Periodic Reporting for period 4 - AlgTateGro (Constructing line bundles on algebraic varieties --around conjectures of Tate and Grothendieck)
Periodo di rendicontazione: 2021-06-01 al 2022-11-30
This project aims at a better understanding of some fundamental questions blending arithmetic and geometric phenomena, focusing on the existence and construction of line bundles on algebraic varieties.
We pursue a web of conjectures mostly formulated during the 1960s by Tate, Grothendieck, Ogus, regarding the relationship between geometric and number-theoretic properties of those objects -- projective varieties -- defined by homogeneous equations with integral coefficients in projective space. Most these conjectures are wide open to this day, except in very special -- important -- cases. Following recent work on specific surfaces, we want to develop new tools that should make it possible to tackle them.
The conjectures that this project deal with seem related at a formal level, but our current understanding of them does not make it possible to use any kind of mathematically precise relationship that would make it possible to pass from one to another -- and they are set in significantly different areas of algebraic geometry. An important part of our project is to transfer methods from one of these settings to others, e.g. allowing us to use geometric methods in the theory of transcendental numbers.
The hope of these projects WAS that mathematical concepts developed in the past decades have reached sufficient maturity that these important conjectures might be tackled directly, and that the mathematics we introduce in doing so will shed a new light on the interaction between geometry, complex analysis, and number theory.
The main focus of this project has been to develop new frameworks for geometry of numbers, with applications to new foundations and new concepts in Arakelov geometry. In summary, we have developed flexible tools and concepts that make it possible to adapt the classical setting of geometry of numbers, originated by Dirichlet and Minkowski, to the setting of arbitrary countable abelian groups. We introduce new numerical invariants and relate them to concrete objects. As a consequence, we have a working theory that adapts to the setting of quasi-coherent sheaves the analogy between vector bundles on projective curves and Hermitian lattices. We are able, through the introduction of the notion of A-schemes which we believe to be central in arithmetic geometry, to prove new diophantine results on integral points. On a related note, we introduce the notion of a formal-analytic arithmetic surface, a new object blending complex and formal geometry, and use it to prove new results on arithmetic fundamental groups, as well as some effective algebraization results. In this direction, new results regarding classical questions on arithmetic ampleness and arithmetic Bertini theorems have been obtained.
On a slightly different path, new uniform finiteness results have been obtained for cohomological invariants of certain families of varieties over number fields.
We pursue a web of conjectures mostly formulated during the 1960s by Tate, Grothendieck, Ogus, regarding the relationship between geometric and number-theoretic properties of those objects -- projective varieties -- defined by homogeneous equations with integral coefficients in projective space. Most these conjectures are wide open to this day, except in very special -- important -- cases. Following recent work on specific surfaces, we want to develop new tools that should make it possible to tackle them.
The conjectures that this project deal with seem related at a formal level, but our current understanding of them does not make it possible to use any kind of mathematically precise relationship that would make it possible to pass from one to another -- and they are set in significantly different areas of algebraic geometry. An important part of our project is to transfer methods from one of these settings to others, e.g. allowing us to use geometric methods in the theory of transcendental numbers.
The hope of these projects WAS that mathematical concepts developed in the past decades have reached sufficient maturity that these important conjectures might be tackled directly, and that the mathematics we introduce in doing so will shed a new light on the interaction between geometry, complex analysis, and number theory.
The main focus of this project has been to develop new frameworks for geometry of numbers, with applications to new foundations and new concepts in Arakelov geometry. In summary, we have developed flexible tools and concepts that make it possible to adapt the classical setting of geometry of numbers, originated by Dirichlet and Minkowski, to the setting of arbitrary countable abelian groups. We introduce new numerical invariants and relate them to concrete objects. As a consequence, we have a working theory that adapts to the setting of quasi-coherent sheaves the analogy between vector bundles on projective curves and Hermitian lattices. We are able, through the introduction of the notion of A-schemes which we believe to be central in arithmetic geometry, to prove new diophantine results on integral points. On a related note, we introduce the notion of a formal-analytic arithmetic surface, a new object blending complex and formal geometry, and use it to prove new results on arithmetic fundamental groups, as well as some effective algebraization results. In this direction, new results regarding classical questions on arithmetic ampleness and arithmetic Bertini theorems have been obtained.
On a slightly different path, new uniform finiteness results have been obtained for cohomological invariants of certain families of varieties over number fields.
