0 Chern characters via connections up to homotopy ∗

1 Introduction: The aim of this note is to point out that Chern characters can be computed using curvatures of " connections up to homotopy " , and to present an application to the vanishing theorem for Lie algebroids. Classically, Chern characters are computed with the help of a connection and its curvature. However, one often has to relax the notion of connection so that one gains more freedom in representing these characteristic classes by differential forms. A well known example is Quillen's notion of super-connection [8]. Here we remark that one can weaken the notion of (super-)connections even further, to what we call " up to homotopy ". Our interest on this type of connections comes from the theory of characteristic classes of Lie algebroids [2, 5] (hence, in particular of Poisson manifolds [6]). From our point of view, the intrinsic characteristic classes are secondary classes which arise from a vanishing result: the Chern classes of the adjoint representation vanish (compare to Bott's approach to characteristic classes for foliations). We have sketched a proof of this in [2] for a particular class of Lie algebroids (the so called regular ones). The problem is that the adjoint representation is a representation up to homotopy only [4]. For the general setting, we have to show that Chern classes can be computed using connections up to homotopy. Since we believe that this result might be of larger interest, we have chosen to present it at the level vector bundles over manifolds. In [3] we will describe the secondary characteristic classes which arise in the flat case.