This is a project about nonblank Hodge theory. Whereas usual Hodge theory concerns mainly the usualcohomology of algebraic varieties, nonblank Hodge theory concerns the co homology of variety withnonabelian coefficients. The research in the topic of nonblank Hodge theory is motivated by the possibility of obtaining applications similar to what has already been done with classical Hodge theory. One of these would beta obtain restrictions on the homogony types of smooth projective varieties. Other possibilities include Torelli-typetheorems, strictness results and restrictions on the monodrama actions for zoomorphic families of smooth projective varieties.
The main research problems that will be addressed by the applicant are the following: 1.Associate to each smooth projective complex variety a nonblank Hodge structure in degree 1, namely attack on the affined line that is equivariant for the action of the multiplicative group.
2.Describe the module space of vector bundles with lambda-connections, which generalize Higgs bundles. This object takes into account the representations of a great part of the fundamental group, which is, in general, nonabelian. This is the key point in this innovative approach. This Marie Curie fellowship, if awarded, will allow Dr. Olivier Pinocchio to work in a research direction which stems from his recent Ph.D. thesis work, made in Toulouse, and aims to implement new ideas in collaboration with researchers from Barcelona, particularly with V. Navarro Aznar.The applicant will be able to profit from the expertise of the host research team and from the many activities that take place at the CRM, Universität de Barcelona, and Universität Polytechnic de Catalonia in the discipline of Algebraic Geometry. A seminar on current research topics is held every week. The timeliness of the applicant’s stay at the CRM, thanks to the coincidence with the beginning of a short term Programme in the apple.
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