From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: Modular forms
Date: 3 Jul 2000 17:45:50 -0400
Newsgroups: sci.math,sci.fractals,alt.fractals
Summary: [missing]

ROGER BAGULA <tftn@earthlink.net> writes:

>Dear Timothy Murphy,
>Do you have a definition of a " Lefschetz Pencil"?
>I've run into it in connection with modular forms, but I can't find it
>in my topology books or on the net with two search engines.

To a topologist (I think the algebraic geometers have a slightly
different definition), a Lefschetz pencil is a smooth mapping 
from  an oriented 4-dimensional manifold to an oriented
2-dimensional manifold with only finitely many critical points, 
near each of which coordinates can be chosen (in the domain and 
the range) so that the mapping locally looks like one of the two 
mappings 
     (x,y,u,v) |-> (x^2-y^2,2xy,u^2-v^2,2uv)
or 
     (x,y,u,v) |-> (x^2-y^2,2xy,u^2-v^2,-2uv)
(that is, in complex coordinates z=x+iy, w=u+iv, like either
z^2+w^2 or z^2+conj(w)^2).  Often other hypotheses are
also imposed: the manifolds might have to be compact,
the values at distinct critical points might have to 
be distinct, and (particularly in algebraic geometry,
and so probably also in whatever application to modular
forms you've encountered) the second local mapping might
not be allowed.  Even stronger (again, suitable for
algebraic geometry), you might require that the mapping
be complex-algebraic (rational, holomorphic, whatever).

Lefschetz pencils are like "Morse functions" in the 
complex realm, if that analogy helps any.

Given a Lefschetz pencil f:M->S, say with M and S compact,
the set B contained in S of critical values of f (values of f
at the critical points in M) is finite, and the restriction of
f to the inverse image of S-B is a submersion and therefore
a fiber bundle (as explained recently by B. Moonen)
whose fiber is again a 2-manifold F.  With trivial exceptions,
the fundamental group of S-B is a free group, and so
topologically (at least) the inverse image of S-B is
described entirely by the "monodromy" of the fibration
which is a representation of this free group in the
so-called "mapping class group" of F.  Moreover,
the hypotheses on local forms of the singularities of
the pencil f mean that the standard generators of the
free group are each mapped to very particular elements 
of this mapping class group--in fact, so-called "Dehn
twists" (of two different signs, depending on the sign
in the local form of the corresponding singularity),
and this means that the complete manifold M can be
built up in a fairly straightforward manner from
the honest fiber bundle part and standard non-fiber-bundle
pieces over the points of B.

I think that the applications to modular forms come from
looking at the action of the monodromy on various 
invariants of F.  But I'm very likely wrong, and I'm
certainly out of date on this, and non-algebro-geometrically
savvy.

Lee Rudolph