18 research outputs found
Schubert calculus and singularity theory
Schubert calculus has been in the intersection of several fast developing
areas of mathematics for a long time. Originally invented as the description of
the cohomology of homogeneous spaces it has to be redesigned when applied to
other generalized cohomology theories such as the equivariant, the quantum
cohomology, K-theory, and cobordism. All this cohomology theories are different
deformations of the ordinary cohomology. In this note we show that there is in
some sense the universal deformation of Schubert calculus which produces the
above mentioned by specialization of the appropriate parameters. We build on
the work of Lerche Vafa and Warner. The main conjecture there was that the
classical cohomology of a hermitian symmetric homogeneous manifold is a Jacobi
ring of an appropriate potential. We extend this conjecture and provide a
simple proof. Namely we show that the cohomology of the hermitian symmetric
space is a Jacobi ring of a certain potential and the equivariant and the
quantum cohomology and K-theory is a Jacobi ring of a particular deformation of
this potential. This suggests to study the most general deformations of the
Frobenius algebra of cohomology of these manifolds by considering the versal
deformation of the appropriate potential. The structure of the Jacobi ring of
such potential is a subject of well developed singularity theory. This gives a
potentially new way to look at the classical, the equivariant, the quantum and
other flavors of Schubert calculus
A description based on Schubert classes of cohomology of flag manifolds
We describe the integral cohomology rings of the flag manifolds of types B_n,
D_n, G_2 and F_4 in terms of their Schubert classes. The main tool is the
divided difference operators of Bernstein-Gelfand-Gelfand and Demazure. As an
application, we compute the Chow rings of the corresponding complex algebraic
groups, recovering thereby the results of R. Marlin.Comment: 25 pages, AMS-LaTeX; typos corrected, an error concerning the
Schubert classes corrected, Remarks 4.4 and 4.8 adde
The even Clifford structure of the fourth Severi variety
The Hermitian symmetric space appears in the classification
of complete simply connected Riemannian manifolds carrying a parallel even
Clifford structure. This means the existence of a real oriented Euclidean
vector bundle over it together with an algebra bundle morphism
mapping
into skew-symmetric endomorphisms, and the existence of a metric connection on
compatible with . We give an explicit description of such a vector
bundle as a sub-bundle of . From this we construct a
canonical differential 8-form on , associated with its holonomy
, that represents
a generator of its cohomology ring. We relate it with a Schubert cycle
structure by looking at as the smooth projective variety
known as the fourth Severi variety
Schubert calculus and Intersection theory of Flag manifolds
Hilbert's 15th problem called for a rigorous foundation of Schubert's
calculus, in which a long standing and challenging part is Schubert's problem
of characteristics. In the course of securing the foundation of algebraic
geometry, Van der Waerden and Andr\'{e} Weil attributed the problem to the
determination of the intersection theory of flag manifolds.
This article surveys the background, content, and resolution of the problem
of characteristics. Our main results are a unified formula for the
characteristics, and a system description for the intersection rings of flag
manifolds. We illustrate the effectiveness of the formula and the algorithm via
explicit examples.Comment: 24 pages, 1 figur