Abstracts

Abstracts

■ Omid Amini:

On a complex projective variety, the canonical measure is defined using an orthonormal basis of the global sections of the canonical sheaf. On a Riemann surface, it coincides with the Bergman-Arakelov measure. In the nineties, Shou-Wu Zhang defined the analog of Bergman-Arakelov measures for non-Archimedean curves. Since then, it has remained an open question how to extend the definition to higher-dimensional non-Archimedean varieties. I will present our solution to this question. We introduce a framework that allows to construct canonical measures on various types of geometric spaces: polyhedral, non-archimedean, and hybrid (with both complex and polyhedral components). I will explain how these measures arise as limits of canonical measures in one-parameter families of complex algebraic varieties. This uses the formalism of higher rank inner products introduced in a companion work. Based on joint works with N. Nicolussi.

A hybrid Riemann surface is a multiscale geometric object with both complex and tropical components. It features a "complex analytic geometry" analoguous to that of a Riemann surface: Riemann-Roch, Abel-Jacobi, Poincaré-Lelong, etc. hold in the hybrid setting. I will present this geometry and explain how it arises as a multiscale limit of complex geometry. Based on joint works with N. Nicolussi.


■ Keita Goto: On a relation between NA SYZ fibrations and SYZ fibrations 

SYZ fibrations have been studied for long years in a context of so-called SYZ mirror symmetry. In particular, this framework of mirror symmetry is expected to work for degenerating families of polarized Calabi-Yau varieties. Because Non-Archimedean geometry can also treat such families, we may consider SYZ fibrations from the perspective of Non-Archimedean geometry. That is, we may consider Non-Archimedean analogs of SYZ fibrations, which are called NA SYZ fibrations. This idea was originally brought by Kontsevich and Soibelman. Currently, we adopt the ones that also take Non-Archimedean Monge–Ampère equations into account.  In this talk, I will define such NA SYZ fibrations and explain a relation between these fibrations and the original SYZ fibrations for degenerating families of polarized abelian varieties. This talk is based on a joint work with Yuji Odaka.


■ Paul Helminck: Toric ranks and component groups of Jacobians of modular curves

Let p be a prime number not equal to 2 or 3, and let H be a congruence subgroup in SL2(Z) with modular curve X_{H}/K. In this talk, I will show how to explicitly reconstruct a deformation retract of the Berkovich analytification of X_{H} at a finite place of K over p in terms of glued Hecke double coset spaces. This will allow us to give the semistable toric rank and component group of the Jacobian of X_{H}, as well as its non-abelian l-Selmer groups for l coprime to p.  In the talk, I will explicitly determine these retracts for the modular curves X_{0}(N). For N=p^{n} and n ≤ 4, this in particular recovers results by Deligne-Rapoport, Edixhoven, Coleman-McMurdy and Tsushima. Using these descriptions, I will then show how to obtain the prime-to-6 parts of the component groups of X_{0}(N) over the field extension given by Krir’s theorem. This generalizes results by Mazur and Rapoport from the squarefree case to general N. 


■ Hiroshi Iritani: Quantum Duistermaat-Heckman measure

Given a (real) symplectic manifold with a Hamiltonian torus action, the Duistermaat-Heckman (DH) measure is defined as the push-forward of the Liouville measure by the moment map. Duistermaat and Heckman proved that it is a piecewise polynomial measure on a vector space. I will introduce a quantum cohomology version of the DH measure and give a conjectural relation to the (quantum) volume of the GIT quotients. Conjecturally, the quantum DH measure is a real-analytic smoothing of the classical DH measure, which arises in the tropical limit. 

■ Yohsuke Matsuzawa: Preimages question of self-morphisms on projective varieties

Pulling back an invariant subvariety by a self-morphism on projective variety, you will get a tower of increasing closed subsets. Working over a number field, we expect that the set of rational points (of bounded degree) contained in this increasing subsets eventually stabilizes. After introducing the problem, I will focus on a special case where the variety is the product of two P^1's. For the case, we proved the affirmative. Our proof uses p-adic local analysis, namely analysis of self-morphisms of strictly k-affionid algebras. This work is based on a joint work with Matt Satriano and Jason Bell, and recent work with Kaoru Sano.


