Divisors and Euler sparks of atomic sections (Q2758103)

This comprehensive but well-arranged article is both a surprising treatment of the basic theory of atomic sections of bundles with their zero currents or divisors (the classical theory of characteristic classes is reformulated within the geometric measure theory) and a deep analysis of the Cheeger-Simons theory of differential characters (homomorphisms \(f\) from the space of smooth singular \((n-1)\)-dimensional cycles to \(\mathbb{R}/\mathbb{Z}\) such that \(f(\partial C)=\int_C\omega \bmod Z\) for appropriate \(n\)-form \(\omega)\) from the analytical point of view. The full contents can be indicated only very briefly: alternative Euler class represented by a current supported on the zero set, transgression formulae relating to Euler-Pfaff currents are identified with the Gauss-Bonnet theorem, the secondary Euler class of a connection, the associative intersection for zero currents, the Thom forms leading to the Chern potential on the total space of the vector bundle, and other topics. The article is self-contained (and it was read with much pleasure).NEWLINENEWLINENEWLINEFew basic concepts. A smooth \(\mathbb{R}^n\)-valued function \(u=(u_1,\dots,u_n)\) on a manifold \(X\) is said to be atomic if \(\log|u|\in L^1_{\text{loc}}(X)\) and \(du^I/|u|^p=du_{i_1}\wedge\dots\wedge du_{i_p}/|u|^p\in L^1_{\text{loc}}(X)\) if \(|I|=p\leq n-1\). Assuming \(u\) atomic, then the divisor of \(u\) is defined by the current equation \(\text{Div}(u)=du^*(\Theta)\) on \(X\), where \(\Theta=y\lfloor dy/|y|^n\) (\(n\dim X\)) is the solid angle kernel. The definitions easily extend to sections of vector bundles \(\pi:E\to X\). In this case, let \(\varphi\) be a smooth \(d\)-closed form on \(E\) with compact vertical support. We introduce \(\text{Res} \varphi=\pi_*\varphi\) (the fiber integral) and then the \(\varphi\)-spark \(T^\varphi\) is defined by NEWLINE\[NEWLINET^\varphi=-\int^\infty_0\frac{dt}{t} y\cdot \frac{\partial}{\partial y} L\varphi_s,NEWLINE\]NEWLINE where \(\varphi_s\) \((0< s\leq \infty)\) is the pull-back of \(\varphi\) with the \(1/s\) homothety.