Compact Kähler manifolds with elliptic homotopy type

Simply connected compact Kähler manifolds of dimension up to three with elliptic homotopy type are characterized in terms of their Hodge diamonds. For surfaces there are only two possibilities, namely h1,1⩽2 with hp,q=0 for p≠q. For threefolds, there are three possibilities, namely h1,1⩽3 with hp,q=0 for p≠q. This characterization in terms of the Hodge diamonds is applied to explicitly classify the simply connected Kähler surfaces and Fano threefolds with elliptic homotopy type.