Abstracts
José Ignacio Burgos Gil (ICMAT)
"Equidistribution of small points on toric varieties"
Abstract: As the culmination of work of many mathematicians, Yuan has obtained a very general equidistribution result for small points in arithmetic varieties. Roughly speaking Yuan's theorem states that given a « very » small generic sequence of points, with respect to a positive metrized divisor, the associated sequence of measures converges weakly to the Monge-Ampere measure of the divisor. Here very small means that the height of the points converges to the lower bound of the essential minimum given by Zhang inequalities.
The existence of a very small generic sequence is a strong condition on the arithmetic variety because it implies that the essential minimum attains its lower bound. We will say that a sequence is small if the height of the points converges to the essential minimum. By definition every arithmetic variety contains small generic sequences.
We study the equidistribution property in the setting of toric line bundles on toric varieties. In this talk we introduce the notion of quasi-canonical metrized divisors and toric monocritical metrized divisors.
We show that, for toric line bundles on toric varieties arithmetic Yuan's theorem can be split in two parts.
A) A toric metrized divisor D is monocritical if and only if for every D-small generic sequence of points, the associated sequence of measures converges weakly to a measure.
B) A toric metrized divisor D is quasi-canonical if and only if or every D-small generic sequence of points, the associated sequence of measures converges weakly to the Monge Ampere measure associated to the divisor.
We give criteria for a toric metrized divisor to be monocritical or quasi-canonical.In particular we show examples of positive line bundles that are non-monocritical and hence where equidistribution does not hold.
Huayi Chen (Univ. Joseph Fourier - Grenoble)
"Relative isoperimetric inequality"
Abstract: In this talk, a relative version of isoperimetric inequality in Arakelov geometry will be explained. The proof uses a construction of Knothe on the transportation of measures between two convex bodies.
Javier Fresan (ETH Zurich)
"Some remarks on Colmez's conjecture"
Abstract: Colmez's conjecture relates the Faltings height of CM abelian varieties to logarithmic derivatives of Artin L-functions at s=0. Very recently, Yuan-Zhang and Andreatta-Goren-Howard-Madapusi-Pera proved an averaged version obtained by summing over all CM types of a fixed CM field. I will first survey on certain aspects of their proofs, then add some small remarks –joint with T. Yang– which allow to derive low-dimensional cases of the original conjecture from the averaged version.
Henri Gillet (Univ. Illinois at Chicago)
"Singular Arithmetic Riemann Roch (Joint with C. Soulé)"
Abstract: The "traditional" arithmetic Riemann Roch theorem applies only to proper morphisms between non-singular arithmetic varieties. However given the lack of resolution of singularities for arithmetic varieties, if one has a smooth projective variety over a number field, there is no guarantee that it has a non-singular model over the ring of integers. Thus it is natural to ask how one may extend the arithmetic Riemann Roch theorem to the category of arithmetic varieties that while having non-singular generic fibers, may be singular over finite places. I shall describe a proof of such a "singular" arithmetic Riemann Roch theorem. Our approach is based on a new proof of the "classical" Baum-Fulton-MacPherson singular Riemann-Roch, which uses de Jong's theorem on the existence of non-singular alterations of schemes.
We also make use of a new approach to arithmetic intersection theory which uses de Jong's theorem together with deformation to the normal cone, avoiding any appeal to the moving lemma.
Souvik Goswami (Univ. Alberta)
"Business of height pairing"
Abstract: In algebraic geometry there is the notion of a height pairing of algebraic cycles, which lies at the confluence of arithmetic, Hodge theory and topology. After explaining the basic notions, we introduce new directions in this subject. Specifically, we address an extension of Beilinson's height pairing to the (still conjectural) graded pieces of a Bloch-Beilinson filtration. Along the way, we use machineries from arithmetic intersection theory to provide an example computation.
