Hermitian Yang–Mills Metrics on Higgs Bundles over Asymptotically Cylindrical K?hler Manifolds
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摘要
Let V be an asymptotically cylindrical K?hler manifold with asymptotic cross-section ■.Let ■ be a stable Higgs bundle over ■, and(E, φ) a Higgs bundle over V which is asymptotic to ■. In this paper, using the continuity method of Uhlenbeck and Yau, we prove that there exists an asymptotically translation-invariant projectively Hermitian Yang–Mills metric on(E, φ).
Let V be an asymptotically cylindrical K?hler manifold with asymptotic cross-section ■.Let ■ be a stable Higgs bundle over ■, and(E, φ) a Higgs bundle over V which is asymptotic to ■. In this paper, using the continuity method of Uhlenbeck and Yau, we prove that there exists an asymptotically translation-invariant projectively Hermitian Yang–Mills metric on(E, φ).
Let V be an asymptotically cylindrical K?hler manifold with asymptotic cross-section ■.Let ■ be a stable Higgs bundle over ■, and(E, φ) a Higgs bundle over V which is asymptotic to ■. In this paper, using the continuity method of Uhlenbeck and Yau, we prove that there exists an asymptotically translation-invariant projectively Hermitian Yang–Mills metric on(E, φ).
引文
[1]Biswas,I.:Stable Higgs bundles on compact Gauduchon manifolds.Comptes Rendus Math.,349,71-74(2011)
[2]Biswas,I.,Schumacher,G.:Yang-Mills equation for stable Higgs sheaves.Inter.J.Math.,20,541-556(2009)
[3]Bando,S.:Einstein-Hermitian metrics on noncompact K?hler manifolds.Einstein metrics and Yang-Mills connections(Sanda,1990),Vol.145,Lecture Notes in Pure and Appl.Math.Dekker,New York,1993
[4]Bando,S.,Siu,Y.T.:Stable sheaves and Einstein-Hermitian metrics.Geometry and analysis on complex manifolds,39,39-50(1994)
[5]Bruzzo,U.,Otero,B.G.:Metrics on semistable and numerically effective Higgs bundles.J.Reine Angew.Math.,612,59-79(2007)
[6]Bruzzo,U.,Otero,B.G.:Approximate Hermitian-Yang-Mills structures on semistable principal Higgs bundles.Ann.Global Anal.Geom.,47,1-11(2015)
[7]Cardona,S.A.H.:Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles.I:generalities and the one-dimensional case.Ann.Global Anal.Geom.,42,349-370(2012)
[8]Donaldson,S.K.:Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles.Proc.London Math.Soc.,50,1-26(1985)
[9]Donaldson,S.K.:Infinite determinants,stable bundles and curvature.Duke Math.J.,54,231-247(1987)
[10]Gilbarg,D.,Trudinger,N.S.:Elliptic partial differential equations of second order.Classics in Mathematics.Reprint of the 1998 edition.Springer-Verlag,Berlin,2001
[11]Guo,G.Y.:Yang-Mills fields on cylindrical manifolds and holomorphic bundles.I,II.Comm.Math.Phys.,179,737-775,777-788(1996)
[12]Hitchin,N.J.:The self-duality equations on a Riemann surface.Proc.London Math.Soc.,55,59-C126(1987)
[13]Hong,M.C.:Heat flow for the Yang-Mills-Higgs field and the Hermitian Yang-Mills-Higgs metric.Ann.Global Anal.Geom.,20,23-46(2001)
[14]Haskins,M,Hein,H.J.,Nordstrom,J.:Asymptotically cylindrical Calabi-Yau manifolds.J.Differ.Geom.,101,213-265(2015)