While investigating fundamental questions in transcendental number theory, we have realized the need for a better understanding of basic positivity questions in Arakelov geometry. This has led us to a long undertaking of new fundations for parts of the subject, that would allow for both new results and concepts, and a significant simplification of and generalization of known results around arithmetic ampleness, which gives them much more flexibility in applications. Specific works related to this subject comprise:
-new equidistribution results on Shimura varieties, with applications to K3 surfaces (Tayou, Math Research Letters, Crelle, Forum of Math. Pi)
-an arithmetic Bertini irreducibility theorem (Charles, ASENS) and versions of an arithmetic Bertini theorem (Wang, JEP)
-new proofs and generalizations of arithmetic Hilbert-Samuel (Ni, 2 preprints)
-a monograph introducing theta-invariants for infinite-dimensional vector bundles (Bost, Progress in Mathematics (Birkhaüser))
-the development of geometry of numbers in infinite rank, with applications to a new class of geometric objects called A-schemes, giving a way of dealing with the cohomology of coherent sheaves in Arakelov geometry, and replacing classical methods of L2 analysis by "softer" functional analysis of nuclear spaces, with applications to arithmetic ampleness, affine objects in Arakelov geometry, Fekete-like theorems in higher dimension, and approximation results for holomorphic functions (4 manuscripts in finalization, totalling roughly 1000 pages)
-the introduction of the notion of formal-analytic arithmetic surfaces, with applications to fundamental groups and algebraization theorems (preprint to be published as a monography).
Regarding other aspects of the projects, both Bost and Charles have given Bourbaki seminars on topics that should play an important role (stability conditions and geometry of numbers). With Cadoret (Algebraic Geometry) and Pirutka (preprint), Charles has investigated uniform boundedness phenomena over number fields. Charles also investigated topics in complex geometry, regarding rational curves on hyperkahler varieties (with Mongardi and Pacienza, Compositio Math.) and Hodge structures on complex tori (Math. Z.).
-new equidistribution results on Shimura varieties, with applications to K3 surfaces (Tayou, Math Research Letters, Crelle, Forum of Math. Pi)
-an arithmetic Bertini irreducibility theorem (Charles, ASENS) and versions of an arithmetic Bertini theorem (Wang, JEP)
-new proofs and generalizations of arithmetic Hilbert-Samuel (Ni, 2 preprints)
-a monograph introducing theta-invariants for infinite-dimensional vector bundles (Bost, Progress in Mathematics (Birkhaüser))
-the development of geometry of numbers in infinite rank, with applications to a new class of geometric objects called A-schemes, giving a way of dealing with the cohomology of coherent sheaves in Arakelov geometry, and replacing classical methods of L2 analysis by "softer" functional analysis of nuclear spaces, with applications to arithmetic ampleness, affine objects in Arakelov geometry, Fekete-like theorems in higher dimension, and approximation results for holomorphic functions (4 manuscripts in finalization, totalling roughly 1000 pages)
-the introduction of the notion of formal-analytic arithmetic surfaces, with applications to fundamental groups and algebraization theorems (preprint to be published as a monography).
Regarding other aspects of the projects, both Bost and Charles have given Bourbaki seminars on topics that should play an important role (stability conditions and geometry of numbers). With Cadoret (Algebraic Geometry) and Pirutka (preprint), Charles has investigated uniform boundedness phenomena over number fields. Charles also investigated topics in complex geometry, regarding rational curves on hyperkahler varieties (with Mongardi and Pacienza, Compositio Math.) and Hodge structures on complex tori (Math. Z.).
Besides the specific results outlined above, the main impact of the project is the development, together with Jean-Benoît Bost, of a new framework for Arakelov geometry and infinite-dimensional geometry of numbers which we believe will have several significant applications beyond the ones we already obtained.
For the first time, we obtain a working theory of quasi-coherent sheaves in the setting of Arakelov geometry, and obtain new functoriality results that make it possible to work with numerical invariants of such objects as with the dimension of cohomology groups. We introduce some new basic objects that occurred in a hidden manner in mathematics, namely A-schemes (schemes over Z equipped with a complex-conjugation invariant compact subset of their complex points) and, dually, formal-analytic schemes, and show that the infinite-dimensional geometry of numbers we introduce makes it possible to obtain new concrete diophantine results. We also recover in a more general setting the main results on positivity for Hermitian line bundles on arithmetic schemes.
We expect that these novel objects and methods will have applications to diophantine geometry as well as transcendence questions, as emphasized for instance by their appearance in our simplified and generalized proof of the arithmetic holonomicity theorem of Calegari-Dimitrov-Tang in their proof of the bounded denominators conjecture for modular forms.
For the first time, we obtain a working theory of quasi-coherent sheaves in the setting of Arakelov geometry, and obtain new functoriality results that make it possible to work with numerical invariants of such objects as with the dimension of cohomology groups. We introduce some new basic objects that occurred in a hidden manner in mathematics, namely A-schemes (schemes over Z equipped with a complex-conjugation invariant compact subset of their complex points) and, dually, formal-analytic schemes, and show that the infinite-dimensional geometry of numbers we introduce makes it possible to obtain new concrete diophantine results. We also recover in a more general setting the main results on positivity for Hermitian line bundles on arithmetic schemes.
We expect that these novel objects and methods will have applications to diophantine geometry as well as transcendence questions, as emphasized for instance by their appearance in our simplified and generalized proof of the arithmetic holonomicity theorem of Calegari-Dimitrov-Tang in their proof of the bounded denominators conjecture for modular forms.