Unitary grassmannians

We study projective homogeneous varieties under an action of a projective unitary group (of outer type). We are especially interested in the case of (unitary) grassmannians of totally isotropic subspaces of a hermitian form over a field, the main result saying that these grassmannians are 2-incompressible if the hermitian form is generic. Applications to orthogonal grassmannians are provided.Comment: 25 page

On the Chow ring of certain hypersurfaces in a Grassmannian

This small note is about Pl\"ucker hyperplane sections $X$ of the Grassmannian $\operatorname{Gr}(3,V_{10})$. Inspired by the analogy with cubic fourfolds, we prove that the only non-trivial Chow group of $X$ is generated by Grassmannians of type $\operatorname{Gr}(3,W_{6})$ contained in $X$. We also prove that a certain subring of the Chow ring of $X$ (containing all intersections of positive-codimensional subvarieties) injects into cohomology.Comment: 11 pages, to appear in Le Matematiche, comments welcom

Compactifications of subvarieties of tori

We study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non-archimedean amoebas. These compactifications have many nice properties, for example any k boundary divisors intersect in codimension k. We consider some examples including $M_{0,n}\subset\bar M_{0,n}$ (and more generally log canonical models of complements of hyperplane arrangements) and compact quotients of Grassmannians by a maximal torus.Comment: 14 pages, submitted versio

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