## 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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