Supersymmetry of IIA warped flux AdS and flat backgrounds

We identify the fractions of supersymmetry preserved by the most general warped flux AdS and flat backgrounds in both massive and standard IIA supergravities. We find that AdS(n) x(w) M10-n preserve 2([n/2])k for n <= 4 and 2([n/2]+1)k for 4 < n <= 7 supersymmetrics, k is an element of N->0. In addition we show that, for suitably restricted fields and M10-n, the killing spinors of AdS backgrounds are given in terms of the zero modes of Dirac like operators on M10-n. This generalizes the Lichnerowicz theorem for connections whose holonomy is included in a general linear group. We also adapt our results to R-1,R-n-1 x(w) M10-n backgrounds which underpin flux compactifications to R-1,R-n-1 and show that these preserve 2([n/2])k for 2 < n <= 4, 2([n+1-2])k for 4 < n <= 8, and 2([n/2])k for n = 9,10 supersymmetries.