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A vector bundle version of the Monge-Ampère equation. (English) Zbl 1452.32047

The aim of this paper is to introduce a vector bundle version of the complex Monge-Ampère equation motivated by the study of stability conditions involving higher Chern forms. Let \(M\) be a compact complex \(n\)-dimensional manifold, \(h\) be a metric on a holomorphic vector bundle \(E\) over \(M\). The vector bundle Monge-Ampère equation is \[ \left(\frac {i\Theta_h}{2\pi}\right)^n=\eta \mathrm{Id}, \] where \(\Theta_h\) is the curvature of the Chern connection of \((E,h)\) and \(\eta\) is a given volume form. Motivated by the Riemann surface case, the existence of a solution of the vector bundle complex Monge-Ampère equation would require some positivity condition on \(i\Theta_h\).
The first main result of the paper provides some consequences of the existence of a positively curved solution to this equation: stability (involving the second Chern character) and a Kobayashi-Lübke-Bogomolov-Miyaoka-Yau-type inequality. Let \(E\) be a holomorphic rank-\(2\) vector bundle on a smooth projective surface \(M\), \(\eta>0\) is a given volume form. If there exists a smooth metric \(h\) such that \((i\Theta_h)^2=\eta \mathrm{Id}\) and \(i\Theta_h\) is Griffiths positive, then the following hold:
(1)
Stability: If \(E\) is indecomposable then \(E\) is \(MA\)-stable.
(2)
Chern class inequality: \(c_1^2(E)-4c_2(E)\leq 0\) with equality holding if and only if \(\Theta\) is projectively flat.

The second main result of this paper is the following theorem. Let \((L,h_0)\) be a holomorphic line bundle over a compact Riemann surface \(X\) such that its curvature \(\Theta_0\) defines a Kähler form \(\omega_{\Sigma}=i\Theta_0\) over \(M\). Assume that the degree of \(L\) is \(1\). Let \(r_1,r_2\geq 2\) be two integers, and \(\phi\in H^0(X,L)\) which is not identically \(0\). Then the following are equivalent.
(1)
Stability: \(r_1>r_2\).
(2)
Existence: There exists a smooth metric \(h\) on \(L\) such that the curvature \(\Theta_h\) of its Chern connection \(\nabla_h\) satisfies the Monge-Ampère Vortex equation \[ i\Theta_h=(1-|\phi|_h^2)\frac {\mu\omega_{\Sigma}+i\nabla_h^{1,0}\phi\wedge\nabla^{0,1}\phi^{*_h}}{(2r_2+|\phi|_h^2)(2+2r_2-|\phi|_h^2)}, \] where \(\mu=2(r_2(r_1+1)+r_1(r_2+1))\) and \(\phi^{*_h}\) is the adjoint of \(\phi\) with respect to \(h\) when \(\phi\) is considered as an endomorphism of the trivial line bundle to \(L\).
Moreover, if a solution \(h\) of the above equation satisfying \(|\phi|_h^2\leq 1\) exists, then it is unique.

MSC:

32W20 Complex Monge-Ampère operators
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58J05 Elliptic equations on manifolds, general theory
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References:

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