[15]Jacob,A.,Walpuski,T.:Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical K?hler manifolds.Communications in Partial Differential Equations,101,1566-1598(2018)
[16]Li,J.Y.,Zhang,X.:Existence of approximate Hermitian-Einstein structures on semi-stable Higgs bundles.Calc.Var.,52,783-795(2015)
[17]Li,J.Y.,Zhang,C.,Zhang,X.:Semi-stable Higgs sheaves and Bogomolov type inequality.Calc.Var.,56,1-33(2017)
[18]Lockhart,R.B.,Mc Owen,R.C.:Elliptic differential operators on noncompact manifolds.Ann.Scuola Norm.Sup.Pisa Cl.Sci.,12,409-447(1985)
[19]Lubke,M.,Teleman,A.:The universal Kobayashi-Hitchin correspondence on Hermitian manifolds.Mem.Amer.Math.Soc.,2006
[20]Lubke,M.,Teleman,A.:The Kobayashi-Hitchin Correspondence.World Scientific Publishing Co.,Inc.,River Edge,NJ,1995
[21]Mochizuki,T.:Kobayashi-Hitchin correspondence for tame harmonic bundles and an application.Asterisque,309,Soc.Math.France,Paris,2006
[22]McDuff,D.,Salamon,D.:J-holomorphic curves and symplectic topology.American Mathematical Society,Providence,2012
[23]Nie,Y.,Zhang,X.:Semistable Higgs bundles over compact Gauduchon manifolds.J.Geom.Anal.,28,627-642(2018)
[24]Ni,L.:The Poisson equation and Hermitian-Einstein metrics on holomorphic vector bundles over complete noncompact K?hler manifolds.Indiana Univ.Math.J.,51,679-704(2002)
[25]Ni,L.,Ren,H.:Hermitian-Einstein metrics for vector bundles on complete K?hler manifolds.Trans.Amer.Math.Soc.,353,441-456(2001)
[26]Narasimhan,M.S.,Seshadri,C.S.:Stable and unitary vector bundles on a compact Riemann surface.Ann.Math.,82,540-567(1965)
[27]Owens,B.Instantons on cylindrical manifolds and stable bundles.Geom.Topol.,5,761-797(2001)
[28]Sa Earp,H.N.:G2-instantons over asymptotically cylindrical manifolds.Geom.Topol.,19,61-111(2015)
[29]Simpson,C.T.:Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization.J.Amer.Math.Soc.,1,867-918(1988)
[30]Siu,Y.T.:Lectures on Hermitian-Einstein metrics for stable bundles and K?hler-Einstein metrics.DMVSeminar 8,Birkhauser Verlag,Basel,1987
[31]Uhlenbeck,K.K.,Yau,S.T.:On the existence of Hermitian-Yang-Mills connections in stable vector bundles.Comm.Pure Appl.Math.,39S,S257-S293(1986)
[32]Zhang,C.,Zhang,P.,Zhang,X.:Higgs bundles over non-compact Gauduchon manifolds.arXiv:1804.08994(2018)
1)Throughout this paper, we denoteΔ=d*d+dd*.
[2]Biswas,I.,Schumacher,G.:Yang-Mills equation for stable Higgs sheaves.Inter.J.Math.,20,541-556(2009)
[3]Bando,S.:Einstein-Hermitian metrics on noncompact K?hler manifolds.Einstein metrics and Yang-Mills connections(Sanda,1990),Vol.145,Lecture Notes in Pure and Appl.Math.Dekker,New York,1993
[4]Bando,S.,Siu,Y.T.:Stable sheaves and Einstein-Hermitian metrics.Geometry and analysis on complex manifolds,39,39-50(1994)
[5]Bruzzo,U.,Otero,B.G.:Metrics on semistable and numerically effective Higgs bundles.J.Reine Angew.Math.,612,59-79(2007)
[6]Bruzzo,U.,Otero,B.G.:Approximate Hermitian-Yang-Mills structures on semistable principal Higgs bundles.Ann.Global Anal.Geom.,47,1-11(2015)
[7]Cardona,S.A.H.:Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles.I:generalities and the one-dimensional case.Ann.Global Anal.Geom.,42,349-370(2012)
