Summer School: Regulators and Differential Algebraic K-Theory what the school is about

Graduiertenkolleg 1821 Cohomological Methods in Geometry
Summer School 2013
Regulators and Differential Algebraic K-Theory

What the summer school is about

The aim of the summer school is to give a new explicit description of the Beilinson regulator from algebraic K-theory to Deligne cohomology, also known as absolute Hodge cohomology. This new description is well-behaved with respect to multiplication.

Basic objects

Algebraic K-theory is a fine invariant of a ring/an algebraic variety/a scheme built from the category of vector bundles on the space. It turns out to be very hard to compute. A first step is the introduction of invariants in cohomology: to every vector bundle we attach its Chern class in say singular cohomology. This is still much to coarse. It factors over a refined Chern class with values in Deligne cohomology. Deligne cohomology is attached to complex algebraic varieties and uses the information coming from the corresponding analytic space, most importantly the Hodge decomposition theorem. The definition of Deligne cohomology can be rewritten in terms of the Hodge structure on singular cohomology and then goes by the name of absolute Hodge cohomology. The Chern class is also called Beilinson regulator because it was introduced by Beilinson.

History of regulators

The Beilinson regulator is a generalization of a construction that was known before in special cases. It turns out that the information in the Beilinson regulator is enough to understand algebraic K-theory of number fields - finite extensions of the field of rational numbers.

The first case is the Dirichlet regulator used to study the units of the ring of integers of a number field, i.e., its K_1. A unit u is mapped to the tuple of the logarithms of the absolute values of u under all embeddings of the field into the complex numbers. The famous Dirichlet unit theorem states that the image of this map is a lattice, in particular that the units form a finitely generated abelian group. Borel in his seminal paper was able to prove the analogous result for higher algebraic K-groups of these rings of integers. His adhoc regulator was translated by Beilinson and later Burgos into the general Beilinson regulator.

Why this is important

As described above, explicit computations of higher algebraic K-theory are very scarce. A better understanding of the Beilinson regulator should eventually lead to a better understanding of algebraic K-groups.

On the topological side, the K-theory spectra of varieties can be used to define cohomology theories. Refinements of these cohomology theories to manifolds should be useful to analyze differentiable stuctures of fibre bundles.

There are also (at least) two number theoretic applications. The Beilinson regulator is a main ingredient of the Beilinson conjecture on special values of L-functions, which generalizes the class number formula (the case of the units of ring of integers of a number field) and the conjecture of Birch and Swinnerton-Dyer.

It also occurs in the definition of arithmetic K-groups generalizing the theory of arithmetic Chow groups in Arakelov theory.

Scope of the summer school

During the summer school, there will be foundational lecture series introducing higher algebraic K-theory, absolute Hodge cohomology and the Beilinson regulator. In the second half of the week, the aim will be to give a new conceptual description of the Beilinson regulator. This construction uses the language of infinity categories to be introduced in a third series of foundational lectures. The new construction of the regulator will be phrased in the language of differential cohomology. The theory will be illustrated by presenting certain constructions in arithmetic geometry (Heights, Bloch group).