Abstract -- Nonarchimedean 2020 Kyoto
Canceled due to COVID-19 Virus
Omid Amini (Ecole Polytechnique)
"Combinatorial Chow rings and tropical geometry"
Abstract: The aim of the talk is to give an interpretation of different combinatorial Chow rings, defined by generators and relations in association with compactifications or degenerations of algebraic varieties, in terms of cohomology of tropical varieties.
Huayi Chen (Université Paris 7)
"Arakelov theory of arithmetic surfaces over a trivially valued field"
Abstract: In this talk, I will explain a work in collaboration with Atsushi Moriwaki on an Arakelov theory of arithmetic surfaces over a field equipped with the trivial absolute value, including arithmetic intersection of adelic divisors, Hilbert-Samuel theorem and positivity of adelic divisors.
Hiroshi Iritani (Kyoto University)
"Asymptotics of periods via tropical geometry"
Abstract: In the context of mirror symmetry, it has been observed that the asymptotics of periods near the large complex structure limit can be expressed in terms of the Gamma class of the mirror. In this talk, I will explain a method to compute the asymptotics of periods using tropical geometry and compare it with the Gamma class. This is based on joint work with Mohammed Abouzaid, Sheel Ganatra and Nick Sheridan.
Reimi Irokawa (Tokyo Institute of Technology / RIKEN)
"Stabilities for families of dynamics over non-archimedean fields"
Abstract: Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line over non-archimedean fields, we study the stability (or passivity) of critical points of families of polynomials parametrized by analytic curves. We construct the activity measure of a critical point of a family of rational functions, and study its properties. For a family of polynomials, we study more about the activity locus such as its relation to boundedness locus (or Mandelbrot set) and to the normality of the sequence of the forward orbit.
Keita Goto (Kyoto University)
"On the Berkovich double residue fields and birational models"
Abstract: For each point of Berkovich analytic spaces, we can consider the residue field in the similar way as we do it for each point of algebraic varieties. Since Berkovich analytic spaces are 'analytic', in general, their residue fields are far from those for varieties. However this Berkovich residue field is a valuation field. Then we can take the residue field of it as a valuation field. This is called the Berkovich double residue field and is more similar to the residue field for a point of some variety. In this talk, we identify the double residue field at each point which is called a 'monomial valuation' as the residue field of a point of some variety.
Walter Gubler (Universität Regensburg)
"Local Volumes and Monge-Ampère measures over a non-archimedean field"
Abstract: In algebraic geometry, the volume measures the size of the space of global sections of a line bundle. Similarly, the arithmetic volume measures the number of small global sections in Arakelov geometry. There is a local version of this volume over any non-archimedean field. We will show differentiability of such volumes and give a connection to the non-archimedean Monge-Ampère problem. This is joint work with Sébastien Boucksom and Florent Martin.
Mattias Jonsson (University of Michigan)
"Non-Archimedean aspects of K-stability"
Abstract: The notion of K-stability was introduced by Tian and Donaldson as a conjectural algebraic criterion for the existence of special Kähler metrics on various classes of complex manifolds. The last few years have seen a great deal of work involving K-stability: not only have the conjectures been proven in important cases, but our understanding of this stability notion and its role in algebraic geometry has drastically improved. In my talk I will explain how notions from non-Archimedean geometry can be used to study K-stability.
Holly Krieger (Cambridge University)
"Degenerations and uniformity in complex dynamics"
Abstract: For a degenerating one-parameter family of rational maps of the Riemann sphere, works of DeMarco-Faber and of Favre describe the limit behavior of the measure of maximal entropy in terms of the associated dynamical measure on the Berkovich projective line. I will discuss joint work with DeMarco and Ye in which we use these degeneration estimates to deduce uniform bounds on the number of common preperiodic points for any two maps in some one-parameter families of rational maps, and an application to proving a case of the uniform Manin-Mumford conjecture.
