TY - JOUR

T1 - Schubert calculus and singularity theory

AU - Gorbunov, Vassily

AU - Petrov, Victor

N1 - Acknowledgements
The authors visited the Max Planck Institute for Mathematics in Bonn while working on this paper. The inspiring discussions with V. Golyshev, Yu. I. Manin and especially with F. Hirzebruch are gratefully acknowledged. The second author is supported by RFBR 09-01-00878, 09-01-91333, 10-01-90016, 10-01-92651.

PY - 2012/2

Y1 - 2012/2

N2 - 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 these authors made 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 the 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.

AB - 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 these authors made 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 the 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.

KW - Schubert calculus

KW - homogeneous spaces

KW - Jacobi rings

KW - Landau–Ginzburg model

KW - mirror symmetry

KW - Frobenius manifolds

U2 - 10.1016/j.geomphys.2011.10.016

DO - 10.1016/j.geomphys.2011.10.016

M3 - Article

VL - 62

SP - 352

EP - 360

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 2

ER -