# Newton-Okounkov Bodies, Test Configurations, and Diophantine Geometry (17w5046)

Arriving in Banff, Alberta Sunday, February 5 and departing Friday February 10, 2017

## Organizers

Tomasz Szemberg (Pedagogical University of Cracow)

Michael Roth (Queen's University)

Alex Küronya (Goethe Universität Frankfurt)

## Objectives

Global sections of line bundles on varieties play a distinguished role in much of geometry. A good way to control all global sections of all multiples of a given bundle is to look at the graded ring they form with respect to multiplication of sections, the so called section ring of the line bundle. A particularly important instance is the ring associated to the canonical line bundle, the canonical ring.

It had been long suspected that the canonical ring is finitely generated; in fact a very intricate theory, Mori's minimal model theory was developed to deal with this issue. Nevertheless, the finite generation of the canonical ring has only been proved recently in full generality in a breakthrough paper by Birkar-Cascini-Hacon-McKernan building on ideas and previous work of Kawamata, Kollár, Mori, Shokurov, Siu, and Tsuji to name but a few. A subsequent simpler proof using results outside MMP was given shortly thereafter by Lazic.

Beside its intrinsic importance, the finite generation of canonical rings and the minimal model program have many applications in and around birational geometry, one notable example being the construction of moduli spaces for canonically polarized varieties by the cumulative work of Alexeev, Hassett, Keel, Kollár, Kovács, Mori, Shepherd-Barron, Viehweg, and others.

Our efforts go one step further, we consider certain important structures on section rings: multiplicative filtrations. The study of these objects has arisen in algebraic, arithmetic, and differential geometry independently in the last few years.

From the point of view of algebraic geometry, multiplicative filtrations lead to concave functions on Newton-Okounkov bodies as pointed out by Boucksom-Chen. In the setting of complex differential geometry a significant source of such filtrations are test configurations introduced by Donaldson to study K-stability. Filtrations of section rings arise in diophantine approximation in connection with the Faltings-Wüstholz approximation theorem in a recent far-reaching generalization of Roth's theorem by McKinnon-Roth.

Newton-Okounkov bodies provide an alternative way to organize all sections of all multiples of a given line bundle via convex geometry. Following ideas of Andrei Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata have recently shown how to associate convex bodies to linear series on varieties. The field has been rapidly extending ever since, with many applications in algebraic geometry and beyond. Of these, arguably the most significant achievement is the construction of integrable systems on projective varieties by Harada and Kaveh; at the same time, its connections to combinatorics and representation theory have become an active area of research by now.

Many asymptotic invariants of linear series are determined by Okounkov bodies, in fact, as pointed out by Jow, the set of Newton-Okounkov bodies can be thought of as 'universal numerical invariants' for line bundles. Thus their study necessarily includes the investigation of such asymptotic notions, including the volume of a divisor or asymptotic orders of vanishing. These concepts play an important role in the recent proofs of the finite generation of canonical or adjoint rings by Birkar-Cascini-Hacon-McKernan and Lazic.

The study of Newton-Okounkov bodies has branched out in various directions: there exist various results concerning the geometric behaviour of these bodies (by Küronya-Lozovanu-Maclean and Anderson-Küronya-Lozovanu), constructions of arithmetic analogues in Arakelov geometry (by Boucksom-Chen and Yuan for instance), and connections have been found to Donaldson's work on test configurations via complex analysis (by Witt-Nyström).

Just very recently, a systematic study of functions on Newton--Okounkov bodies arising from geometric valuations was begun in papers of Boucksom-Küronya-Maclean-Szemberg, which has led to exciting developments as for example the explicit description of the asymptotics of jumping numbers in terms of restricted volumes.

