Real non-Abelian mixed Hodge structures for quasi-projective varieties : formality and splitting / J.P. Pridham.

Includes bibliographical references and index.

* Introduction* Splittings for MHS on real homotopy types* Non-abelian structures* Structures on cohomology* Relative Malcev homotopy types* Structures on relative Malcev homotopy types* MHS on relative Malcev homotopy types of compact Kahler manifolds* MTS on relative Malcev homotopy types of compact Kahler manifolds* Variations of mixed Hodge and mixed twistor structures* Monodromy at the Archimedean place* Simplicial and singular varieties* Algebraic MHS/MTS for quasi-projective varieties I* Algebraic MHS/MTS for quasi-projective varieties II - non-trivial monodromy* Canonical splittings*${\rm SL}_2$ splittings of non-abelian MTS/MHS and strictification* Bibliography.

The author defines and constructs mixed Hodge structures on real schematic homotopy types of complex quasi-projective varieties, giving mixed Hodge structures on their homotopy groups and pro-algebraic fundamental groups. The author also shows that these split on tensoring with the ring $\mathbb{R}[x]$ equipped with the Hodge filtration given by powers of $(x-i)$, giving new results even for simply connected varieties. The mixed Hodge structures can thus be recovered from the Gysin spectral sequence of cohomology groups of local systems, together with the monodromy action at the Archimedean place. As the basepoint varies, these structures all become real variations of mixed Hodge structure.