}\cong \mathcal\{K\}$ on a rank two vector bundle $E$ with $c_1(E)=c_1(K)$ and $c_2(E)=0$ for all minimal class VII surfaces $S$ with $b_2(S)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are non-elliptic and non-Kähler complex surfaces and have recently been completely classified. When $S$ is a half or parabolic Inoue surface, $\mathscr\{M\}^\{\rm pst\}_g(S,E)$ is always a compact one-dimensional complex disc. When $S$ is an Enoki surface, one obtains a complex disc with finitely many transverse self-intersections whose number becomes arbitrarily large when $g$ varies in the space of Gauduchon metrics. $\mathscr\{M\}^\{\rm pst\}_g(S,E)$ can be identified with a moduli space of $\{\rm PU\}(2)$-instantons. The moduli spaces of simple bundles of the above type lead to interesting examples of non-Hausdorff singular one-dimensional complex spaces.},
affiliation = {Université de Provence Centre de Mathématiques et Informatique Laboratoire d’Analyse, Topologie et Probabilités 39 rue F. Joliot Curie 13453 Marseille cedex 13 (France)},
author = {Schöbel, Konrad},
journal = {Annales de l’institut Fourier},
keywords = {Moduli spaces; holomorphic bundles; complex surfaces; instantons; moduli spaces; gauge theory; classification; class VII surfaces},
language = {eng},
number = {5},
pages = {1691-1722},
publisher = {Association des Annales de l’institut Fourier},
title = {Moduli Spaces of $\{\rm PU\}(2)$-Instantons on Minimal Class VII Surfaces with $b_2=1$},
url = {http://eudml.org/doc/10359},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Schöbel, Konrad
TI - Moduli Spaces of ${\rm PU}(2)$-Instantons on Minimal Class VII Surfaces with $b_2=1$
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 5
SP - 1691
EP - 1722
AB - We describe explicitly the moduli spaces $\mathscr{M}^{\rm pst}_g(S,E)$ of polystable holomorphic structures $\mathcal{E}$ with $\det \mathcal{E}\cong \mathcal{K}$ on a rank two vector bundle $E$ with $c_1(E)=c_1(K)$ and $c_2(E)=0$ for all minimal class VII surfaces $S$ with $b_2(S)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are non-elliptic and non-Kähler complex surfaces and have recently been completely classified. When $S$ is a half or parabolic Inoue surface, $\mathscr{M}^{\rm pst}_g(S,E)$ is always a compact one-dimensional complex disc. When $S$ is an Enoki surface, one obtains a complex disc with finitely many transverse self-intersections whose number becomes arbitrarily large when $g$ varies in the space of Gauduchon metrics. $\mathscr{M}^{\rm pst}_g(S,E)$ can be identified with a moduli space of ${\rm PU}(2)$-instantons. The moduli spaces of simple bundles of the above type lead to interesting examples of non-Hausdorff singular one-dimensional complex spaces.
LA - eng
KW - Moduli spaces; holomorphic bundles; complex surfaces; instantons; moduli spaces; gauge theory; classification; class VII surfaces
UR - http://eudml.org/doc/10359
ER -

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