JournalsjemsVol. 13, No. 5pp. 1373–1422

The gradient flow of Higgs pairs

  • Jiayu Li

    Chinese Academy of Sciences, Beijing, China
  • Xi Zhang

    Zhejiang University, Hangzhou, China
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Abstract

In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional of Higgs pairs on a Hermitian vector bundle (E,H0)(E , H_{0}) over a Kähler surface (M,ω)(M , \omega ), and study the asymptotic behavior of the heat flow for Higgs pairs at infinity. The main result is that the gradient flow with initial condition (A0,ϕ0)(A_{0} , \phi_{0}) converges, in an appropriate sense which takes into account bubbling phenomena, to a critical points (A,ϕ)(A_{\infty } , \phi_{\infty}) of this functional. We also prove that the limiting Higgs pair (A,ϕ)(A_{\infty }, \phi_{\infty}) can be extended smoothly to a vector bundle EE_{\infty } over (M,ω)(M , \omega ) , and the isomorphism class of the limiting Higgs bundle (E,A,ϕ)(E_{\infty } , A_{\infty } , \phi_{\infty}) is given by the double dual of the graded Higgs sheaves associate to Harder-Narasimhan-Seshadri filtration of the initial Higgs bundle (E,A0,ϕ0)(E , A_{0} , \phi_{0}).

Cite this article

Jiayu Li, Xi Zhang, The gradient flow of Higgs pairs. J. Eur. Math. Soc. 13 (2011), no. 5, pp. 1373–1422

DOI 10.4171/JEMS/284