48 research outputs found
The adjoint variety of SL_{m+1}(C) is rigid to order three
I show that the adjoint variety of the complex special linear group is rigid
to order three.Comment: 15 pages, v
Singular loci of cominuscule Schubert varieties
Let X = G/P be a cominuscule rational homogeneous variety. (Equivalently, X
admits the structure of a compact Hermitian symmetric space.) I give a uniform
description (that is, independent of type) of the irreducible components of the
singular locus of a Schubert variety Y in X in terms of representation
theoretic data. The result is based on a recent characterization of the
Schubert varieties by an non-negative integer A and a marked Dynkin diagram.
Corollaries include: (1) the variety is smooth if and only if A=0; (2) if G of
Type ADE, then the singular locus occurs in codimension at least three.Comment: Two tables for the exceptional E6 and E7. Version 2: expository edit
Geodesics in Randers spaces of constant curvature
Geodesics in Randers spaces of constant curvature are classified.Comment: 7 small figures added to latest version, which has been accepted by
TAMS. 16 pages, 10 small figure
Schur flexibility of cominuscule Schubert varieties
Let X=G/P be cominuscule rational homogeneous variety. (Equivalently, X
admits the structure of a compact Hermitian symmetric space.) We say a Schubert
class [S] is Schur rigid if the only irreducible subvarieties Y of X with
homology class [Y] = r [S], for an integer r, are Schubert varieties. Robles
and The identified a sufficient condition for a Schubert class to be Schur
rigid. In this paper we show that the condition is also necessary.Comment: Two figures for the exceptional E6 and E7. Version 3: editorial
revision
Characterization of Calabi--Yau variations of Hodge structure over tube domains by characteristic forms
Sheng and Zuo's characteristic forms are invariants of a variation of Hodge
structure. We show that they characterize Gross's canonical variations of Hodge
structure of Calabi-Yau type over (Hermitian symmetric) tube domains
Classification of smooth horizontal Schubert varieties
We show that the smooth horizontal Schubert subvarieties of a rational
homogeneous variety are homogeneously embedded cominuscule , and
are classified by subdiagrams of a Dynkin diagram. This generalizes the
classification of smooth Schubert varieties in cominuscule .Comment: replaces sections 3 and 4 of arXiv:1407.4507v1, which were eliminated
in its revision; comments welcom
Variations of Hodge structure and orbits in flag varieties
Period domains, the classifying spaces for (pure, polarized) Hodge
structures, and more generally Mumford-Tate domains, arise as open
--orbits in flag varieties . We investigate
Hodge--theoretic aspects of the geometry and representation theory associated
with these flag varieties. In particular, we relate the Griffiths--Yukawa
coupling to the variety of lines on (under a minimal homogeneous
embedding), construct a large class of polarized --orbits in
, and compute the associated Hodge--theoretic boundary components. An
emphasis is placed throughout on adjoint flag varieties and the corresponding
families of Hodge structures of levels two and four.Comment: v.2, substantially revised and shortened; comments welcom
Calibrated associative and Cayley embeddings
Using the Cartan-Kahler theory, and results on real algebraic structures, we
prove two embedding theorems. First, the interior of a smooth, compact
3-manifold may be isometrically embedded into a G_2-manifold as an associative
submanifold. Second, the interior of a smooth, compact 4-manifold K, whose
double has a trivial bundle of self-dual 2-forms, may be isometrically embedded
into a Spin(7)-manifold as a Cayley submanifold. Along the way, we also show
that Bochner's Theorem on real analytic approximation of smooth differential
forms, can be obtained using real algebraic tools developed by Akbulut and
King
Fubini's Theorem in codimension two
We classify codimension two analytic submanifolds X of projective space
having the property that any line through a general point p having contact to
order two with X at p automatically has contact to order three. We give
applications to the study of the Debarre--de Jong conjecture, and of
n-dimensional varieties whose Fano variety of lines has dimension 2n-4.Comment: v.1, 11 page
Quotients of non-classical flag domains are not algebraic
A flag domain D = G/V for G a simple real non-compact group G with compact
Cartan subgroup is non-classical if it does not fiber holomorphically or
anti-holomorphically over a Hermitian symmetric space. We prove that any two
points in a non-classical domain D can be joined by a finite chain of compact
subvarieties of D. Then we prove that for F an infinite, finitely generated
discrete subgroup of G, the analytic space F\D does not have an algebraic
structure