Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Hodge filtration on nonabelian cohomology   This is partly a survey article on nonabelian Hodge theory, but we also give
proofs of results that have only been announced elsewhere. In the introduction
we discuss a wide range of recent work on this subject and give some
references. In the body of the paper, we discuss Corlette's nonabelian Hodge
theorem, Hitchin's quaternionic structure on the moduli space of
representations of $\pi _1(X)$ (for a compact K\"ahler manifold $X$), and
Deligne's construction of the resulting twistor space. We then mention the
interpretation of Deligne's space of $\lambda$-connections as the Hodge
filtration of the nonabelian de Rham cohomology $M_{DR}(X,G)= H^1(X,G)$. We go
on to prove several properties of this Hodge filtration, such as Griffiths
transversality and regularity of the Gauss-Manin connection; a local triviality
property coming from Goldman and Millson's analysis of the singularities; and a
weight property coming from Langton's theory. We construct a compactification
of $M_{DR}$ using the Hodge filtration with its weight property. At the end, we
define a nonabelian version of the Noether-Lefschetz locus, and prove that it
is algebraic if the base of the fibration is compact. We pose the open problem
of studying degenerations of nonabelian Hodge structure sufficiently well to be
able to prove algebraicity of the Noether-Lefshetz locus even when the base is
quasiprojective.

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The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th