[Paper Review] Characteristic classes of flat bundles and determinant of the Gauss-Manin connection

Hélène Esnault|ArXiv.org|Jun 13, 2002

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper establishes a framework for characteristic classes of flat bundles using algebraic differential characters, linking them to the determinant of the Gauss-Manin connection. It proves that the $ε$-connection on $ε$-lines, constructed via local invariants and trace residues, is compatible with the Gauss-Manin connection, providing a global formula for the determinant of cohomology in terms of local $ε$-factors.

ABSTRACT

The purpose of this note is to give a survey on recent progress on characteristic classes of flat bundles, and how they behave in a family.

Motivation & Objective

• To develop a cohomological framework for characteristic classes of flat bundles using algebraic differential characters.
• To relate the determinant of the Gauss-Manin connection to local $ε$-factors via a product formula.
• To construct a connection on $ε$-lines that is compatible with the Gauss-Manin connection.
• To generalize the Chern-Cheeger-Simons differential character construction to algebraic and arithmetic settings.
• To unify algebraic, analytic, and $ε$-factor approaches to the determinant of cohomology in families.

Proposed method

• Uses the ring $AD(X) = \oplus_n \mathbb{H}^n(X, \mathcal{K}_n^M \xrightarrow{d\log} \Omega_{X/k}^n \to \cdots \to \Omega_{X/k}^{2n-1})$ to define characteristic classes of flat bundles with connection.
• Applies the Gersten resolution of $\mathcal{K}_n^M$ to relate $AD^n(X)$ to Chow groups and de Rham cohomology.
• Constructs the $\epsilon$-line as a product of discrete and compact determinants via $\lambda_d$ and $\lambda_c$, using the local monodromy at singularities.
• Introduces a relative connection on $\epsilon$-lines over $\omega_{K((t))/K}^\times$, compatible with the Gauss-Manin connection via functoriality.
• Derives a formula for the $\epsilon$-connection in terms of trace and residue: $\epsilon(\nabla, dt/t^m) = \mathrm{Tr} \mathrm{Res}_0 da a^{-1} A + \frac{m}{2} d\log \det(a(0))$.
• Uses Grothendieck's definition of connection and integrable liftings to lift the relative connection to an integrable connection over $k$.

Experimental results

Research questions

• RQ1 How can characteristic classes of flat bundles be defined algebraically using differential characters and extended to families?
• RQ2 What is the precise relationship between the determinant of cohomology and local $\epsilon$-factors in the context of the Gauss-Manin connection?
• RQ3 How can the $\epsilon$-connection on $\epsilon$-lines be constructed to be compatible with the Gauss-Manin connection?
• RQ4 Can the Chern-Cheeger-Simons differential character be generalized to algebraic and arithmetic settings via algebraic differential characters?
• RQ5 How do the algebraic $\epsilon$-factor constructions relate to the polarized Fredholm line method in higher rank?

Key findings

• The characteristic classes $c_n((E,\nabla)) \in AD^n(X)$ are functorial, additive, and map to the Chern class $c_n(E)$ in the Chow group $CH^n(X)$.
• The curvature evaluation $d(c_n((E,\nabla)))$ equals the Chern-Weil form $c_n(\nabla^2)$, linking algebraic and differential geometry.
• The restriction of $c_n((E,\nabla))$ to the generic point gives the algebraic Chern-Simons invariant in $H^0(X, \Omega_{X/k}^{2n-1}/d\Omega_{X/k}^{2n-2})$.
• For $k \subset \mathbb{C}$, the image $\iota(c_n((E,\nabla)))$ in $D^n(X)$ corresponds to the differential character $c_n(E_{\mathrm{an}}^{\nabla}) \in H^{2n-1}(X_{\mathrm{an}}, \mathbb{C}/\mathbb{Z}(n))$.
• The $\epsilon$-connection on $\epsilon$-lines is given by $\epsilon(\nabla, dt/t^m) = \mathrm{Tr} \mathrm{Res}_0 da a^{-1} A + \frac{m}{2} d\log \det(a(0))$, a formula involving trace and residue.
• The product formula $\det H^\ast(X/K, E_{\mathrm{min}}) = \otimes_{x \in \bar{X} \setminus X} \epsilon_x(\nabla_{K((t))/K, \nu})$ holds, with $\epsilon_x$ defined via local invariants.