Potential theory and Lefschetz theorems for arithmetic surfaces

We prove an arithmetic analogue of the so-called Lefschetz theorem which as serts that, if D is an effective divisor in a projective normal surface X w hich is nef and big, then the inclusion map from the support \D\ of D in X induces a surjection from the (algebraic) fondamental group of \D\ onto the one of X. In the arithmetic setting, X is a normal arithmetic surface, qua si-projective over Spec Z, D is an effective divisor in X, proper over Spec Z, and furthermore one is given an open neighbourhood Omega of \D\(C) on t he Riemann surface X(C) such that the inclusion map \D\(C) --> Omega is a h omotopy equivalence, Then we may consider the equilibrium potential g(D,Ome ga) of the divisor D(C) in Omega and the Arakelov divisor (D,(gD,Omega)), a nd we show that if the latter is nef and big in the sense of Arakelov geome try, then the fundamental group of \D\ still surjects onto the one of X. Th is results extends an earlier theorem of Ihara, and is proved by using a ge neralization of Arakelov intersection theory on arithmetic surfaces, based on the use of Green functions which, up to logarithmic singularities, belon g to the Sobolev space L-1(2). (C) Elsevier, Paris.