Restriction of stable sheaves and representations of the fundamental group

Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization

“…Donaldson gave a simplified proof for projective varieties in [8]. The application to constructing flat connections is based on the fact that if the Chern classes of E vanish then any Hermitian-Yang-Mills metric is flat [34,35,36]. In fact only certain classes have to be checked: it is enough that c 1 (E) = 0 and c 2 …”

Perspectives on geometric analysis

“…(That polystability of bundle is a consequence of existence of Hermitian Yang-Mills connection was first observed by Lüber [482]. Donaldson [196] was able to make use of the theorem of Mehta-Ramanathan [505] and ideas of above two papers to prove the theorem for projective manifold). It was generalized by C. Simpson [617], using ideas of Hitchin [331], to bundles with Higgs fields.…”

Birational Invariance of the S-fundamental Group Scheme

“…When k = C, the Narasimhan-Seshadri theorem [12] and its generalizations [11], establish an equivalence between unitary representations of the topological fundamental group of X(C) and polystable vector bundles on X with vanishing Chern classes. While the topological fundamental group does not make sense if char(k) > 0, the category of vector bundles with various other conditions does make sense.…”