The existence of Kähler-Einstein metric has been a central question of complex differential geometry for at least thirty years, with major contributions coming from Yau, Tian, and Donaldson among others. The notion of K-stability was introduced by Tian as a tool to understand this problem, and has been proved to be an extremely useful concept. Although originally of analytic nature, Donaldson has found a way to define Futaki invariants and K-stability purely in terms of algebraic geometry. A little bit more concretely, Donaldson introduced test configurations - certain one-parameter deformations equipped with one-dimensional torus actions that correspond to one-parameter subgroups in traditional geometric invariant theory - and used them to achieve this.

The recent proof of the Donaldson-Tian-Yau conjecture by Donaldson et al. and Tian at about the same time is widely considered to be a milestone in complex differential geometry. Witt-Nyström observed that test configurations give rise to filtrations of section rings of ample line bundles, and studied some simple cases. Nevertheless we find it highly likely that this line of thought can be continued and one can establish further connections between test configurations, K-stability and functions on Newton-Okounkov bodies.

The notion of the volume of a line bundle has been used implicitly in diophantine approximation since K. F. Roth's 1955 paper on approximating points on the affine line. In this argument and all later extensions to higher dimensional varieties a key step is to ensure that one can find sections of line bundles of relatively low height but which vanish to high order at a specified number of points. This is accomplished through a type of counting argument with the global sections which, when taken asymptotically leads to a comparison between volumes.

The most general result of this type is the flexible and powerful approximation theorem of Faltings-Wüstholz. One of the inputs to the theorem is a weighted filtration of the space of global sections. Taken asymptotically this leads to a function which may be computed as the pushforward of the Liouville measure on a Newton-Okounkov body via a convex function on that body. This type of function was recently used by McKinnon-Roth to give a connection between diophantine approximation and the Seshadri constant, a local algebro-geometric measure of positivity. This paper however stopped short of further study of the resulting functions on Newton-Okounkov bodies, or their diophantine implications.

We believe that the connections among the three areas that have been brought to light so far have barely scratched the surface, and we expect serious novel relationships between geometrically defined functions on Newton--Okounkov bodies and K-stability, and Okounkov functions and diophantine approximation, respectively.

In more practical terms the goal is to bring together specialists in all three areas, exchange information on recent developments, and work together on concrete problems. One of the expected products of the workshop will be a list of important open problems and a road map to explore them. We also believe that an important aim of the workshop should be to introduce young scholars to modern concepts and current problems attached to notions of positivity for linear series and vector bundles. One of our aims is to initiate a systematic study of filtrations on sections rings capitalizing on recent advances in various subfields of geometry.

It had been long suspected that the canonical ring is finitely generated; in fact a very intricate theory, Mori's minimal model theory was developed to deal with this issue. Nevertheless, the finite generation of the canonical ring has only been proved recently in full generality in a breakthrough paper by Birkar-Cascini-Hacon-McKernan building on ideas and previous work of Kawamata, Kollár, Mori, Shokurov, Siu, and Tsuji to name but a few. A subsequent simpler proof using results outside MMP was given shortly thereafter by Lazic.

Beside its intrinsic importance, the finite generation of canonical rings and the minimal model program have many applications in and around birational geometry, one notable example being the construction of moduli spaces for canonically polarized varieties by the cumulative work of Alexeev, Hassett, Keel, Kollár, Kovács, Mori, Shepherd-Barron, Viehweg, and others.

Our efforts go one step further, we consider certain important structures on section rings: multiplicative filtrations. The study of these objects has arisen in algebraic, arithmetic, and differential geometry independently in the last few years.

From the point of view of algebraic geometry, multiplicative filtrations lead to concave functions on Newton-Okounkov bodies as pointed out by Boucksom-Chen. In the setting of complex differential geometry a significant source of such filtrations are test configurations introduced by Donaldson to study K-stability. Filtrations of section rings arise in diophantine approximation in connection with the Faltings-Wüstholz approximation theorem in a recent far-reaching generalization of Roth's theorem by McKinnon-Roth.

Newton-Okounkov bodies provide an alternative way to organize all sections of all multiples of a given line bundle via convex geometry. Following ideas of Andrei Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata have recently shown how to associate convex bodies to linear series on varieties. The field has been rapidly extending ever since, with many applications in algebraic geometry and beyond. Of these, arguably the most significant achievement is the construction of integrable systems on projective varieties by Harada and Kaveh; at the same time, its connections to combinatorics and representation theory have become an active area of research by now.

