Deligne I Section 4

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 11 November 2011

The Lefschetz Theory: local theory

(4.1) On the local Lefschetz results are the following. Let D={z | |z|<1} the unit disc, D*=D-{0} and f:XD a morphism of analytic spaces. Suppose that

  1. X is nonsingular, and of pure dimension n+1;
  2. f is proper;
  3. f is smooth outside of a point x of the special fiber X0 =f-1(0);
  4. in x, f presents a nondegenerate quadratic point.

Let t0 in D and Xt =f-1(t); "the" general fiber. To the above data associate:

  1. the specialiation morphisms sp: Hi(X0,) Hi(Xt,) : X0 is a deformation retract of X, and sp is the composite arrow Hi(X0,) Hi(X,) Hi(Xt,) ;
  2. the monodromy transformations T: Hi(Xt,) Hi(Xt,) , which describe the effect of the singular cycles of Xt as "t turns around 0". This is also the action on Hi(Xt, ), the fibre in t of the local system Rif* _|D*, of a positive generator of π1 (D*,t).

The Lefschetz theory writes α) and β) in terms of the vanishing cycle δHn( Xt,). This cycle is well defined up to sign as follows. For in,n+1, one has Hi(X0,) Hi(Xt,) (in,n+1). For i=n,n+1, one has an exact sequence 0 Hn(X0,) Hn(Xt,) x (x,δ) Hn+1 (X0,) Hn+1 (Xt,) 0 . For in the monodromy T is the identity. For i=n, one has Tx=x±(x,δ) δ. The values of the ±, of Tδ, and of (δ,δ) are the following: nmod4 0 1 2 3 Tx=x± (x,δ)δ SPACE - - + + (δ,δ) 2 0 -2 0 Tδ -δ δ -δ δ

The monodromy transformation T respects the intersection form Tr(xy) on Hn(Xt, ). For n odd, this is a symplectic transvection. For n even, this is an orthogonal symmetry.

(4.2) Here is the analogue of (4.1) in abstract algebraic geometry. The disc D is replaced by the spectrum of a discrete henselien valuation ring A with algebraically closed residue field. Let S be this spectrum, η its generic point (spectrum of the field of fractions of A), s its closed point (spectrum of the residue field). The role of t is played by a generic geometric point η (spectrum of the algebraic closure of the field of fractions of A).

Let F:XS be a proper morphism, with X purely regular of dimension n+1. We assume that f is smooth, except at an ordinary quadratic point x in the special fiber Xs. Let be a pime number different from the characteristic p of the residue field of S. Denoting the generic geometric fiber by X η, we construct again a specialisation morphism

sp: Hi(Xs, ) Hi(X, ) Hi(X η, ) . (4.2.1)
The role of T is played by the action of the intertia group I= Gal(η/η) , acting on Hi(X η, ) by transport of structure (cf. (1.15)):
I= Gal(η/η) GL( Hi(X η, ) ). (4.2.2)

The maps in (4.2.1) and (4.2.2) completly describe the sheaves Rif* on S.

(4.3)We put n=2m for n even, and n=2m+1 for n odd. (4.2.1) and (4.2.2) can again be described in terms of a vanishing cycle

δ Hn(X η, ) (m). (4.3.1)
This cycle is well defined up to sign as follows.

For in,n+1 we have

Hi(Xs, ) Hi(X η, ) (in, n+1). (4.3.1)
For i=n,n+1, we have an exact sequence
0 Hn(Xs, ) Hn( Xη, ) x Tr(xδ) (m-n) Hn+1 (Xs, ) Hn+1 ( Xη, ) 0 . (4.3.3)

The action (4.2.2) of I (the local monodromy) is trivial if in. For i=n, it is described as follows.

A) n odd. -- We use the canonical homomorphism t:I (1), and the action of σI is xx±t (σ)(x,δ) δ.

B) n even. -- We will not use this case. We will note only that, if p2, there exists a unique character of order two ε:I{±1} , for which we have σx=x, ifε(σ) =1, σx=x ±(x,δ)δ, ifε(σ) =-1.

(4.4) These results carry the following information on the Rif* .

  1. If δ0:
    1. For in, the sheaf Rif* is constant.
    2. Let j be the inclusion of η in S. We have Rnf* = j*j* Rnf* .
  2. If δ=0: (This is an exceptional case. Since (δ,δ)= ±2 for n even, it cannot occur except when n is odd.)
    1. For in,n+1, the sheaf Rif* is constant.
    2. Let (m-n) s be the sheaf (m-n) on {s}, extended by zero on S. We have an exact sequence 0 (m-n) s Rn+1f* j*j* Rn+1f* 0 , where j*j* Rn+1f* is a constant sheaf.

Notes and References

This is an attempt to translate Section 4 of [DeI].


[DeI] P. Deligne, La Conjecture de Weil I, Publ. Math. IHÉS (1974) 273-307. MR0340258.

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