DS10 - Défi des autres savoirs

Fundamental Groups, Hodge Theory and Motives – Hodgefun

Submission summary

Hodgefun is a collaborative research project in pure Mathematics around the topology of complex algebraic varieties.
It lies at the crossroads of Topology and Algebraic Geometry
and aims at studying the relationship between the topology of a complex algebraic variety and its algebraic
structure.

The project will focus on three hot topics: the Serre problem, Mixed Hodge Modules and Motives,
their interactions and applications

The Serre problem aims at identifying the fundamental groups of smooth complex algebraic varieties among finitely presented groups and admits several variants (the projective case, the Kähler case and their orbifold versions).
Considerable progress on the Kähler case has been achieved in the last 30 years. One of the main obstacles to
further progress is the lack of examples.

The constraints governing the Serre problem or more generally the homotopy type of smooth complex algebraic varieties originate typically in Hodge theory: many natural homotopical invariants carry mixed Hodge structures. M. Saito's theory of Mixed Hodge Modules is a theory of coefficients stable by Grothendieck's six operations developped in the 1980s. It was developped as the natural Hodge theoretic analogue of the theory of mixed l-adic
complexes abstracted by Gabber from Deligne's solution to the Weil conjectures. As such, it is the ultimate formulation of the cohomological aspects of Mixed Hodge Theory. The theory first found applications in singularity theory or in the topology of hypersurface complements by Saito, Dimca
and others. But a new wave of applications began recently to hit birational algebraic geometry and we expect this trend to have a
major impact . The technical difficulty of Saito's theory is however
a major stumbling block and a lot of work has to be done in the direction of
popularizing these methods.

Ultimately, Mixed Hodge theory should be a realization of a deeper theory of Mixed Motives, precisely envisioned by
Grothendieck and Beilinson:
Hodge structures are just the simplest of the special structures that live on homological invariants of algebraic varieties,
Galois module structures being another instance.
A fundamental problem on the interaction of Hodge Modules and Motives
is the motivic decomposition conjecture, which states that there is a motivic version
of the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber.

We aim at building a research group bringing together algebraic geometers of various horizons, both
confirmed and in earlier stages of their career, with a common interest in the topology of complex algebraic varieties and
its ramifications.

The Serre problem, the Hodge conjecture and the
motivic decomposition conjecture are not likely to be solved in the next few years but the members of the group do have
realistic research projects in these directions generally in collaboration with other members of the group.

In order to develop a common culture, train doctoral students and young researchers, we plan to organize 1 workshop per
year mixing the three main topics above.
We would study the main new results (and the important older results forming the theoretical basis of these
developpements)
on these topics.

The project would oragnize an international conference in 2019 and the closing conference in 2019 would be the third week
of a Summer School in Grenoble in June 2020.

The project would also fund missions and equipments for the participants of the group and their doctoral students and
invitations of international experts in order to facilitate collaborative research projects.

Its activities will be valorized by scientific publications.

Project coordinator

Institut Fourier (Laboratoire public)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

Institut Mathématique de Jussieu-Paris Rive Gauche
Université Rennes 1
Institut Fourier
Institut Elie Cartan de Lorraine

Help of the ANR 196,938 euros
Beginning and duration of the scientific project: December 2016 - 48 Months

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