Periodic Reporting for period 5 - BirNonArchGeom (Birational and non-archimedean geometries)
Période du rapport: 2024-05-01 au 2025-04-30
Smooth geometric shapes (manifolds) are extensively studied and used in mathematics, physics, computer science, etc. Often a geometric shape (variety), for example, the one given by an equation f(x,y,z) in the three-dimensional space, is singular: it may contain self-intersections, cusps, pinch points, etc. Resolution of singularities is a classical branch of algebraic geometry which studies how a variety can be modified to a smooth manifold, and such modifications are a very useful tool for working with general varieties. When achieved, resolution results have numerous applications in mathematics and related areas (such as math physics), so any serious advance in constructing new methods, improving old ones, etc. is of high importance.
The first resolution of singularities in all dimensions over fields containing the rational numbers (the characteristic zero case) was obtained by Hironaka in 1964 and awarded him a Fields medal. Until very recently, Hironaka's method was tremendously polished and improved, but mathematicians have known essentially a unique resolution algorithm. Over fields containing a finite subfield (the so-called positive characteristic case) resolution is only known in dimensions 2 and 3, and this is a major roadblock in study of various questions about algebraic varieties in positive characteristic.
The goal of this project was to design new resolution methods both in the classical setting of varieties and in other settings, such as resolution of maps between varieties, resolution of varieties with boundaries, etc. and to try to achieve any progress over fields of positive characteristics.
The first resolution of singularities in all dimensions over fields containing the rational numbers (the characteristic zero case) was obtained by Hironaka in 1964 and awarded him a Fields medal. Until very recently, Hironaka's method was tremendously polished and improved, but mathematicians have known essentially a unique resolution algorithm. Over fields containing a finite subfield (the so-called positive characteristic case) resolution is only known in dimensions 2 and 3, and this is a major roadblock in study of various questions about algebraic varieties in positive characteristic.
The goal of this project was to design new resolution methods both in the classical setting of varieties and in other settings, such as resolution of maps between varieties, resolution of varieties with boundaries, etc. and to try to achieve any progress over fields of positive characteristics.
The project achieved a great success in the characteristic zero case, and a whole series of new methods was discovered and worked out In a series of papers of the PI with D. Abramovich, J. Wlodarczyk and A. Belotto (some have already been published, other ones are available on archive and submitted for publication in top journals). This includes logarithmic methods applying to varieties with boundaries and maps between them, weighted methods which use no memory and just visibly improve the variety at each basic step of the algorithm (the feature which was known to be unachievable in the classical setting), and a foliated method which resolves a variety with a vector field (establishing a link to the theory of ordinary differential equations). This progress will certainly vitalize the area which was considered "finished" by some experts.
In addition, in a paper of the PI with Adiprasito and Liu (available on archive and submitted for publication) we established the best possible combinatorial resolution of maps, proving in characteristic zero the semistable conjecture formulated in 2000 by Abramovich and Karu.
The achievements in the case of positive characteristics are more modest. It seems that the new methods, especially the memoryless approach, have to be incorporated in any attempt to construct a canonical resolution in positive characteristic, but these methods alone are still not enough and some new ideas should be designed.
I and my co-authors actively disseminated our knowledge and discoveries. We have already run a learning seminar at Oberwolfach for early career mathematicians and published a lecture series book on these newly discovered resolution methods. We expect not only the results, but also the methods (the use of the so-called orbifolds and orbifold weighted blow ups) to become an important tool of birational algebraic geometry and related fields.
In addition, together with Katharina Huebner I organized a spring school on resolution and non-archimedean geometry at Heidelberg, which consisted of four lecture series and was attended by about 40 graduate students and postdocs, and a conference on birational and non-archimedean geometries at Caen, which was attended by about 30 senior experts and contained 16 lectures, including the talks by my co-authors on the new resolution methods.
In addition, in a paper of the PI with Adiprasito and Liu (available on archive and submitted for publication) we established the best possible combinatorial resolution of maps, proving in characteristic zero the semistable conjecture formulated in 2000 by Abramovich and Karu.
The achievements in the case of positive characteristics are more modest. It seems that the new methods, especially the memoryless approach, have to be incorporated in any attempt to construct a canonical resolution in positive characteristic, but these methods alone are still not enough and some new ideas should be designed.
I and my co-authors actively disseminated our knowledge and discoveries. We have already run a learning seminar at Oberwolfach for early career mathematicians and published a lecture series book on these newly discovered resolution methods. We expect not only the results, but also the methods (the use of the so-called orbifolds and orbifold weighted blow ups) to become an important tool of birational algebraic geometry and related fields.
In addition, together with Katharina Huebner I organized a spring school on resolution and non-archimedean geometry at Heidelberg, which consisted of four lecture series and was attended by about 40 graduate students and postdocs, and a conference on birational and non-archimedean geometries at Caen, which was attended by about 30 senior experts and contained 16 lectures, including the talks by my co-authors on the new resolution methods.
Both the new algorithms developed with Abramovich and Wlodarczyk (logarithmic and weighted), with Abramovich, Belotto and Wlodarczyk (foliated) and the semistable conjecture solved with Adiprasito and Liu have taken the field of resolution beyond what was known in the characteristic zero case and introduced essentially new techniques.
The positive characteristic case is even more challenging and many mathematicians think it is out of reach. We have tested the new methods, found in characteristics zero, on the positive characteristic case too. It seems that the weighted method has most positive input and it enabled other groups to solve some particular cases that were open. It still seems that to do a real breakthrough the weighted method should be combined with new ideas/techniques in the positive characteristic, such as usage of non-tame stacks, and this will certainly be one of the central research topics of me and my collaborators.
The positive characteristic case is even more challenging and many mathematicians think it is out of reach. We have tested the new methods, found in characteristics zero, on the positive characteristic case too. It seems that the weighted method has most positive input and it enabled other groups to solve some particular cases that were open. It still seems that to do a real breakthrough the weighted method should be combined with new ideas/techniques in the positive characteristic, such as usage of non-tame stacks, and this will certainly be one of the central research topics of me and my collaborators.