Periodic Reporting for period 4 - NEDAG (New Directions in Derived Algebraic Geometry)
Période du rapport: 2022-03-01 au 2022-08-31
The research area of this project lies at the interface between several subjects: algebraic geometry, algebraic topology and singularity theory. Its main objects of study are varietes and schemes, defined as solutions of polynomial equations, and the central question is to describe their topology using various methods: algebraic, categorical etc. In the last decades, a new subject called derived algebraic geometry has emerged and provides today a powerful tool for the study of algebraic varieties and schemes.
In this project we propose to give a new impulsion on derived algebraic geometry by exploring new domains of applicability as well as developing new ideas and fundamental results. For this, we propose to focus on the, still very much unexplored, interactions of derived algebraic geometry with an extremely rich domain: singularity theory (to be understood in a broad sense, possibly in positive and mixed characteristics, but also singularities of meromorphic flat connections and of constructible sheaves). We plan to use the fruitful interactions between these two subjects in a two-fold manner: on the one hand derived techniques will be used in order to prove long standing open problems, and on the other hand we propose new developments in derived algebraic itself and thus open new research directions. The project has three major objectives, interacting with each other in a coherent manner.
1 - This first part of the proposal focuses on the Bloch's conductor formula conjecture, its variants and its consequences. This is a conjectural formula expressing the variation of the topology of a continuous families or algebraic varieties around a singularities. We propose to prove this conjecture by means of a new trace formula in the non commutative setting and the motivic study of the category of matrix factorizations. The method uses the full strength of the motivic homotopy theory, the homotopy of dg-categories , derived algebraic geometry, as well as an important totally new idea of introducing an E_2 algebra which controls arithmetic phenomenon in mixed characteristics. This part of the proposal is definitely on the solving problem side with direct impact: the proofs of oldstanding conjectures. It also introduces a novelty of techniques in the setting of arithmetic geometry, namely the theory of E 2 algebras and matrix factorizations which will undoubtedly find some further applications in the future and will al low us to approach other open problems.
2 - The main theme of this second part is a proof of an index formula in two different but related contexts: algebraic foliations and l adic sheaves of non commutative origin. Algebraic foliations are recast using a more general setting of “derived foliations”. The index formula for such derived foliations is a Riemann Roch type formula that computes Euler characteristics of sheaves endowed with connections along the leaves, and interpolates between the classical Riemann Roch theorem and the well known index formula for algebraic D modules. On the l adic side we will be interested in constructible l adic sheaves of “non commutative origin”, which are obtained by realization of certain categories. The “characteristic cycles” of such l adic sheaves will be introduced in terms of algebraic invariants of the corresponding sheaves of categories. This will lead to an index formula for l-adic sheaves of non commutative origin, valid for varieties in positive characteristic but also for schemes in the mixed characteristic setting which is a comp letely new aspect of the theory.
3 - In this last part of the proposal we will explore a new landscape in the study of meromorphic connections and their moduli. We will use the techniques from derived algebraic and shifted Poisson geometry in order to extend the beautiful geometry of moduli spaces of connections on (possibly open) Riemann surfaces to the higher dimensional setting. We develop a program which will tackle the construction of (deriv ed) moduli spaces of flat connections on open smooth algebraic varieties, in the regular and irregular setting. We will study the Poisson geometry of these moduli spaces and its interactions with non-abelian Hodge theory. Unlike the first part of this proposal, concerned with proving an old conjecture with new methods, this third section is truly about exploring new mathematical territories and extending the
common knowledge on moduli of flat connections beyond its present limits.
These three objectives represent together an important goal, on the one side by the open problems that will be solved along the way, and on the other side by the new interactions they provide with different subjects in mathematics: number theory, singularity theory, geometry and to some extend mathematical physics.
In this project we propose to give a new impulsion on derived algebraic geometry by exploring new domains of applicability as well as developing new ideas and fundamental results. For this, we propose to focus on the, still very much unexplored, interactions of derived algebraic geometry with an extremely rich domain: singularity theory (to be understood in a broad sense, possibly in positive and mixed characteristics, but also singularities of meromorphic flat connections and of constructible sheaves). We plan to use the fruitful interactions between these two subjects in a two-fold manner: on the one hand derived techniques will be used in order to prove long standing open problems, and on the other hand we propose new developments in derived algebraic itself and thus open new research directions. The project has three major objectives, interacting with each other in a coherent manner.
1 - This first part of the proposal focuses on the Bloch's conductor formula conjecture, its variants and its consequences. This is a conjectural formula expressing the variation of the topology of a continuous families or algebraic varieties around a singularities. We propose to prove this conjecture by means of a new trace formula in the non commutative setting and the motivic study of the category of matrix factorizations. The method uses the full strength of the motivic homotopy theory, the homotopy of dg-categories , derived algebraic geometry, as well as an important totally new idea of introducing an E_2 algebra which controls arithmetic phenomenon in mixed characteristics. This part of the proposal is definitely on the solving problem side with direct impact: the proofs of oldstanding conjectures. It also introduces a novelty of techniques in the setting of arithmetic geometry, namely the theory of E 2 algebras and matrix factorizations which will undoubtedly find some further applications in the future and will al low us to approach other open problems.
