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A Hitchin-Kobayashi correspondence for Kähler fibrations. (English) Zbl 1002.53057

Let \(X\) be a compact Kähler manifold, \(E\to X\) a \(K\)-principal fibre bundle, where \(K\) is a connected compact Lie group, and let \(\mathcal A^{1,1}\) denote the space of \(K\)-connections \(A\) on \(E\), whose curvature form \(F_A\) is of type \((1,1)\). These connections correspond to integrable complex structures on the associated \(G\)-principal bundle \(E_G\), where \(G = K(\mathbf C)\). The gauge group \(\mathcal G_G = \Gamma(E\times_{\text{Ad}}G)\) acts on \(\mathcal A^{1,1}\).
Suppose that a Hamiltonian action of \(K\) by biholomorphic transformations of a Kähler manifold \(F\) is given and denote by \(\mathcal S\) the set of smooth sections of the associated bundle on \(X\) with fibre \(F\). Identifying the Lie algebra \(\mathfrak k\) of \(K\) with \(\mathfrak k^*\), the author considers the equation \(\Lambda F_A + \mu(\Phi) = c\), where \(A\in\mathcal A^{1,1}, \Phi\in\mathcal S\), and \(c\in\mathfrak k\) is a constant central element. Here \(\Lambda\) is the operator on forms adjoint to the wedging with the Kähler form on \(X\), and \(\mu\) is the moment map. The following problem is studied: which pairs \((A,\Phi)\in\mathcal A^{1,1}\times\mathcal S\) can be mapped by a gauge transformation onto a solution of the above equation? The author solves this problem for the so-called simple pairs \((A,\Phi)\), i.e., those pairs which are not left fixed by a non-trivial semisimple 1-parameter subgroup of \(\mathcal G_G\). The main theorem describes the simple pairs lying on \(\mathcal G_G\)-orbits of solutions of the equation in terms of the so-called \(c\)-stability. It also claims that if \((A,\Phi)\) is a simple pair and if \((g(A),g(\Phi))\) and \((g'(A),g'(\Phi))\), \(g,g'\in\mathcal G_G\), are solutions of the equation, then \(g'g^{-1}\) lies in the real gauge group \(\mathcal G_K\).
Another important result is the construction of the generalized Yang-Mills-Higgs functional, whose local minima coincide with the solutions of the equation.

MSC:

53D20 Momentum maps; symplectic reduction
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
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