Mounir Hajli (Inst. Math. Academia Sinica)
"On the notion of the canonical Kähler metrics on the projective toric manifolds"
Abstract: In this talk, I will introduce the notion of singular Kähler metrics on compact Kähler manifolds and an associated spectral theory which generalizes partially the smooth case. As an application, when X is a projective toric manifold we define the notion of the canonical Kähler metric on X, as an analogue of the notion of canonical metrics on equivariant line bundles on X. Also, I will show that the holomorphic analytic torsion can be extended to a large class of singular Kähler metrics.
Hideaki Ikoma (Univ. Tokyo)
"Remarks on the arithmetic restricted volumes and the arithmetic base loci"
Abstract: I would like to report some fundamental properties of arithmetic restricted volumes or arithmetic multiplicities of adelically metrized line bundles. The arithmetic restricted volume has the concavity property and characterizes the arithmetic augmented base locus as the null loci. As a consequence, we can obtain an easy proof of a weak version of the arithmetic Nakai-Moishezon theorem.
Klaus Künnemann (Univ. Regensburg)
"Metrics and delta-forms in non-archimedean analytic geometry"
Abstract: We report on joint work with Walter Gubler from Regensburg. We consider metrics on line bundles over the non-archimedean analytification of an algebraic variety. Extending work by Chambert-Loir and Ducros we introduce delta-forms, discuss their basic properties, describe briefly their use in the construction of first Chern forms, Monge-Ampère measures, and local heights, and report on recent work about their positivity properties.
Yuji Odaka (Kyoto Univ.)
"Generalised Faltings heights"
Abstract: We discuss an extension of Faltings (modular) heights for arithmetic varieties.
Aurélien Rodriguez (Univ. Essen)
"Arithmetic Cobordism"
Abstract: In the early 2000's, Levine and Morel built an algebraic cobordism theory, extending to the case of arbitrary algebraic varieties over any field the construction and properties of the complex cobordism ring studied by Milnor and Quillen. We will show how we can refine their construction to build a weak arithmetic cobordism group in the context of Arakelov geometry. The general strategy is to define the notion of homological theory of arithmetic type, encapsulating the common properties of arithmetic K-theory and arithmetic Chow groups, and to build a universal such theory. We will show how we can generalize some structural results about arithmetic K-theory and Chow theory, and if we have time left we'll also explain some future possible developments of the theory.
Emmanuel Ullmo (IHES)
"Algebraic flows on homogeneous varieties"
Abstract: Let S be an abelian variety or a Shimura variety. The Ax-Lindemann conjecture predicts that the Zariski closure of an algebraic flow in S is a weakly special subvariety of S. We will discuss some conjectures and a few results on the usual topological closure of an algebraic flow in the same context.
Kazuhiko Yamaki (Kyoto Univ.)
"Bogomolov conjecture for curves over any function field"
Abstract: Let $K$ be a number field or the function field of a normal projective variety over an algebraically closed field $k$. Let $C$ be a smooth projective curve over $K$ of genus $g ¥geq 2$. When $K$ is a function field, assume that $C$ is non-isotrivial. Fixing a divisor on $C$ of degree $1$, we embed $C$ in its Jacobian $J$. Then the Bogomolov conjecture claims that $C$ has only a finite number of small points with respect to the N¥'eron--Tate height.
In 1998, Ullmo proved that the Bogomolov conjecture holds when $K$ is a number field. In 2011, Cinkir proved that the conjecture holds when $K$ is the function field of a curve of characteristic $0$. In this talk, we report our recent result that the Bogomolov conjecture holds over any function fields of any characteristic.
Ken-Ichi Yoshikawa (Kyoto Univ.)
"Analytic torsion for K3 surfaces with involution"
Abstract: About ten years ago, I introduced a holomorphic torsion invariant of K3 surfaces with involution and proved its automorphy viewed as a function on the moduli space. Very recently, its explicit formula is completely determined. It is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. In my talk, I will report this progress. This is a joint work with Shouhei Ma.
Tong Zhang (Univ. Alberta)
"Effective bounds of linear series on arithmetic varieties"
Abstract: I will talk about an inequality bounding the number of effective sections of arithmetic line bundles from above. This can be viewed as an effective version of the arithmetic Hilbert-Samuel formula. It is a joint work with Xinyi Yuan.