International Congress of Mathematicians

478 Hélène Esnaultno longer the case that j*E necessarily contains a rank r bundle such that in a localdti ßiformal frame of this bundle, the local equation has the shape Ai = a,—- H ^-j-,ti t iwith a, G GL(r,K[[ti]]),ßi G ifi K ® M(r,K[[ti]]). We call an integrable connection(F, V) with this existence property an admissible connection.Even if (E,V) is admissible, its determinant connection det(F, V) might havemuch lower order poles (for example trivial). This indicates that one can not extenddirectly in this form the formula 4.1. However, assuming (E,V) to be admissibleand choosing some v £ OJ X /K ® ^(X) which generates i^C^,i m iPi)a t Pi as f° rformula 4.3, the right hand side of 4.3 makes sense, if one replaces dloga, byd log det (a»). Using global methods inspired by the Higgs correspondence betweenHiggs fields and connections on complex smooth projective varieties ([21]), one isable to prove the "same" formula as 4.3 in the higher rank case on P 1 .Theorem 5.1 ([8], Theorem 1.3). If (E,V) is admissible and has at least oneirregular point, and if v £ OJ X /K ® ^(X) generates uCJ^^iPi) at the points pi,thendetlj2(^y(H i (X,(n x/K ®L,Vx/K),GM i (Vj))= (-1)V| K („) + ^2 ( SU P C 1 » -y)dlogdet(ai(pij) + Tr Res^dt^1 A AA .ìThe connection Res Tr^dg,^1 A A t £ Q 1 K/dlogK x is well defined, as well asTH, 'the 2-torsion connection sup (l, —-)dlogdet(a,(p,)).However, one needs a different method in order to understand the contributionof singularities in which (F, V) is not admissible.We describe now the origin of the method contained [1]. It is based on theidea that Tate's method ([22]) applies for connections.Locally formally over the Laurent series field K((tj), E becomes a r-dimensionalvector space over K((tj). The relative connection VK((T))/K '• E —¥ i^K((t))/K ® Eis a Fredholm operator. This means that FF(VX/K),ì = 0,1 are finite dimensionalFJ-vector spaces, and that VX/K carries compact lattices to compact lattices.Let F = ®\K((ij) be the choice of a local frame. A compact lattice is aFJ-subspace of F which is commensurable to ®iFJ[[r]]. Given 0 fi^ v ^G ^K((t))lK,one composes VK((t))/K,v '•= l/^1 ° ^K((t))/K '• E —t E to obtain a Fredholmendomorphism. To a Fredholm endomorphism A : E —t E, one associates a1-dimensional FJ-vector space X(A) = det(F°(A)) ® det(F 1 (A)) _1 together withthe degree x(A) = dimF°(A) — dimF 1 (A). We call this a super-line. It doesnot refer to the topology defined by compact lattices. Then one measures how Amoves a compact lattice F c E. First for 2 lattices F and F', one takes a smallercompact lattice N c F n F' and defines det(F : F') := det(L/N) • det(F'/A r ) _1 ,where • is the tensor product of super-lines and det(L/N) has degree dim(L/N).This does not depend on the choice of N. Then one defines asymptotic superlines.The compact one is X C (A) = det(A(F) : F) -det(FnKer(A)) and the discrete one isXd(A) = det(F : A^1^)) • det(V/(L + A(\'j). They do not depend on the choice of