Canonical Metrics And The Related Heat Flows On Reflexive Sheaves

In this thesis,we study the existence of canonical metrics and the asymptotic be-havior of the Hermitian-Yang-Mills flow on a reflexive sheaf.In the first part,we study semi-stable Higgs sheaves over compact Kahler mani-folds.We prove that there is an admissible approximate Hermitian-Einstein structure on a semi-stable reflexive Higgs sheaf and consequently,the Bogomolov type inequality holds on a semi-stable reflexive Higgs sheaf.In the second part,we study the asymptotic behavior of the Hermitian-Yang-Mills flow on a reflexive sheaf.We prove that for any sequence of Chern connections with respect to evolving metrics along the Hermitian-Yang-Mills flow must converge sub-sequently to,modulo complex gauge transformations,a limiting connection in Clob∞－topology outside a closed subset with Hausdorff codimension at least 4,the holomorphic bundle determined by the limiting connection can be extended to a reflexive sheaf on the whole manifold.Furthermore,we prove that the limiting reflexive sheaf is isomor-phic to the double dual of the graded sheaves associated to Harder-Narasimhan-Seshadri filtration of the initial sheaf,this answers a question by Bando and Siu.