Rough outline of lectures
In the lectures we will discuss how a family of varieties gives rise to a
variation of Hodge structures (Lecture 1). In Lecture 2 we discuss
examples of variations of Hodge structure coming from certain explicit
families of curves. In Lecture 3 we discuss a family of surfaces and its
associated variation of Hodge structures. A common feature of the examples
of Lectures 2 and 3 will be the existence of a Shimura variety parametrizing
the geometric objects. In the final lecture we describe the variation of
Hodge structure on H^{3} of a family of threefolds and we formulate a
question which will hopefully be addressed by the students.
Prerequisites
Everybody should know what a Hodge structure is (this will not be
explained in the lectures). There are several kinds; for us ZHodge
structures and QHodge structures will be the important ones. Students should
know what a morphism of Hodge structures is, and how to take tensor
products of Hodge structures, duals etc. It is also important
to know what a polarization of a Hodge structure is.
Everybody should know what an algebraic variety is, and what a family of
varieties is. (This will not be explained in the lectures.)
Useful to know: The cohomology on a nonsingular complex projective
variety carries a natural polarizable Hodge structure. An abelian variety
corresponds to a polarizable ZHodge structure of weight 1.
References
This is a little tricky since I do not know a short
introduction to Hodge structures. You really don't have to know Hodge
theory, you just have to know what a Hodge structure is. You can look at
the original articles by Deligne, Griffiths for example. There is a book
by Fouad El Zein ``Introduction à la théorie de Hodge mixte'' where
you can find the definitions in Section 1.2. You can also look at
Wells, R. O., Jr. Differential analysis on complex manifolds. Second
edition. Graduate Texts in Mathematics, 65. SpringerVerlag, New
YorkBerlin, 1980. In general talking about Hodge structures with your
friends might be a good preparation for this course.
Project description
We will take a particular variation of Hodge structures corresponding to a
family of threefolds and we will try to determine whether the Hodge
structures in question ever are reducible. This will be related to
geometrical questions on the 3folds and we will try to solve those instead.
As an experiment, to guess the answers, we are going to do
some counting of points on the 3folds modulo p! (At least if there are
any students who know how to use computers to count points in finite
fields having p, p^{2},... elements.)
Additional, latebreaking details
Here is the counting of points that I would like to see (some of you,
hopefully at least one) do:
For small primes p not equal to 5, say p=2,3,7,11,... and any s∈F_{p}
∪∞ count how many points the hypersurface
Z_{s} : s(Y_{0}+Y_{1}+Y_{2}+Y_{3}+Y_{4})^{5} = Y_{0}Y_{1}Y_{2}Y_{3}Y_{4}
in P^{4} has over F_{p} and and over F_{p2}. (You can also try to count over
F_{p3} and F_{p4}; we'll be able to use this information to check your
results.)
For F_{p} points:
I suggest to just try to use the ``stupid'' method where you simply make
variable N_{s} for s=0,1,2,3,...,p1 and s=∞. Then you just loop
through all points (y_{0}:y_{1}:y_{2}:y_{3}:y_{4}) of P^{4}(F_{p}) and you increment
N_{s} by one if your point lies on Z_{s}. (Namely, if ∑ y_{i} ≠ 0
then you just have to increment N_{s} where s is the quotient (∏
y_{i})/(∑ y_{i})^{5}. If the sum is zero you have to be careful since there
may be more than one N_{s} that has to be incremented.)
For F_{p2}points:
Again you can use the stupid method, but now you have to write a little
program that allows you to compute in F_{p2}. It might be interesting to
know the number of points on Z_{s} also when s ∈ F_{p2}. (But in this
case I would like to know the number of points for these Z_{s} over F_{p4}
also.)
Please save the data as we are going to convert the numbers into
characteristic polynomials of Frobenii.
I myself know absolutely ZERO about computers and algorithms. So it is
quite possible that you're going to be able to find something smarter than
what I propose above.
