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Content archived on 2022-12-23

Topological and Geometric Invariants of Algebraic Varieties

Objective

The project is divided into four themes:
1) Fundamental groups and universal coverings of Kahler manifolds, braid monodromy invariants of algebraic and symplectic manifolds;
2) Birational geometry, terminal singularities and Mori's elementary birational maps;
3) Moduli spaces of sheaves on Fano varieties.
4) Quantization of algebraic varieties, its application to deformation of moduli spaces of coherent sheaves, Poisson brackets, B-field deformations of Hodge structures in view of mirror symmetry.

The main objectives are:
- to study the topology of the complements of hypersurfaces in projective space, in particular, the complements of branch curves of generic coverings of the projective plane, their homotopy groups and applications to moduli spaces of the surfaces;
- to study the topology and combinatorics of the complements of line arrangement sand the connections between the two ( first counter examples?)
- to establish which cuspidal braid monodromy types can be attached to algebraic curves;
- to find out a finite algorithm that recognizes different cuspidal braid monodromy types;
- to investigate the structure of fibrations on uniruled varieties belonging to a large class of Fano varieties and fibrations, and describe their groups of birational automorphisms;
- to investigate local and global geometry of linear systems of divisors on Fano fibrations with respect to singularities of such systems;
- to describe the structure of three-dimensional Mori contractions in terms of inductive blowups, classify log terminal exceptional singularities and study their higher-dimensional generalizations;
- to study birational and enumerative geometry of certain Hilbert schemes and moduli spaces;
- to describe moduli spaces of vector bundles with small second Chern classes on Fano 3-folds with Picard number 1;
- determine the geography of indecomposable rank-2 vector bundles with natural cohomology on Fano 3-folds;
- to produce new constructions of symplectic structures on moduli spaces of sheaves on 4-folds;
- to study quantization of symplectic groupoids and Poisson brackets with application to constructing invariants of algebraic varieties;
- to investigate moduli spaces of coherent sheaves on commutative and noncommutative surfaces with applications to physical S-duality, MacKay correspondence and Calogero-Moser space;
- to construct and study Hodge structures for noncommutative varieties with application to mirror duality.

We expect numerous results to be published in mathematical journals and reported at scientific seminars and mathematical conferences.

Call for proposal

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Funding Scheme

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Coordinator

Bar-Ilan University
EU contribution
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Address

52900 Ramat Gan
Israel

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Total cost
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Participants (6)