Deligne I Section 4
Last updates: 11 November 2011
The Lefschetz Theory: local theory
On the local Lefschetz results are the following.
the unit disc,
and a morphism of analytic spaces.
- is nonsingular, and of pure dimension ;
- is proper;
- is smooth outside of a point of the special fiber
- in , presents a nondegenerate quadratic point.
Let in and
"the" general fiber. To the above data associate:
- the specialiation morphisms
: is a deformation retract of
, and is the composite arrow
- the monodromy transformations
, which describe the effect of the singular cycles of
as " turns around ". This is also the action on
, the fibre in of the local system
of a positive generator of .
The Lefschetz theory writes α) and β) in terms of the vanishing cycle
. This cycle is well defined
up to sign as follows. For ,
For , one has an exact
For the monodromy is the identity.
For , one has
The values of the , of , and of
are the following:
The monodromy transformation respects the intersection form
. For odd, this is a symplectic
transvection. For even, this is an orthogonal symmetry.
(4.2) Here is the analogue of (4.1) in abstract algebraic geometry.
The disc is replaced by the spectrum of a discrete henselien valuation
ring with algebraically closed residue field. Let
be this spectrum, its generic point (spectrum of the field
of fractions of ), its closed point
(spectrum of the residue field). The role of is played by a
generic geometric point
(spectrum of the algebraic closure of the field of fractions of ).
Let be a proper
morphism, with purely regular of dimension
. We assume that is smooth, except
at an ordinary quadratic point in the special fiber
. Let be a pime number
different from the characteristic of the residue field of .
Denoting the generic geometric fiber by , we construct again a
The role of
is played by the action of the intertia group
, acting on
by transport of structure (cf. (1.15)):
The maps in (4.2.1) and (4.2.2) completly describe the sheaves
for even, and
(4.2.1) and (4.2.2) can again be described in terms of a vanishing cycle
This cycle is well defined up to sign as follows.
we have an exact sequence
The action (4.2.2) of (the local monodromy) is trivial if
. For ,
it is described as follows.
A) odd. -- We use the canonical homomorphism
and the action of is
B) even. -- We will not use this case. We will note only that,
if , there exists a unique character of order two
for which we have
(4.4) These results carry the following information on the
- If :
- For , the sheaf
- Let be the inclusion of in .
- If : (This is an exceptional case.
Since for even, it cannot occur except
when is odd.)
- For ,
- Let be the sheaf
on , extended by zero
on . We have an exact sequence
is a constant sheaf.
Notes and References
This is an attempt to translate Section 4 of [DeI].
La Conjecture de Weil I,
Publ. Math. IHÉS (1974) 273-307.