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A Brill-Noether stack is an algebraic very presentable stack whose homotopytype has two nontrivial homotopy groups. We consider one with a fundamentalgroup --- a reductive algebraic group-scheme S and one higher homotopy group,represented by a vector space V. The homotopy type also defines an action of Son V. This stack is used as coefficient space for nonabelian cohomologicalspace on a smooth algebraic variety X.  We define nonabelian MHS on cohomological spaces of this type in the contextof the work of C. Simpson related to MHS on the space of local systems. It isdefined via an action of the multiplicative complex group on a appropriatelychosen category. The exhibited structure in fact generalizes Simpson's work.Allowing a more general type of coefficient stack.  Furthermore, the so-defined MHS, when considered at a vicinity of an object,which remains fixed under the structural action of C*, produces the local MHSon Brill-Noether Stacks as defined in our earlier work.  The nonabelian mixed Hodge structure on a Brill-Noether stack is a example ofthe mixed Hodge structure on a schematic homotopy type, studied by Katzarkov,Pantev and Toen. It has the advantage, due to the relative simplicity of thecoefficient stack, that it could be locally written out in terms of iteratedintegrals.

Nonabelian Mixed Hodge Structure on Brill-Noether Stacks