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Search: MSC category 14J60 ( Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] )

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1. CJM 2015 (vol 67 pp. 961)

 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 matricesCategories:14J60, 15B99
 Integrable Systems Associated to a Hopf Surface A Hopf surface is the quotient of the complex surface $\mathbb{C}^2 \setminus \{0\}$ by an infinite cyclic group of dilations of $\mathbb{C}^2$. In this paper, we study the moduli spaces $\mathcal{M}^n$ of stable $\SL (2,\mathbb{C})$-bundles on a Hopf surface $\mathcal{H}$, from the point of view of symplectic geometry. An important point is that the surface $\mathcal{H}$ is an elliptic fibration, which implies that a vector bundle on $\mathcal{H}$ can be considered as a family of vector bundles over an elliptic curve. We define a map $G \colon \mathcal{M}^n \rightarrow \mathbb{P}^{2n+1}$ that associates to every bundle on $\mathcal{H}$ a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map $G$ is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on $\mathcal{M}^n$. We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not K\"ahler, it is an elliptic fibration that does not admit a section. Categories:14J60, 14D21, 14H70, 14J27