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1. CJM Online first
Orthogonal Bundles and Skew-Hamiltonian Matrices Using properties of skew-Hamiltonian matrices and classic
connectedness results, we prove that the moduli space
$M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles
on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial
splitting on the general line, is smooth irreducible of
dimension $(r-2)n-\binom{r}{2}$ for $r=n$ and $n \ge 4$, and
$r=n-1$ and $n\ge 8$. We speculate that the result holds in
greater generality.
Keywords:orthogonal vector bundles, moduli spaces, skew-Hamiltonian matrices Categories:14J60, 15B99 |
2. CJM 2014 (vol 66 pp. 961)
Moduli Spaces of Vector Bundles over a Real Curve: $\mathbb Z/2$-Betti Numbers Moduli spaces of real bundles over a real curve arise naturally
as Lagrangian submanifolds of the moduli space of semi-stable
bundles over a complex curve. In this paper, we adapt the methods
of Atiyah-Bott's ``Yang-Mills over a Riemann Surface'' to compute
$\mathbb Z/2$-Betti numbers of these spaces.
Keywords:cohomology of moduli spaces, holomorphic vector bundles Categories:32L05, 14P25 |
3. CJM 2007 (vol 59 pp. 845)
Representations of the Fundamental Group of an $L$-Punctured Sphere Generated by Products of Lagrangian Involutions |
Representations of the Fundamental Group of an $L$-Punctured Sphere Generated by Products of Lagrangian Involutions In this paper, we characterize unitary representations of $\pi:=\piS$ whose
generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially)
can be decomposed as products of two Lagrangian involutions
$u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. Our main result is that such
representations are exactly the elements of the fixed-point set of an
anti-symplectic involution defined on the moduli space
$\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixed-point set is
non-empty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use
the quasi-Hamiltonian description of the symplectic structure of $\Mod$ and
give conditions on an involution defined on a quasi-Hamiltonian $U$-space
$(M, \w, \mu\from M \to U)$ for it to induce an anti-symplectic involution on
the reduced space $M/\!/U := \mu^{-1}(\{1\})/U$.
Keywords:momentum maps, moduli spaces, Lagrangian submanifolds, anti-symplectic involutions, quasi-Hamiltonian Categories:53D20, 53D30 |