By Daniel Barlet (auth.), Vincenzo Ancona, Alessandro Silva (eds.)

The papers during this wide-ranging assortment record at the result of investigations from a few associated disciplines, together with complicated algebraic geometry, advanced analytic geometry of manifolds and areas, and complicated differential geometry.

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Chapter 1 28 Now we may assume that F$ Gis not normal is E; we get b(F$ G) 5F (F$ G) n bE 5F F$ G (because F normal c1ude that = F rt bE). Since dimc(F$ G/b(F$ G)) = 2, we con- dime F$ G/(F$ G) n bE = dimc(F$ G) n bE/b(F(fy G) = 1. The decreasing filtration bNE is separated (nNbNE = (0)), so there exists No 2:: I such that and (F(fy G) n bNo + IE c b(F$ G). Let bNO ( E F$ G such that bNO ( rf: b(F$ G) (so (rf: bE). We may write bNO ( = A(b)x + B(b)y, with A, BE IC[[b]]. Then A(O) = B(O) = 0 implies bNO ( E b(F(fy G); moreover, A(O) =F 0 and B(O) = 0 give XE bE, contradicting the normality of F.

The subcategory of objects over X, where the morphisms are the morphisms over the identity of X, and similarly for Y; see Ref. 20, Exp VI, for details. Instead of J*(G), GE (f)(Y), we often write G Q9(1)y (9x = G Q9 (9x or Gx . , formal real) spaces and SES is a point, then any object G of (f)(X) induces an object of (f)(X(s», denoted G(s). 1. 1. (f) = 9t o , p = id. 2. , G:o~(X) is the category of aB (9xmodules being 10caBy of finite presentation. Here J * : G:o~( Y) -+ G:o~(X) is the usual base-change functor.

A=du+df/\ vlu, v EflP-1[j-IJ} Then yt'P is a sheaf of C[[S]][S-I] finite-dimensional vector spaces on Y (endowed with a regular C[[s]][s-I]-connection), and HP is a sheaf of C[[s]]modules on Y that is a lattice in yt'P (in each fiber), thanks to Malgrange's positivity theorem (see Ref. 1 for the isolated singularity case). Thus, HP is a sheaf of finitely generated C[[s]]-modules on Y. Define 4>: HP -+ EP by 4> (a) = [da]. Then 4> is C-linear and surjective. Actually 4> is bijective because [da] = 0 in EP implies da = df /\ dß with ß E flP-I[f-I], so Va = dß = 0 in yt'P (where V is the Gauss-Manin connection).