The non-abelian Hodge correspondence on some non-K?hler manifolds  

作　　者：Changpeng Pan Chuanjing Zhang Xi Zhang 

机构地区：[1]School of Mathematics and Statistics,Nanjing University of Science and Technology,Nanjing 210094,China [2]School of Mathematical Sciences,University of Science and Technology of China,Hefei 230026,China

出　　处：《Science China Mathematics》2023年第11期2545-2588,共44页中国科学：数学（英文版）

基　　金：supported by the National Key R&D Program of China(Grant No.2020YFA0713100);National Natural Science Foundation of China(Grant Nos.12141104,11801535,11721101and 11625106)。

摘　　要：The non-abelian Hodge correspondence was established by Corlette(1988),Donaldson(1987),Hit chin(1987)and Simpson(1988,1992).It states that on a compact Kahler manifold(X,ω),there is a one-to-one correspondence between the moduli space of semi-simple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers.In this paper,we extend this correspondence to the projectively flat bundles over some non-Kahler manifold cases.Firstly,we prove an existence theorem of Poisson metrics on simple projectively flat bundles over compact Hermitian manifolds.As its application,we obtain a vanishing theorem of characteristic classes of projectively flat bundles.Secondly,on compact Hermitian manifolds which satisfy Gauduchon and astheno-K?hler conditions,we combine the continuity method and the heat flow method to prove that every semi-stable Higgs bundle withΔ(E,?E)·[ωn-2]=0 must be an extension of stable Higgs bundles.Using the above results,over some compact non-Kahler manifolds(M,ω),we establish an equivalence of categories between the category of semi-stable(poly-stable)Higgs bundles(E,?E,φ)withΔ(E,?E)·[ωn-2]=0 and the category of(semi-simple)projectively flat bundles(E,D)with(-1)(1/2)FD=α■IdE for some real(1,1)-formα.

关 键 词：projectively flat bundle Higgs bundle non-Kahler the Hermitian-Yang-Mills flow e-regularity theorem 

分 类 号：O186.1[理学—数学]