 
Summary: COMPACT KšAHLER MANIFOLDS WITH ELLIPTIC HOMOTOPY
TYPE
JAUME AMORŽOS AND INDRANIL BISWAS
Abstract. Simply connected compact Kšahler manifolds of dimension up to three with
elliptic homotopy type are characterized in terms of their Hodge diamonds. For sur
faces 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 Kšahler surfaces and Fano threefolds with elliptic homotopy type.
1. Introduction
A finite, simply connected CWcomplex has elliptic homotopy type if the total rank of
its homotopy groups i2 dim i(X) Z Q is finite.
A simply connected compact homogeneous manifold has elliptic homotopy type. From
the homotopytheoretic point of view simply connected manifolds with elliptic homotopy
type constitute an extension of the class of 1connected homogeneous manifolds (this is
discussed in Section 2).
