## Regulators and Heights in Algebraic Geometry

- Start Date: 04/12/2008
- End Date: 04/17/2008

Marie-José Bertin

Francis Brown

José Ignacio Burgos Gil

Rob de Jeu

Elisenda Feliu

Herbert Gangl

Henri Gillet

Junmyeong Jang

Matt Kerr

Steven Lu

Kumar Murty

Greg Pearlstein

Wayne Raskind

Ramesh Sreekantan

Sampei Usui; Xiaowei Wang

University of Alberta

**The Mahler measure of curves and surfaces**

by Marie José Bertin Université Pierre et Marie Curie (Paris 6), Institut de Mathématiques de Jussieu

I report on some new examples of explicit logarithmic Mahler measures of multivariate polynomials.

When the polynomial defines a parametrizable curve, its Mahler measure is expressed in terms of Bloch-Wigner dilogarithms of an element of the Bloch group of an imaginary quadratic field ( Thus a link with hyperbolic varieties). When the polynomial defines a singular K3-surface, I give several examples of the Mahler measure expressed in terms of the L-series of the K3-surface for s=3. Dedekind zeta motives for totally real fields by Francis Brown CNRS, Institut de Mathématiques de Jussieu, IHES

**On singular Bott-Chern classes**

by José Ignacio Burgos Gil Universidad de Barcelona

The singular Bott-Chern classes measure the failure of an exact Riemann-Roch theorem for closed immersions at the level of currents. They are the key ingredient in the definition of direct images of hermitian vector bundles under closed immersions and in the proof of the arithmetic Riemann-Roch theorem in Arakelov geometry for closed immersions. There are two definitions of singular Bott-Chern classes. The first due to Bismut, Gillet and Soulé uses the formalism of super connections. The second, due to Zha, is an adaptation of the original definition of Bott-Chern classes by Bott and Chern.In this talk we will give an axiomatic characterization of singular Bott-Chern classes, which is similar to the characterization of Bott-Chern forms, but that depends on the choice of an arbitrary characteristic class. This characterization allow us to give a new definition of singular Bott-Chern forms by means of the deformation to the normal cone technique and to compare the previous definitions of singular Bott-Chern forms. Moreover we will give an explicit computation of the characteristic class associated to Bismut-Gillet-Soulé definition of singular Bott Chern currents.

**Generic p-rank of semi-stable fibration**

by Junmyeong Jang Purdue University

In this presentation, I will be concerned with two pathological phenomenons of positive characteristic, the failure of Miyaoka-yau inequality and the failure of semi-positivity theorem. Szpiro showed that a Frobenius base change of non-isotrivial smooth fibration violates Miyaoka-Yau inequality. For such a fibration, if the p-rank of the generic fiber is maximal, the dimension of the Lie algebra of Picard scheme is stable after the Frobenius base change. Using this fact and a reduction argument we can construct a counter example of Miyaoka-Yau inequality with smooth Picard scheme, which is a counterexample of Parshin's expectation. And we will see for a semi-stable fibration p : X ? C of a proper smooth surface to a proper smooth curve, if the p-rank of the generic fiber is maximal, the semi-positivity theorem holds and if the p-rank of the generic fiber is 0, some Frobenius base change of p violates the semi-positivity theorem. This result may be applied to a problem of the distribution of p-ranks of reductions of a certain non-closed point in the moduli space of curves over Q¯.

**The Abel-Jacobi map on the Einsestein symbol**

by Matthew Kerr Durham University

In this talk we consider two different constructions of motivic cohomology classes on families of toric hypersurfaces and on Kuga varieties. Under certain modularity conditions on the former we say how the constructions "coincide", obtaining a complete explanation of the phenomenon observed by Villegas, Stienstra, and Bertin in the context of Mahler measure. (This is where the AJ computation on the Kuga varieties, done using our formula with J. Lewis and S. Mueller-Stach, will be summarized). We also look at an application of the toric construction in the non-modular case, to limits of normal functions for families of Calabi-Yau 3-folds.

**Moduli of polarized logarithmic Hodge structures and period maps**

by Sampei Usui Osaka University

**Height and GIT weight**

by Xiaowei Wang The Chinese University of Hong Kong

In this talk, we will establish a new connection between the weight in the geometric invariant theory and the height introduced by Cornalba and Harris CH and Zhang Z. Then I will explains two applications of this connection.

*Talks will be held at CAB 269 (April 12, 14, 15, 16) and ETL E1 008 (April 13). We have booked the computer lab at CAB 341. map
*

The study of regulators and that of heights, are both highly developed and intricate subjects, that thrive through energetic interactions with arithmetic algebraic geometry, number theory, algebraic K-theory, and Hodge theory. For a variety defined over a number field, the height of a given point is a measure of the complexity of that point. The notion of heights in Algebraic Geometry lies in the interpretation of geometric information being translated into arithmetic datum. The role of heights in the literature gained prominence after Faltings announced his proof of the celebrated Mordell-Weil conjecture, which stems from the fact that there are only a finite number of points with bounded height. The subject of heights interacts naturally with the subject of algebraic cycles and regulators, Arakelov geometry and Mahler measure. While initially defined as a height on polynomials, Mahler measure can also be seen, in favorable cases, as periods of regulators (thus leading to special values of L-functions).

One of the classical examples of heights involves the canonical heights of Abelian varieties (Néron, Tate) over number fields. Related to this is the height regulator and Néron-Tate pairing, which is a forerunner to the height pairings introduced by Bloch and Beilinson, as well as their works on regulators.

Heights can also be defined over function fields. Although the statements regarding heights over function fields are in general much easier to prove than the corresponding statements over number fields, their solutions usually shed some light on the number theory situation. The study of heights over function fields typically involves areas such as Nevalinna theory, Bogomolov stability theory, which are interesting in themselves.

The immense recent progress on regulators and on heights, based on so many interactions with so many other areas of mathematics (not unlike algebraic geometry itself), has contributed to a considerable degree of inaccessibility, especially for graduate students and non-specialists. This is also true for the two camps of specialists in regulators versus those working on heights.

The purpose of this conference is to bring together leading experts in the areas mentioned above to interact and discuss the latest developments in the field.

- James D. Lewis (lewisjd@ualberta.ca) is a full professor at the

University of Alberta. His research interests include regulators of

algebraic cycles and Hodge theory. - Xi Chen (xiche@math.ualberta.ca)
- Matilde Lalín (mlalin@math.ualberta.ca) is an assistant

professor at

the University of Alberta. Her research interests include regulators and Mahler measures.

**Participants**

Stephanie Belcher

Matthew Greenberg

Harish Krishnamurthy

Ayobami Omololu Opajobi

**Registration**

To register, please click here.