Final Report Summary - TGASS (Topological, Geometric and Analytical Study of Singularities)
The research project address several topics in Singularity Theory, Algebraic Geometry and Homological Algebra which are interconnected. We have made emphasis in understanding topological and analytic properties of singularities, study complements of algebraic varieties and advance in simplicial techniques in homological algebra. In Singularity Theory some attention has been paid to the real analytic case (the complex analytic case is by far the most widely studied)
Perhaps the most important achievement of the project is the solution of Nash problem for surfaces (dating from 1968), which relates the structure of the space of arcs of a singularity with the structure of the exceptional divisor of its resolution. After several preliminary results which showed first that the problem is of topological nature, and proved the conjecture in the challenging case of quotient singularities, we have succeeded to solve this problem with a new approach based on topological methods. The techniques and ideas introduced for the solution were completely novel in the field and probably will be useful for further research.
The solution to Nash problem was motivated by and opens a way to study topological equisingularity of surfaces with algebro-geometric methods (topological methods have proved to be very difficult in this topic), and also introduces techniques which will probably reveal deeper properties of arc spaces.
Another algebro-geometic attack to topological equisingularity is based in a generalisation of McKay correspondence to any surface singularities. This generalisation involves moduli spaces of reflexive sheaves and divisors in a resoltion of the surface singularity, and in the later stages of the project we contributed in this direction.
Besides this we have solved two other equisingularity problems: the 1-dimensional critical set case of a problem proposed by B. Teissier on polynomial representability of topologycal types of analytic germs, and a counterexample in dimension 2 of Zariski's conjecture B (only higher dimensional couterexamples were known).
In the study of vanishing topology for non-isolated singularities we made progress in the study of the homotopy type of Milnor fibres in two opposite directions. On one hand we have proved that under restrictions on the structure of the critical set, the Milnor fibre is a bouquet of spheres. On the other we have shown that in the general case there are not many restrictions on the homotopy type of the Milnor fibre: the homotopy type of any complement of local analytic set can be realised. Using these methods we have found the first example of simply-connected non-formal Milnor fibre, which motivates now the study of vanishing homotopy with its Hodge structure.
In the particular study of the topology of complements of hyperplane arrangements, we have understood completely the relationship between the existence of pencils of curves embedded in the arrangement, and the vanishing cohomology locus of its complement. We have also given a first step in the generalization towards higher dimensions, showing that the vanishing products in cohomology are the right object to generalize. Besides, these techniques have been proved to be useful for the case of general curves, for which an analogous theorem has been proved.
We have also deepened our understanding of the the topology of singularity links (that is, the boundary of a small neighborhood) and its embedding in the ambient space, both for the case of isolated surface singularities, and complex line arrangements. We have developped algorithmic methods for our research, whose implementation has lead to the improvement of software of widespread use.
In the field of Homological Algebra, our contributions concern the theory of homotopy limits. We have proved that homotopy limits of any shape may be constructed only using homotopically well-behaved products and homotopical simplicial equalizers. As an application of the previous result, we show that Deligne's cosimplicial construction for mixed Hodge complexes is a homotopy limit and therefore homotopically unique.
A converse to Quillen-Maltsiniotis derived adjunction theorem has been obtained and as a corollary we have deduced an equivalence between the two notions of homotopy limits corresponding to Grothendieck derivators and derived functor of limits.
Perhaps the most important achievement of the project is the solution of Nash problem for surfaces (dating from 1968), which relates the structure of the space of arcs of a singularity with the structure of the exceptional divisor of its resolution. After several preliminary results which showed first that the problem is of topological nature, and proved the conjecture in the challenging case of quotient singularities, we have succeeded to solve this problem with a new approach based on topological methods. The techniques and ideas introduced for the solution were completely novel in the field and probably will be useful for further research.
The solution to Nash problem was motivated by and opens a way to study topological equisingularity of surfaces with algebro-geometric methods (topological methods have proved to be very difficult in this topic), and also introduces techniques which will probably reveal deeper properties of arc spaces.
Another algebro-geometic attack to topological equisingularity is based in a generalisation of McKay correspondence to any surface singularities. This generalisation involves moduli spaces of reflexive sheaves and divisors in a resoltion of the surface singularity, and in the later stages of the project we contributed in this direction.
Besides this we have solved two other equisingularity problems: the 1-dimensional critical set case of a problem proposed by B. Teissier on polynomial representability of topologycal types of analytic germs, and a counterexample in dimension 2 of Zariski's conjecture B (only higher dimensional couterexamples were known).
In the study of vanishing topology for non-isolated singularities we made progress in the study of the homotopy type of Milnor fibres in two opposite directions. On one hand we have proved that under restrictions on the structure of the critical set, the Milnor fibre is a bouquet of spheres. On the other we have shown that in the general case there are not many restrictions on the homotopy type of the Milnor fibre: the homotopy type of any complement of local analytic set can be realised. Using these methods we have found the first example of simply-connected non-formal Milnor fibre, which motivates now the study of vanishing homotopy with its Hodge structure.
In the particular study of the topology of complements of hyperplane arrangements, we have understood completely the relationship between the existence of pencils of curves embedded in the arrangement, and the vanishing cohomology locus of its complement. We have also given a first step in the generalization towards higher dimensions, showing that the vanishing products in cohomology are the right object to generalize. Besides, these techniques have been proved to be useful for the case of general curves, for which an analogous theorem has been proved.
We have also deepened our understanding of the the topology of singularity links (that is, the boundary of a small neighborhood) and its embedding in the ambient space, both for the case of isolated surface singularities, and complex line arrangements. We have developped algorithmic methods for our research, whose implementation has lead to the improvement of software of widespread use.
In the field of Homological Algebra, our contributions concern the theory of homotopy limits. We have proved that homotopy limits of any shape may be constructed only using homotopically well-behaved products and homotopical simplicial equalizers. As an application of the previous result, we show that Deligne's cosimplicial construction for mixed Hodge complexes is a homotopy limit and therefore homotopically unique.
A converse to Quillen-Maltsiniotis derived adjunction theorem has been obtained and as a corollary we have deduced an equivalence between the two notions of homotopy limits corresponding to Grothendieck derivators and derived functor of limits.