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Deformation of complex structures and the coupled Kähler-Yang-Mills equations. (English) Zbl 1300.53026

The first author introduced the notion of coupled Kähler-Yang-Mills system in his Ph.D thesis. This is a generalization of both the Yang-Mills equation and the constant scalar curvature equation. Given a triple \((X,L,E)\) where \(X\) is a complex manifold polarized by \(L\) and \(E\) a holomophic principal \(G^c\) bundle, one considers the system of non-linear PDEs in \((\omega, H)\) given by \[ \begin{aligned} \Lambda_\omega F &= z ,\\ \mathrm{Scal}(\omega) - \alpha \Lambda^2_\omega \mathrm{tr}(F\wedge F)&=c,\end{aligned} \] where \(\omega\) is a Kähler metric, \(F\) the curvature of the Chern connection associated to \(H\), \(z\) is an element of the center of the Lie algebra of \(G\) and \(c\) is a topological constant. The system depends also on a real constant \(\alpha\). It arises from a symplectic construction, i.e., as the zero of a certain moment map associated to the action of a group \(\tilde{\mathcal{G}}\) on a certain parameter space. Here, \(\tilde{\mathcal{G}}\) is given by a non-trivial extension of the gauge group \(\mathcal{G}\) and the group of Hamiltonian symplectomorphisms. This explains why this system is a natural generalization of both the constant scalar curvature equation and the Hermitian-Einstein equation. We refer to [L. Álvarez-Cónsul et al., Geom. Topol. 17, No. 5, 2731–2812 (2013; Zbl 1275.32019)] or the Ph.D thesis of the first author for details. Non-trivial examples of coupled Kähler-Yang-Mills equations have been found over ruled surfaces in [the reviewer and C. W. Tønnesen-Friedman, Cent. Eur. J. Math. 10, No. 5, 1673–1687 (2012; Zbl 1269.53049)].
In the paper, the authors provide a deformation theory for the system above. The main theorem is the following: Assume that \((X,L,E)\) admits a solution with \(\alpha>0\), then any small deformation \((X',L',E')\) with close \(K^c\) orbit in \(H^1(X,L_\omega^*)\) also admits a solution, and in particular if \(K\) is finite, the existence of a solution is an open condition under deformations of the complex structure.
Here, \(K\) is the group of holomorphic automorphisms of \(E\) lying in the extended gauge group \(\tilde{\mathcal{G}}\); it admits a complexification denoted \(K^c\) and a finite-dimensional representation encoding infinitesimal deformations of the complex structure on \(E\) and \(X\).
The proof is a nice application of Kuranishi’s method and of some work of G. Székelyhidi relating a moment map problem on a vector space to its linearization. As an application, the authors find some new examples of solutions to the system.

MSC:

53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
32Q15 Kähler manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58E11 Critical metrics
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References:

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