[Paper Review] Second Chern numbers of vector bundles and higher adeles

Denis Vasilievich Osipov|arXiv (Cornell University)|Jun 22, 2017

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper presents a novel, K-theory-free construction of the second Chern number for vector bundles on a smooth projective surface over a perfect field using adelic transition matrices and central extensions of GL_n(𝔸_X) by ℤ. The method relies on canonical ℤ-torsors from locally linearly compact vector spaces and lifts transition matrices to a central extension, where their product yields the second Chern number as an integer, offering a new approach analogous to analytic Riemann-Roch proofs.

ABSTRACT

We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an algebraic $K$-theory and depends on the canonical $\mathbb{Z}$-torsor of a locally linearly compact $k$-vector space. Analogs of certain auxiliary results for the case of an arithmetic surface are also discussed.

Motivation & Objective

• To provide an alternative, elementary construction of the second Chern number of a vector bundle on a smooth projective surface without relying on algebraic K-theory.
• To establish a connection between the second Chern number and the commutator of lifts in a central extension of GL_n(𝔸_X) by ℤ.
• To extend the construction to arithmetic surfaces by incorporating Archimedean components via real-valued torsors.
• To generalize the method to the arithmetic case using central extensions by ℝ₊*, enabling applications to arithmetic Riemann-Roch and Noether formula.

Proposed method

• Construct the Parshin-Belinson adelic ring 𝔸_X for a smooth projective surface X over a perfect field k.
• Utilize the canonical ℤ-torsor of dimension theories for locally linearly compact k-vector spaces arising from the filtration by divisors on X.
• Define a central extension 𝔾𝕃̃_n(𝔸_X) of GL_n(𝔸_X) by ℤ using the torsor structure on 𝔸_X^n.
• Construct a second central extension 𝔾𝕃̂_n(𝔸_X) by ℤ, compatible with the first, and prove canonical splittings over specific subgroups of GL_n(𝔸_X).
• Use trivializations of the vector bundle at scheme points to obtain transition matrices in GL_n(𝔸_X), satisfying the cocycle condition.
• Lift these matrices to 𝔾𝕃̂_n(𝔸_X) using the canonical splittings; their product lies in ℤ, yielding the second Chern number as an integer.

Experimental results

Research questions

• RQ1 Can the second Chern number of a vector bundle on a smooth projective surface be constructed without algebraic K-theory, using only adelic transition matrices and torsor structures?
• RQ2 How can the central extension of GL_n(𝔸_X) by ℤ be canonically defined using the dimension theory of locally linearly compact vector spaces?
• RQ3 What is the role of canonical splittings of the central extension over subgroups of GL_n(𝔸_X) in defining the second Chern number?
• RQ4 How can this construction be extended to arithmetic surfaces, including Archimedean fibers, via real-valued torsors and central extensions by ℝ₊*?
• RQ5 Can this method be used to prove the Noether formula in the spirit of analytic Riemann-Roch theorems?

Key findings

• The second Chern number of a vector bundle on a smooth projective surface over a perfect field is constructed as an integer via lifts of adelic transition matrices to a central extension of GL_n(𝔸_X) by ℤ.
• The construction relies on the canonical ℤ-torsor of dimension theories for locally linearly compact k-vector spaces, arising from the filtration of 𝔸_X by divisors.
• Canonical splittings of the central extension 𝔾𝕃̂_n(𝔸_X) over certain subgroups of GL_n(𝔸_X) allow the lifting of transition matrices, whose product gives the second Chern number.
• For arithmetic surfaces, the central extensions 𝔾𝕃̂_n(𝔸_X^ar) by ℝ₊* canonically split over subgroups like GL_n(𝔸_X,12) × GL_n(𝔸_X,∞(0)), GL_n(𝔸_X,02), and GL_n(𝔸_X,01), generalizing the algebraic case.
• The central extension 𝔾𝕃̂_n(𝔸_X^ar) is the Baer sum of the extensions over GL_n(𝔸_X) and GL_n(𝔸_X,∞), establishing compatibility with the arithmetic setting.
• The method provides a foundation for proving the Noether formula and generalizing analytic Riemann-Roch theorems to higher Chern classes in an adelic framework.