■ Yuji Odaka: 

The K-stability is known to give a criteria of existence of canonical Kahler metrics, such as Kahler-Einstein metrics. The classical examples include hyperbolic metrics, Ricci-flat Kahler metrics, etc. Now the K-stability theory is also developed in the disguise of non-archimedean pluri-potential theory, but we show that there are much more relations between the canonical Kahler metrics with non-archimedean geometry to expect, rather than just an existence problem. For instance, if we take their metric limits, the obtained underlying spaces will be controlled by non-archimedean geometry and also give a control of moduli compactifications e.g. K-moduli, tropical geometric compactifications (O-Oshima) etc. There is also an arithmetic counterpart, which provides extension of Faltings heights to general arithmetic varieties. We give some short  survey talk around this direction. 

In my 2nd talk, we’ll continue discussions on the asymptotic of CY metrics and introduce the notion of "Galaxy models" over Puiseux series ring, an infinite version of semistable reduction, to geometrically realize/visualize the asymptotic behavior of CY metrics and non-archimedean CY metrics. Then we explain that at least if we see only a part of them, the compact ("tropical") limits, we observe that it often gives a certain "hybrid-type” compactification of the moduli spaces. The latter is proven for curves, abelian varieties, and K3 surfaces by bypassing some Lie theory+hyperKahler rotation. Time permitting, I may mention some more developments. Based on joint works with Keita Goto and Yoshiki Oshima. 


■ Jérôme Poineau: 

Consider a dynamical system on the projective line over a valued field k, which may be either Archimedean (R, C), or non-Archimedean (Q_p, C((t)), etc.). In this setting, one may define an equilibrium measure: a Radon measure, which is preserved by the dynamical system up to scalar multiplication, that lives either on the complex projective line (possibly up to conjugation), or a projective Berkovich line. We prove a general continuity statement for such measures, where the base field k, is also allowed to vary. We formulate our statement in the setting of Berkovich spaces over Z. Such spaces may be seen as fibrations containing complex analytic spaces as well as p-adic analytic spaces, for every prime number p. We will provide an introduction to those spaces.

We will explain how the continuity statement for equilibrium measures from the first talk may be used in an arithmetic context to study the distance between two dynamical systems (some Arakelov intersection number) varying in a family.  As an application, following a strategy by DeMarco-Krieger-Ye, we will give a proof of a conjecture of Bogomolov-Fu-Tschinkel on uniform bounds on the number of common images on the projective line of torsion points of two elliptic curves. 


■ Joseph Rabinoff: Semipositivity on Berkovich spaces I, II

We introduce a notion of a piecewise-linear semipositive function on a Berkovich analytic space, and we develop the theory of such functions.  For instance, semipositivity of a PL function can be checked on locally embedded curves; a PL function that is a pointwise limit of semipositive PL functions is again semipositive; the maximum principle holds for semipositive PL functions; and so on.  A key technical result is a local lifting theorem for a subscheme of the special fiber of an admissible formal scheme. This work is joint with Walter Gubler.


■ JuAe Song: Finitely generated congruences on tropical rational function semifields defined by subsets of tropical tori

We can define a tropical analogue of rational function fields, i.e., tropical rational function semifields. In tropical settings, we frequently deal with congruences instead of ideals. As in classical algebraic geometry, subsets of tropical tori define congruences in tropical rational function semifields (and in fact, vise versa). In this talk, we give a necessary and sufficient condition for which for a subset of a tropical torus, the congruence defined by it is finitely generated as a congruence. If time allows, we introduce results so far related to tropical rational function semifields.


■ Yuto Yamamoto: Non-archimedean SYZ fibrations via tropical contractions

For a maximally degenerate Calabi--Yau variety, the Berkovich retraction associated with a (good) minimal dlt model is regarded as an SYZ fibration in non-archimedean geometry. In general, the integral affine structure induced on the base space of the fibration differs from the one defined for the dual intersection complex of a toric degeneration in the Gross--Siebert program. In this talk, using tropical geometry, we construct non-archimedean SYZ fibrations whose bases are integral affine manifolds appearing in the Gross--Siebert program for Calabi--Yau complete intersections of Batyrev--Borisov.


■ Tony Yue Yu: Moduli space of non-archimedean holomorphic disks I, II

I will describe the moduli space of non-archimedean holomorphic disks in affine log Calabi-Yau varieties, which is foundational to the non-archimedean mirror symmetry program. I will discuss boundary conditions, smoothness, dimension and properness. Smoothness relies on the non-archimedean deformation theory joint with M. Porta. Properness relies on formal models and Temkin’s theory of reduction of germs. Work in progress with S. Keel.