Regulators and Heights in Algebraic Geometry
Date

Topic
The Mahler measure of curves and surfaces
by Marie JosÃ© Bertin UniversitÃ© Pierre et Marie Curie (Paris 6), Institut de MathÃ©matiques 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 BlochWigner 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 K3surface, I give several examples of the Mahler measure expressed in terms of the Lseries of the K3surface for s=3. Dedekind zeta motives for totally real fields by Francis Brown CNRS, Institut de MathÃ©matiques de Jussieu, IHES
On singular BottChern classes
by JosÃ© Ignacio Burgos Gil Universidad de Barcelona
The singular BottChern classes measure the failure of an exact RiemannRoch 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 RiemannRoch theorem in Arakelov geometry for closed immersions. There are two definitions of singular BottChern classes. The first due to Bismut, Gillet and SoulÃ© uses the formalism of super connections. The second, due to Zha, is an adaptation of the original definition of BottChern classes by Bott and Chern.In this talk we will give an axiomatic characterization of singular BottChern classes, which is similar to the characterization of BottChern forms, but that depends on the choice of an arbitrary characteristic class. This characterization allow us to give a new definition of singular BottChern forms by means of the deformation to the normal cone technique and to compare the previous definitions of singular BottChern forms. Moreover we will give an explicit computation of the characteristic class associated to BismutGilletSoulÃ© definition of singular Bott Chern currents.
Generic prank of semistable fibration
by Junmyeong Jang Purdue University
In this presentation, I will be concerned with two pathological phenomenons of positive characteristic, the failure of Miyaokayau inequality and the failure of semipositivity theorem. Szpiro showed that a Frobenius base change of nonisotrivial smooth fibration violates MiyaokaYau inequality. For such a fibration, if the prank 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 MiyaokaYau inequality with smooth Picard scheme, which is a counterexample of Parshin's expectation. And we will see for a semistable fibration p : X ? C of a proper smooth surface to a proper smooth curve, if the prank of the generic fiber is maximal, the semipositivity theorem holds and if the prank of the generic fiber is 0, some Frobenius base change of p violates the semipositivity theorem. This result may be applied to a problem of the distribution of pranks of reductions of a certain nonclosed point in the moduli space of curves over QÂ¯.
The AbelJacobi 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. MuellerStach, will be summarized). We also look at an application of the toric construction in the nonmodular case, to limits of normal functions for families of CalabiYau 3folds.
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
by Marie JosÃ© Bertin UniversitÃ© Pierre et Marie Curie (Paris 6), Institut de MathÃ©matiques 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 BlochWigner 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 K3surface, I give several examples of the Mahler measure expressed in terms of the Lseries of the K3surface for s=3. Dedekind zeta motives for totally real fields by Francis Brown CNRS, Institut de MathÃ©matiques de Jussieu, IHES
On singular BottChern classes
by JosÃ© Ignacio Burgos Gil Universidad de Barcelona
The singular BottChern classes measure the failure of an exact RiemannRoch 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 RiemannRoch theorem in Arakelov geometry for closed immersions. There are two definitions of singular BottChern classes. The first due to Bismut, Gillet and SoulÃ© uses the formalism of super connections. The second, due to Zha, is an adaptation of the original definition of BottChern classes by Bott and Chern.In this talk we will give an axiomatic characterization of singular BottChern classes, which is similar to the characterization of BottChern forms, but that depends on the choice of an arbitrary characteristic class. This characterization allow us to give a new definition of singular BottChern forms by means of the deformation to the normal cone technique and to compare the previous definitions of singular BottChern forms. Moreover we will give an explicit computation of the characteristic class associated to BismutGilletSoulÃ© definition of singular Bott Chern currents.
Generic prank of semistable fibration
by Junmyeong Jang Purdue University
In this presentation, I will be concerned with two pathological phenomenons of positive characteristic, the failure of Miyaokayau inequality and the failure of semipositivity theorem. Szpiro showed that a Frobenius base change of nonisotrivial smooth fibration violates MiyaokaYau inequality. For such a fibration, if the prank 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 MiyaokaYau inequality with smooth Picard scheme, which is a counterexample of Parshin's expectation. And we will see for a semistable fibration p : X ? C of a proper smooth surface to a proper smooth curve, if the prank of the generic fiber is maximal, the semipositivity theorem holds and if the prank of the generic fiber is 0, some Frobenius base change of p violates the semipositivity theorem. This result may be applied to a problem of the distribution of pranks of reductions of a certain nonclosed point in the moduli space of curves over QÂ¯.
The AbelJacobi 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. MuellerStach, will be summarized). We also look at an application of the toric construction in the nonmodular case, to limits of normal functions for families of CalabiYau 3folds.
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
Speakers
Details
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 Ktheory, 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 MordellWeil 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 Lfunctions).
One of the classical examples of heights involves the canonical heights of Abelian varieties (NÃ©ron, Tate) over number fields. Related to this is the height regulator and NÃ©ronTate 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 nonspecialists. 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.
One of the classical examples of heights involves the canonical heights of Abelian varieties (NÃ©ron, Tate) over number fields. Related to this is the height regulator and NÃ©ronTate 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 nonspecialists. 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.
Additional Information
Participants
Stephanie BelcherMatthew Greenberg
Harish Krishnamurthy
Ayobami Omololu Opajobi
Registration
To register, please click here.
MarieJosÃ© BertinJosÃ© Ignacio Burgos Gil Rob de JeuElisenda FeliuHerbert GanglHenri GilletJunmyeong Jang Matt KerrSteven LuKumar MurtyGreg Pearlstein Wayne RaskindRamesh SreekantanSampei Usui; Xiaowei Wang
This is a Past Event
Event Type
Scientific, Seminar
Date
April 12â€“17, 2008
Time

Location