Research of Chad Schoen in Algebraic Geometry
The Geometry of Algebraic Varieties
Algebraic varieties are the solution sets to systems of polynomial equations.
Any closed complex submanifold of complex projective space may be viewed as
a projective algebraic variety over the field of complex numbers. These
are among the most studied geometric objects in mathematics, because a
wide range of mathematical techniques apply.
In items 2 and 17 on the publication list I study the geometry of interesting
Abelian varieties are the group objects in the category of projective
varieties. They are among the most important algebraic varieties. Items
5, 7, and 10 on the publicaton list concern abelian vareities.
The study of curves is the most classical part of algebraic geometry. Its
roots go back at least as far as Gauss, Abel, Jacobi, and Riemann. Today
arithmetic questions are a focus for a great deal of work. The basic
problem is to find all rational solutions (x,y) to an equation f(x,y)=0,
where f is a polynomial with rational coefficients. Item 9 on my
publications list is written in this spirit. Item 14 relates the study of
curves over the real numbers to the study of non-orientable minimal surfaces
in three space. Sharad Chaudhary's thesis, written under my direction, was
about curves over the real numbers.
Given an algebraic variety, X, the algebraic cycles on X are the formal linear
combinations of subvarieties of X. By modding out by an equivalence
relations one gets the Chow groups of X. These function as a sort of
homology theory of the variety X. Generally the Chow groups are more subtle
and contain more information than ordinary homology. Much of the work in
the field of algebraic cycles is organized around three major conjectures:
the Hodge conjecture, the Tate conjecture and the generalized
Birch-Swinnerton Dyer conjecture. Each of these is described briefly below.
The Hodge Conjecture
Let M be a closed complex submanifold of complex projective space. One
would like to be able to classify all closed complex submanifolds of
M. It is known that this problem is extremely involved and perhaps
intractable. A simpler problem would be to describe the cohomology
classes of all closed complex submanifolds of M. The Hodge Conjecture
suggests a possible answer to this question. It has been a leading open
question in complex analytic geometry since Hodge posed it about 1950.
Although it guides much research in the field today, one still does not
have strong evidence either that it is true or that it is false.
Items 1,5,7,22 on my publication list relate to this conjecture.
The Tate Conjecture
The Tate conjecture is similar to the Hodge conjecture described above,
the difference being that the Tate conjecture is concerned with
cohomology classes of algebraic subvarieties of a smooth projective
variety M defined over an arbitrary finitely generated field. The
Tate conjecture is especially interesting when the base field is
a finite field. Item 21 on my publication list relates to this conjecture.
The Generalized Birch-Swinnerton Dyer Conjecture
An elliptic curve is a variety defined by the equation, y^2=x^3+ax+b=0.
If a and b are rational numbers, it is generally a very difficult
problem to describe all pairs of rational numbers (x,y) which solve
the equation. A remarkable and important fact is that the solution set
(with a point at infinity added) has a natural stucture of finitely
generated Abelian group. Thus all solutions to this equation are
easily generated by a finite collection. In practice determining
the torsion subgroup of the solution set is not difficult. Determining
the rank is a deep open problem which is the object of the conjecture
of Birch and Swinnerton Dyer. Much progress has been made here in the
past 15 years. My work pertains to generalizations of this conjecture
to Chow groups on varieties of higher dimension. See items 3, 4, 12, 16, 18
on my publication list.
Abel Jacobi maps
Abel Jacobi maps are a generalization of the map from a Riemann surface to
its Jacobian. They provide a tool for studying algebraic cylces whose
homology class is zero. The Abel Jacobi maps for cycles of codimension
greater than one are presently quite mysterious. Classical Abel Jacobi
maps are used extensively in items 3, 4, 11, and 12 in the publications list.
An l-adic analog of the classical Abel Jacobi map plays an important role
in items 11, 15, 18, 19, and 20.
Algebraic Geometry and theoretical physics have found common interest in the
study of "Calabi-Yau Manifolds". These may be viewed as three dimensional
complex projective varieties with trivial canonical bundle. One of the
first problems which arose when this area developed was to find out how
large and diverse the class of all Calabi-Yau manifolds is. Items 2 and
6 on my publication list make a contribution to this problem.