2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79223In this paper, we first prove a local family version of the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, then we extend the famous Witten's rigidity Theorems to the family case. Several family vanishing theorems for elliptic genera are also proved.22 pages. Duke Math. J. to appearDifferential Geometry57R20, 58G10, 11F27text