Regulators of canonical extensions are torsion:the case of two transversally intersecting smooth divisors

This note extends the main result of \cite{IS} 2007 — torsion of the extended Chern–Simons (regulator) classes of the Deligne canonical extension of a flat bundle with unipotent monodromy at infinity — from the case of a smooth irreducible boundary divisor to the case of a boundary divisor $D=D_1\cup D_2$ with two smooth irreducible components meeting transversally along a smooth center $Z=D_1\cap D_2$. Let $X$ be a smooth projective variety defined over $\mathbb{C}$, and $U:=X-D$. Given a flat bundle $(E,\mathrm{\nabla })$ on $U$ with unipotent monodromy around the components of $D$ consider Deligne’s canonical extension $(\bar{E},\bar{\mathrm{\nabla }})$ on $X$. Then the extended Chern-Simons classes $$c_p(\bar{E},\bar{\mathrm{\nabla }})\in H^{2p-1}(X,\mathbb{C}\mathrm{/}\mathbb{Z})$$ are torsion, for $p\ge 2$. These notes were prepared in 2009-2010, and the preprint \cite[2026]{IS2} treats the full normal crossing case via a different approach.