2 - The main theme of this second part is a proof of an index formula in two different but related contexts: algebraic foliations and l adic sheaves of non commutative origin. Algebraic foliations are recast using a more general setting of “derived foliations”. The index formula for such derived foliations is a Riemann Roch type formula that computes Euler characteristics of sheaves endowed with connections along the leaves, and interpolates between the classical Riemann Roch theorem and the well known index formula for algebraic D modules. On the l adic side we will be interested in constructible l adic sheaves of “non commutative origin”, which are obtained by realization of certain categories. The “characteristic cycles” of such l adic sheaves will be introduced in terms of algebraic invariants of the corresponding sheaves of categories. This will lead to an index formula for l-adic sheaves of non commutative origin, valid for varieties in positive characteristic but also for schemes in the mixed characteristic setting which is a comp letely new aspect of the theory.
3 - In this last part of the proposal we will explore a new landscape in the study of meromorphic connections and their moduli. We will use the techniques from derived algebraic and shifted Poisson geometry in order to extend the beautiful geometry of moduli spaces of connections on (possibly open) Riemann surfaces to the higher dimensional setting. We develop a program which will tackle the construction of (deriv ed) moduli spaces of flat connections on open smooth algebraic varieties, in the regular and irregular setting. We will study the Poisson geometry of these moduli spaces and its interactions with non-abelian Hodge theory. Unlike the first part of this proposal, concerned with proving an old conjecture with new methods, this third section is truly about exploring new mathematical territories and extending the
common knowledge on moduli of flat connections beyond its present limits.
These three objectives represent together an important goal, on the one side by the open problems that will be solved along the way, and on the other side by the new interactions they provide with different subjects in mathematics: number theory, singularity theory, geometry and to some extend mathematical physics.
During a first period, the members of the project have been working towards objective 1. The various trace formulas involved have been established, as well as interesting generalizations. The Bloch's conjecture has been intensively studied in the unipotent case, following the strategy that it should be obtained
as a consequence of the trace formulas. A categorical version of the Bloch's conjecture has been proven. On the way to these results, an exciting construction, which asscoiate a motive to a category, has been discovered and used in order to relate vanishing cycles with categories of matrix factorizations.
During a second period of the project, objective 3 has been intensively studied. The moduli of local systems and flat connexions on open varieties have been constructed and it has been shown that they carry shifted Poisson structures as expected. For this, a new foundamental object has been introduced, the formal boundary of a smoooth variety. The geometric nature of this new object is rather non-standard and studying the moduli of flat connections on it has been challenging.
Finally, the last couple of years of the project has been devoted to make progress on objective 2. The general framework of derived foliations has been established. It has been used successfuly in order to established to foliated index formula.
On the way, positive characteristic versions have been investigated, and some new results on characteristic classes of foliations in characterictic p have been obtained.
These results are the content of 15 manuscripts published in reviewed international journals, as well as 11 manuscripts still under review for publications.
The achievements of the project have been disseminated regularly at conferences, seminars and workshops. The members of the project have organized two major international conferences (Lisboa, fall 2018 and Toulouse June 2022), as well as a
two advanced schools (Lisboa, fall 2018 and Aussois 2021) aimed at younger colleagues.
as a consequence of the trace formulas. A categorical version of the Bloch's conjecture has been proven. On the way to these results, an exciting construction, which asscoiate a motive to a category, has been discovered and used in order to relate vanishing cycles with categories of matrix factorizations.
During a second period of the project, objective 3 has been intensively studied. The moduli of local systems and flat connexions on open varieties have been constructed and it has been shown that they carry shifted Poisson structures as expected. For this, a new foundamental object has been introduced, the formal boundary of a smoooth variety. The geometric nature of this new object is rather non-standard and studying the moduli of flat connections on it has been challenging.
Finally, the last couple of years of the project has been devoted to make progress on objective 2. The general framework of derived foliations has been established. It has been used successfuly in order to established to foliated index formula.
On the way, positive characteristic versions have been investigated, and some new results on characteristic classes of foliations in characterictic p have been obtained.
These results are the content of 15 manuscripts published in reviewed international journals, as well as 11 manuscripts still under review for publications.
The achievements of the project have been disseminated regularly at conferences, seminars and workshops. The members of the project have organized two major international conferences (Lisboa, fall 2018 and Toulouse June 2022), as well as a
two advanced schools (Lisboa, fall 2018 and Aussois 2021) aimed at younger colleagues.
Some results obtained during the project have open completely new and unexpected research directions, and has allowed to go beyond the state of the art. The discovery of the so-called 'filtered circle', introduced by the members of the project for the study of loop spaces, has lead to some very exciting new interactions between
loop spaces and Lubin-Tate theory. The filtered circle has also been used in order to provide a complete solution to the 'Schematization problem' of Grothendieck.
loop spaces and Lubin-Tate theory. The filtered circle has also been used in order to provide a complete solution to the 'Schematization problem' of Grothendieck.