Endomorphisms of logarithmic schemes (Q1878659)

Consider a log. smooth log. scheme \((X,M)\), defined over a discrete valuation ring \(V\), with residue field \(k\) of characteristic \(p\). Fix a uniformizer \(\pi\) of \(V\). The author studies the extensions of automorphisms \(\sigma\) of the special fiber \(X_s\) of \(X\) to the whole family \(X\). Assuming that the smooth locus is dense in \(X_s\), the author proves the existence of an integer \(m\) (actually computable) such that \(\sigma\) lifts to the log. structure \((X_s,M_s)\) over the special fiber if and only it lifts locally to the infinitesimal neighbourhood \(X/\pi^m\). This explains in greater detail the known fact that sheaves of nearby cycles to the special fiber depend only on the completion of \(X\). In the case of (relative) curves, the author shows a trace formula for the number of points collapsing to fixed points of \(\sigma\), in its lifting to \(X\).