Johannes Nicaise (Imperial College London / University of Leuven)
"Stable rationality of complete intersections"
Abstract: I will explain an ongoing project with John Christian Ottem to establish several new classes of stably irrational complete intersections. Our results are based on degeneration techniques and a birational version of the nearby cycles functor that was developed in collaboration with Evgeny Shinder.
Takeo Nishinou (Rikkyo University)
"Deformations of maps of codimension one"
Abstract: In deformation theory, the calculation of higher order obstructions to deformations is usually difficult. The classical notion of semiregularity of subvarieties plays a crucial role in such calculation. We extend this notion to the case of local embeddings of codimension one, and using a differential geometric technique, prove the vanishing of higher order obstructions. We will also mention the relative case.
Yuji Odaka (Kyoto University)
"Collapsing K3 surfaces, Tropicalizations, through Moduli-theoretic framework (Part 1)"
Abstract: For studying degenerations of Calabi-Yau varieties, its non-archimedean analytification, and Ricci-flat Kahler metric behaviours in a compatible manner, our recent joint program (cf., e.g. 1810.07685) provide a moduli-theoretic framework. The work is also interprettable from non-archimedean perspective, although being implicit in the paper.
To be a little more precise, we first provide (non-variety) compactifications of their moduli spaces, together with parametrization of ``tropical geometric” objects by the boundaries, which we conjecture to be all the metrics’ (collapsed) limits. We also give partial proofs, which in particular proves a conjecture by Kontsevich-Soibelman for K3 surface unconditionally, for instance.
Yûsuke Okuyama (Kyoto Institute of Technology)
"A direct translation from degenerate complex dynamics to quantized Berkovich dynamics"
Abstract: In their study of degenerating complex dynamics, DeMarco and Faber introduced a "transfer principle" from degenerating complex dynamics to non-archimedean
dynamics on quantized Berkovich projective lines over the formal Puiseux series fields. Instead of their conceptual way via bimeromorphically modified surface dynamics and semistable models from rigid geometry, we would give a direct (and explicit) translation from degenerating complex dynamics into quantized Berkovich dynamics.
Yoshiki Oshima (Osaka University)
"Collapsing K3 surfaces, Tropicalizations, through Moduli-theoretic framework (Part 2)"
Abstract: For studying degenerations of Calabi-Yau varieties, its non-archimedean analytification, and Ricci-flat Kahler metric behaviours in a compatible manner, our recent joint program (cf., e.g. 1810.07685) provide a moduli-theoretic framework. The work is also interprettable from non-archimedean perspective, although being implicit in the paper.
To be a little more precise, we first provide (non-variety) compactifications of their moduli spaces, together with parametrization of ``tropical geometric” objects by the boundaries, which we conjecture to be all the metrics’ (collapsed) limits. We also give partial proofs, which in particular proves a conjecture by Kontsevich-Soibelman for K3 surface unconditionally, for instance.
Jérôme Poineau (Université de Caen)
"Non-Archimedean compactifications of complex analytic varieties"
Abstract: Let X be a complex algebraic variety and X^h be its analytification. In this talk, I will explain how to compactify X^h in a functorial way by adding a normalized Berkovich space (over the trivially valued field C), following constructions by Thuillier and Fantini. The resulting space is reminiscent of the non-Archimedean degeneration introduced by Berkovich in his study of limit mixed Hodge structures and further investigated by Boucksom and Jonsson.
Joseph Rabinoff (Duke University)
"Abelian schemes are log-Néron models"
Abstract: We will use non-Archimedean geometry in the style of Berkovich and Bosch--Lütkebohmert to prove that abelian schemes satisfy a Néron mapping property with respect to log smooth schemes.
Tony Yue Yu (Université Paris-Sud)
"The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus"
Abstract: We show that the naive counts of rational curves in any affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a compatible multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized log Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel.
Last Modified: February 26, 2020