## TAME HARMONIC BUNDLES AND AN APPLICATION (2006)

### BibTeX

@MISC{Mochizuki06tameharmonic,

author = {Takuro Mochizuki and Takuro Mochizuki and Takuro Mochizuki},

title = {TAME HARMONIC BUNDLES AND AN APPLICATION},

year = {2006}

}

### OpenURL

### Abstract

Key words and phrases. — Higgs bundle, harmonic bundle, Kobayashi-Hitchin correspondence, Hermitian-Einstein metric, Bogomolov-Gieseker inequality, flat bundle, variation of polarized Hodge structure, quasi projective variety.

### Citations

162 |
On the existence of Hermitian YangMills connections in stable vector bundles
- Uhlenbeck, Yau
- 1986
(Show Context)
Citation Context .... Donaldson pioneered the way for the inverse problem ([11] and [12]). He attributed the problem to Kobayashi and N. Hitchin. The definitive result was given by K. Uhlenbeck, S. T. Yau and Donaldson (=-=[62]-=- and [13]). We also remark that V. Mehta and A. Ramanathan ([39]) proved the correspondence in the case where the Chern class is trivial, i.e., the correspondence of flat unitary bundles and stable ve... |

113 |
Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization
- Simpson
- 1988
(Show Context)
Citation Context ...mething related to pluri-harmonic metrics of G-flat bundles. 1.5. Acknowledgement The author owes much thanks to C. Simpson. The paper is a result of an effort to understand his works, in particular, =-=[49]-=- and [51], where the reader can find the framework and many ideas used in this paper. The problem of Bogomolov-Gieseker inequality was passed to the author from him a few years ago. The author express... |

89 |
Connections with L p bounds on curvature
- Uhlenbeck
- 1982
(Show Context)
Citation Context ...ack via the map φγ : X∗ −→ X∗ given by φγ(z) = γ · z, we may assume to have C1-orthonormal frames vn of ϕ∗E |Hn satisfying the following conditions, due to Proposition 7.5 and the Uhlenbeck’s result (=-=[61]-=-): – Let An denote the connection one form of ∇ with respect to vn on Hn. Then the norm of An with respect to g0 is dominated as ∣ An∣ ≤ C · n g0 −2 , where C is a good constant. Proposition 7.9. — Th... |

82 |
Harmonic bundles on noncompact curves
- Simpson
(Show Context)
Citation Context ...following: – The decomposition is orthogonal with respect to both of hi. – The restrictions of hi to Ea are denoted by hi,a. Then there exist positive numbers ba such that h1,a = ba · h2,a. Proof See =-=[50]-=-. We give only a remark on the uniqueness. Let (E, ∂E, θ) be a Higgs bundle on X − D, and hi (i = 1, 2) be harmonic metrics on it. Assume that the induced prolongments cE(hi) are isomorphic. (See Sect... |

53 | The Hodge filtration on nonabelian cohomology. Algebraic geometry - Simpson - 1995 |

33 |
Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves
- Yokogawa
- 1989
(Show Context)
Citation Context ...lic Higgs Bundle 3.1.1. c-Parabolic Higgs sheaf. — Let us recall the notion of parabolic structure and the Chern characteristic numbers of parabolic bundles following [33], [37], [49], [50], [60] and =-=[63]-=-. Our convention is slightly different from theirs. Let X be a connected complex manifold and D be a simple normal crossing divisor with the irreducible decomposition D = ⋃ i∈S Di. Let c = (ci ∣i ∈ S)... |

18 |
Trautmann: Gap–sheaves and extension of coherent analytic subsheaves
- Siu, G
- 1971
(Show Context)
Citation Context ...→ E ′ which is isomorphic outside of the subset Z of codimension two. Let iF1 a denote the subsheaf of E ′ which consists of the sections f of E ′ such that f |X−Z ∈ iFa. Such a subsheaf is coherent (=-=[58]-=-). We have E ′(−Di) ⊂ iF1 a for any a ∈]ci −1, ci]. We have the natural surjection πi,a : E ′ −→ E ′/ iF1 a , and the target is the ODi-module. Let Ti,a denote the torsion part of E ′/ iF1 a as an ODi... |

18 |
The Donaldson-Hitchin-Kobayashi correspondence for parabolic bundles over orbifold surfaces
- Steer, Wren
(Show Context)
Citation Context ...ooth irreducible projective variety and a normal crossing divisor such that Y = X − D. Such a generalization has been studied by several people. In the non-Higgs case, J. Li [33] and B. Steer-A. Wren =-=[60]-=- established the correspondence. In the Higgs case, Simpson established the correspondence in the one dimensional case [50], and O. Biquard established it in the case where D is smooth [4]. Remark 1.3... |

