publication . Preprint . 2014

The limit of the Yang-Mills-Higgs flow on Higgs bundles

Li, Jiayu; Zhang, Xi;
Open Access English
  • Published: 30 Oct 2014
In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional for Higgs pairs on a Hermitian vector bundle $(E, H_{0})$ over a compact K\"ahler manifold $(M, \omega )$. We study the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity, and show that the limiting Higgs sheaf is isomorphic to the double dual of the graded Higgs sheaves associated to the Harder-Narasimhan-Seshadri filtration of the initial Higgs bundle.
arXiv: High Energy Physics::PhenomenologyHigh Energy Physics::LatticeMathematics::Algebraic GeometryHigh Energy Physics::ExperimentMathematics::Symplectic Geometry
free text keywords: Mathematics - Differential Geometry, 53C07, 58E15
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53 references, page 1 of 4

21π RM (T r(√−1Λω(FAj + [φj , φj∗])πj ) − |∂Aj+φj πj |2)

21π RM (T r(√

21π RM (T r(aπ−j )1Λ+ω2(1πFARMj +(T[rφ(j √,φ−j∗1])Λπωj )((FAj + [φj , φj∗]) − a)πj).

Let µ~ 0 = (µ 1, · · · , µ R) be the HN type of Higgs bundle (E, A0, φ0), by (2.4) in

Lemma 2.1, we have

Definition 5.6. Fix δ > 0 and 1 ≤ p ≤ ∞. An Lp-δ-approximate critical

−1 Λω(FAH + [φ, φ∗H ]) − ΨHN (E, φ, H)kLp(ω) ≤ δ,

bundles (Qi, φi) are ωǫ-stable for all 0 < ǫ ≤ ǫ∗, by Simpson's result (Theorem 1

in [36]), we have a Hermitian-Einstein metric Hi,ǫ on Higgs bundle (Qi, φi) respect

(5.8) 2−π1 Λωǫ (F(∂Qi ,Hi,ǫ) + [φi, φi∗]) − µ ǫ(Qi)IdQi = 0. 2δ0 + HY Mα,N (µ~ 0)

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and stable holomorphic chains, Int. J. Math., 2(2001), 159-201. [3] O.Biquard, On parabolic bundles over a complex surface, J. London. Math. Soc., 53(1996),

no.2, 302-316. [4] S.B.Bradlow, Vortices in holomorphic line bundles over closed K¨ahler manifolds, Com-

mun.Math.Phys. 135(1990), 1-17. [5] S.B.Bradlow and O. Garcia-Prada, Stable triples, equivariant bundles and dimensional re-

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