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On a relative Kobayashi-Hitchin correspondence. (English) Zbl 0826.53030

The authors write that the results of S. Donaldson [Commun. Math. Phys. 93, 453-460 (1984; Zbl 0581.14008)] and A. D. King [Instantons and holomorphic bundles on the blown up plane, Ph.D. Thesis, Oxford (1989)] can be formulated in terms of complex projective plane \(P^2\): the moduli space of holomorphic bundles on \(P^2\) which are trivial along a line \(L \subset P^2\) is in one-to-one correspondence with the moduli space of unitary connections which are framed by (i.e. gauge equivalent to) the standard flat connection in the trivial bundle over \(L\), and satisfy on \(P^2 \setminus L\) the Hermitian-Einstein equation with respect to the flat metric \(h_{\text{flat}}\). The aim of the reviewed paper is to study such a correspondence between framed holomorphic and framed Hermitian-Einstein structures in a more conceptual way.
Let \(X\) be a smooth, compact complex surface, \(C \subset X\) a smooth curve, and \(E \to X\) a differentiable complex vector bundle of rank \(r\). In \(Y = X \setminus C\) there is fixed a Kählerian metric \(\omega\), in \(E\) a Hermitian metric \(h\), and in \(E_0 = E|_C\) an integrable \(h_0\) connection \(A_0\) where \(h_0 = h|_{E_0}\). In this situation the authors define Hermitian-Einstein (HE) connections and study their basic properties. Then they define \(A_0\)-framed HE- connections and their moduli space \({\mathcal M}^\omega_X (E, h, A_0)\), construct a natural comparison map \(\rho : {\mathcal M}^\omega_X (E, h, A_0) \to {\mathcal M}^s_X (E, \overline {\partial}_0)\) into the moduli space of simple \(\overline {\partial}_0\)-framed holomorphic structures and investigates the map \(\rho\) in detail, discussing various conditions under which \(\rho\) is a bijection (in this case the authors say that a relative Kobayashi-Hitchin correspondence holds).

MSC:

53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
32G13 Complex-analytic moduli problems
58H15 Deformations of general structures on manifolds

Citations:

Zbl 0581.14008
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