17 |
Techniques of Extension of Analytic Objects
- Siu
- 1974
(Show Context)
Citation Context ...nsion two. It is also the orthogonal projection with respect to h0. Let h0,V and hin,V be the metric of V induced by h0 and hin. J. Li proved the following lemma [29] based on the result of Y. T. Siu =-=[54]-=-. Lemma 4.38. — ∂πV is L 2 with respect to h0 and ωǫ if and only if there exists a coherent subsheaf cV ⊂ cE such that cV |X−D = V . Lemma 4.39 (Li). — Let hV denotes the metric of V induced by h0 or ... |

17 |
Hodge theory with degenerating coefficients: L 2 -cohomology
- Zucker
(Show Context)
Citation Context ...n v satisfying the following: d¯z ∧ dz ∂∂v = f · |z| 2 ∣ ( , ∣v(z) ≤ C · |z| ǫ ǫ (N−1)/2 + |z| 1/2) · ‖f‖L2. The constant C can be independent of ǫ, N and f. Proof We use the argument of S. Zucker in =-=[64]-=-. First let us consider the equation ∂u = f · d¯z/¯z. For the decomposition u = ∑ un(ρ) · e √ −1nθ , it is equivalent to the following equations: We put as follows: ( 1 r 2 ∂ ∂r un ) − n · un = fn, (n... |

16 | Mixed twistor structures - Simpson |

15 |
Private communication
- Simpson
- 2008
(Show Context)
Citation Context ...desired inequality. ∫ Remark 6.12. — Narasimhan posed it to Simpson how to get a Bogomolov-Gieseker inequality for parabolic Higgs bundle in higher dimensional case, and it was passed to the author. (=-=[52]-=-). Corollary 6.13. — Let X be a smooth projective surface, and D be a simple normal crossing divisor. Let (E∗, θ) be a µL-stable parabolic Higgs bundle on (X, D). Assume ∫ X par-ch2(E∗) = par-degω(E∗)... |

10 |
Representations of Fundamental Groups of Algebraic Varieties
- Zuo
- 1999
(Show Context)
Citation Context ...n a π1(X, x)-equivariant map F : ˜ X −→ G/K, where the π1(X, x)-action on G/K is given by the homomorphism11.2. DEFINITIONS 105 π1(X, x) −→ G. If PK ⊂ PG is pluri-harmonic, then F is pluri-harmonic (=-=[65]-=-) in the sense that any restriction of F to holomorphic curve is harmonic. 11.2.3. A tame pure imaginary G-harmonic bundle. — Let G be a linear reductive group over C. Let h denote a Cartan subalgebra... |

8 |
Polarizable twistor D-modules, Astérisque, 300, Société Mathématique de
- Sabbah
- 2005
(Show Context)
Citation Context ...econd claim easily follows from the proof of the uniqueness result in [43]. (See the argument of Proposition 2.6). The third claim follows from the second claim. We also have the following lemma (see =-=[48]-=- or [43]) Lemma 11.14. — If there exists a Corlette-Jost-Zuo metric on a flat bundle (E, ∇), then the flat bundle is semisimple. We have the involution χ ↦−→ χ on Irrep ( Γ ) such that χ ⊗R C = χ ⊕ χ.... |

6 |
Higgs bundles and local systmes, Publ
- Simpson
- 1992
(Show Context)
Citation Context ...V∗, θ) |Y is µL-(semi)stable, where Y denotes a complete intersection of sufficiently ample generic hypersurfaces. We closely follow the arguments of V. Mehta, A. Ramanathan ([38], [39]) and Simpson (=-=[51]-=-). See the papers for more detail. 3.4.2. W-operator. — In the following, let k denote a field of characteristic 0. Let X be a smooth projective variety over k, with an ample line bundle L. Let D be a... |

3 | Formalized proof, computation, and the construction problem in algebraic geometry - Simpson - 2004 |

2 | Parabolic vector bundles and Hermitian-Yang-Mills connections over a Riemann surface - Poritz - 1993 |

2 |
Moduli of representations of the fundametal group of a smooth projective variety
- Simpson
- 1994
(Show Context)
Citation Context ...en if (c ˜ Ei, ˜ F i, ˜ θi) are not µL-stable, the conclusion in the third claim of Proposition 1.9 should be true. In fact, Simpson gave a detailed argument to show it, in the case where D is empty (=-=[54]-=-, [55]). More strongly, he obtained the homeomorphism of the coarse moduli spaces of semistable flat bundles and semistable Higgs bundles. In this paper, we do not discuss the moduli spaces, and hence... |