Automorphisms of Enriques surfaces which act trivially on the cohomology groups

Abstract

Let S be a smooth surface over the complex number field r An a u t o m o r p h i s m q~ of S is cohomologically trivial (resp. numerically trivial) if the induced a u t o m o r p h i s m q~* of the cohomology ring H * (S, 7/) (resp. H * (S, I1~)) is trivial. We denote by A o (S) (resp. A (S)) the quot ient o f the group o fcohomolog ica l ly (resp. numerically) trivial au tomorph i sms of S by its connected component . I t is known tha t A (S) is a finite g roup i f S is Kiihler, [4, 3]. A (S) is even trivial, for example, in the case of ra t ional surfaces, abelian surfaces and K3 surfaces, [9, 5]. In this article, we shall s tudy A (S) for an Enriques surface S. In this case, it is no more true that Ao (S) is always trivial. So far two examples are known.