Many asymptotic invariants of linear series are determined by Okounkov bodies, in fact, as pointed out by Jow, the set of Newton-Okounkov bodies can be thought of as 'universal numerical invariants' for line bundles. Thus their study necessarily includes the investigation of such asymptotic notions, including the volume of a divisor or asymptotic orders of vanishing. These concepts play an important role in the recent proofs of the finite generation of canonical or adjoint rings by Birkar-Cascini-Hacon-McKernan and Lazic.

The study of Newton-Okounkov bodies has branched out in various directions: there exist various results concerning the geometric behaviour of these bodies (by Küronya-Lozovanu-Maclean and Anderson-Küronya-Lozovanu), constructions of arithmetic analogues in Arakelov geometry (by Boucksom-Chen and Yuan for instance), and connections have been found to Donaldson's work on test configurations via complex analysis (by Witt-Nyström).

Just very recently, a systematic study of functions on Newton--Okounkov bodies arising from geometric valuations was begun in papers of Boucksom-Küronya-Maclean-Szemberg, which has led to exciting developments as for example the explicit description of the asymptotics of jumping numbers in terms of restricted volumes.

The existence of Kähler-Einstein metric has been a central question of complex differential geometry for at least thirty years, with major contributions coming from Yau, Tian, and Donaldson among others. The notion of K-stability was introduced by Tian as a tool to understand this problem, and has been proved to be an extremely useful concept. Although originally of analytic nature, Donaldson has found a way to define Futaki invariants and K-stability purely in terms of algebraic geometry. A little bit more concretely, Donaldson introduced test configurations - certain one-parameter deformations equipped with one-dimensional torus actions that correspond to one-parameter subgroups in traditional geometric invariant theory - and used them to achieve this.

The recent proof of the Donaldson-Tian-Yau conjecture by Donaldson et al. and Tian at about the same time is widely considered to be a milestone in complex differential geometry. Witt-Nyström observed that test configurations give rise to filtrations of section rings of ample line bundles, and studied some simple cases. Nevertheless we find it highly likely that this line of thought can be continued and one can establish further connections between test configurations, K-stability and functions on Newton-Okounkov bodies.

The notion of the volume of a line bundle has been used implicitly in diophantine approximation since K. F. Roth's 1955 paper on approximating points on the affine line. In this argument and all later extensions to higher dimensional varieties a key step is to ensure that one can find sections of line bundles of relatively low height but which vanish to high order at a specified number of points. This is accomplished through a type of counting argument with the global sections which, when taken asymptotically leads to a comparison between volumes.

The most general result of this type is the flexible and powerful approximation theorem of Faltings-Wüstholz. One of the inputs to the theorem is a weighted filtration of the space of global sections. Taken asymptotically this leads to a function which may be computed as the pushforward of the Liouville measure on a Newton-Okounkov body via a convex function on that body. This type of function was recently used by McKinnon-Roth to give a connection between diophantine approximation and the Seshadri constant, a local algebro-geometric measure of positivity. This paper however stopped short of further study of the resulting functions on Newton-Okounkov bodies, or their diophantine implications.

We believe that the connections among the three areas that have been brought to light so far have barely scratched the surface, and we expect serious novel relationships between geometrically defined functions on Newton--Okounkov bodies and K-stability, and Okounkov functions and diophantine approximation, respectively.

In more practical terms the goal is to bring together specialists in all three areas, exchange information on recent developments, and work together on concrete problems. One of the expected products of the workshop will be a list of important open problems and a road map to explore them. We also believe that an important aim of the workshop should be to introduce young scholars to modern concepts and current problems attached to notions of positivity for linear series and vector bundles. One of our aims is to initiate a systematic study of filtrations on sections rings capitalizing on recent advances in various subfields of geometry.