[8]Donaldson,S.K.:Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles.Proc.London Math.Soc.,50,1-26(1985)
[9]Donaldson,S.K.:Infinite determinants,stable bundles and curvature.Duke Math.J.,54,231-247(1987)
[10]Gilbarg,D.,Trudinger,N.S.:Elliptic partial differential equations of second order.Classics in Mathematics.Reprint of the 1998 edition.Springer-Verlag,Berlin,2001
[11]Guo,G.Y.:Yang-Mills fields on cylindrical manifolds and holomorphic bundles.I,II.Comm.Math.Phys.,179,737-775,777-788(1996)
[12]Hitchin,N.J.:The self-duality equations on a Riemann surface.Proc.London Math.Soc.,55,59-C126(1987)
[13]Hong,M.C.:Heat flow for the Yang-Mills-Higgs field and the Hermitian Yang-Mills-Higgs metric.Ann.Global Anal.Geom.,20,23-46(2001)
[14]Haskins,M,Hein,H.J.,Nordstrom,J.:Asymptotically cylindrical Calabi-Yau manifolds.J.Differ.Geom.,101,213-265(2015)
[15]Jacob,A.,Walpuski,T.:Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical K?hler manifolds.Communications in Partial Differential Equations,101,1566-1598(2018)
[16]Li,J.Y.,Zhang,X.:Existence of approximate Hermitian-Einstein structures on semi-stable Higgs bundles.Calc.Var.,52,783-795(2015)
[17]Li,J.Y.,Zhang,C.,Zhang,X.:Semi-stable Higgs sheaves and Bogomolov type inequality.Calc.Var.,56,1-33(2017)
[18]Lockhart,R.B.,Mc Owen,R.C.:Elliptic differential operators on noncompact manifolds.Ann.Scuola Norm.Sup.Pisa Cl.Sci.,12,409-447(1985)
[19]Lubke,M.,Teleman,A.:The universal Kobayashi-Hitchin correspondence on Hermitian manifolds.Mem.Amer.Math.Soc.,2006
[20]Lubke,M.,Teleman,A.:The Kobayashi-Hitchin Correspondence.World Scientific Publishing Co.,Inc.,River Edge,NJ,1995
[21]Mochizuki,T.:Kobayashi-Hitchin correspondence for tame harmonic bundles and an application.Asterisque,309,Soc.Math.France,Paris,2006
[22]McDuff,D.,Salamon,D.:J-holomorphic curves and symplectic topology.American Mathematical Society,Providence,2012
[23]Nie,Y.,Zhang,X.:Semistable Higgs bundles over compact Gauduchon manifolds.J.Geom.Anal.,28,627-642(2018)
[24]Ni,L.:The Poisson equation and Hermitian-Einstein metrics on holomorphic vector bundles over complete noncompact K?hler manifolds.Indiana Univ.Math.J.,51,679-704(2002)
[25]Ni,L.,Ren,H.:Hermitian-Einstein metrics for vector bundles on complete K?hler manifolds.Trans.Amer.Math.Soc.,353,441-456(2001)
[26]Narasimhan,M.S.,Seshadri,C.S.:Stable and unitary vector bundles on a compact Riemann surface.Ann.Math.,82,540-567(1965)
[27]Owens,B.Instantons on cylindrical manifolds and stable bundles.Geom.Topol.,5,761-797(2001)
[28]Sa Earp,H.N.:G2-instantons over asymptotically cylindrical manifolds.Geom.Topol.,19,61-111(2015)
[29]Simpson,C.T.:Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization.J.Amer.Math.Soc.,1,867-918(1988)
[30]Siu,Y.T.:Lectures on Hermitian-Einstein metrics for stable bundles and K?hler-Einstein metrics.DMVSeminar 8,Birkhauser Verlag,Basel,1987
[31]Uhlenbeck,K.K.,Yau,S.T.:On the existence of Hermitian-Yang-Mills connections in stable vector bundles.Comm.Pure Appl.Math.,39S,S257-S293(1986)
[32]Zhang,C.,Zhang,P.,Zhang,X.:Higgs bundles over non-compact Gauduchon manifolds.arXiv:1804.08994(2018)
1)Throughout this paper, we denoteΔ=d*d+dd*.