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:
Stability: If \(E\) is indecomposable then \(E\) is \(MA\)-stable.
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.
Stability: \(r_1>r_2\).
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.


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


[1] Bismut, J-M.; Gillet, H.; Soulé, C., Analytic torsion and holomorphic determinant bundles III - Quillen metrics on holomorphic determinants, Comm. Math. Phys., 115, 301-351 (1988) · Zbl 0651.32017
[2] Collins, T.; Jacob, A.; Yau, S. T., \((1, 1)\) forms with specified Lagrangian phase: a priori estimates and algebraic obstructions
[3] Collins, T.; Xie, D.; Yau, S. T., The deformed Hermitian-Yang-Mills equation in geometry and physics · Zbl 1421.35300
[4] Donaldson, S., Infinite determinants, stable bundles and curvature, Duke Math. J., 54, 1, 231-247 (1987) · Zbl 0627.53052
[5] Fine, J., The Hamiltonian geometry of the space of unitary connections with symplectic curvature, J. Symplectic Geom., 12, 1, 105-123 (2014) · Zbl 1304.53082
[6] García-Prada, O., Invariant connections and vortices, Comm. Math. Phys., 156, 527-546 (1993) · Zbl 0790.53031
[7] Jacob, A., Existence of approximate Hermitian-Einstein structures on semi-stable bundles, Asian J. Math., 18, 5, 859-884 (2014) · Zbl 1315.53079
[8] Jacob, A.; Yau, S. T., A special Lagrangian type equation for holomorphic line bundles, Math. Ann., 369, 1-2, 869-898 (2017) · Zbl 1375.32045
[9] Lübke, M.; Teleman, A., The Kobayashi-Hitchin Correspondence (1995), World Scientific · Zbl 0849.32020
[10] Pingali, V., A note on the deformed Hermitian-Yang-Mills PDE, Complex Var. Elliptic Equ., 64, 3, 503-518 (2019) · Zbl 1414.35097
[11] Pingali, V., Quillen metrics and perturbed equations · Zbl 1442.53017
[12] Sibley, B., Asymptotics of the Yang-Mills flow for holomorphic vector bundles over Kähler manifolds: the canonical structure of the limit, J. Reine Angew. Math., 706, 123-191 (2015) · Zbl 1329.58006
[13] Sibley, B.; Wentworth, R., Analytic cycles, Bott-Chern forms, and singular sets for the Yang-Mills flow on Kähler manifolds, Adv. Math., 279, 501-531 (2015) · Zbl 1317.58016
[14] Uhlenbeck, K.; Yau, S. T., On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math., 39, S1, S257-S293 (1986) · Zbl 0615.58045
[15] Yau, S. T., Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci., 74, 5, 1798-1799 (1977) · Zbl 0355.32028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.