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Acta Mathematica Sinica,Chinese Series
2018, Vol.34 Num.3
Online: 20180315
Articles
Articles
297
Preface
2018 Vol. 34 (3): 297298 [
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299
Jacopo GANDINI
Embeddings of Spherical Homogeneous Spaces
We review in these notes the theory of equivariant embeddings of spherical homogeneous spaces. Given a spherical homogeneous space
G/H
, the normal equivariant embeddings of
G/H
are classified by combinatorial objects called
colored fans
, which generalize the fans appearing in the classification of toric varieties and which encode several geometric properties of the corresponding variety.
2018 Vol. 34 (3): 299340 [
Abstract
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4
) [
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1KB] [
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501KB] (
41
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341
Bernhard KRÖTZ, Henrik SCHLICHTKRULL
Harmonic Analysis for Real Spherical Spaces
We give an introduction to basic harmonic analysis and representation theory for homogeneous spaces
Z
=
G/H
attached to a real reductive Lie group
G
. A special emphasis is made to the case where
Z
is real spherical.
2018 Vol. 34 (3): 341370 [
Abstract
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4
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1KB] [
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371KB] (
40
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371
Nicolas PERRIN
Sanya Lectures: Geometry of Spherical Varieties
These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of
B
orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider
B
stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit
B
stable canonical divisor on any spherical variety.
2018 Vol. 34 (3): 371416 [
Abstract
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4
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519KB] (
26
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417
Guido PEZZINI
Lectures on Wonderful Varieties
These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples.
2018 Vol. 34 (3): 417438 [
Abstract
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3
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312KB] (
32
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439
Stéphanie CUPITFOUTOU, Dmitry A. TIMASHEV
Orbits of Real Semisimple Lie Groups on Real Loci of Complex Symmetric Spaces
Let
G
be a complex semisimple algebraic group and
X
be a complex symmetric homogeneous
G
variety. Assume that both
G
,
X
as well as the
G
action on
X
are defined over real numbers. Then
G
(R) acts on
X
(R) with finitely many orbits. We describe these orbits in combinatorial terms using Galois cohomology, thus providing a patch to a result of Borel and Ji.
2018 Vol. 34 (3): 439453 [
Abstract
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5
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246KB] (
28
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454
Kiumars KAVEH, Kiumars KAVEH
Spherical Tropical Geometry: a Survey of Recent Developments
This is a survey of some recent results on spherical tropical geometry.
2018 Vol. 34 (3): 454465 [
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6
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233KB] (
28
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466
ShinYoung KIM, KyeongDong PARK
Standard Embeddings of Smooth Schubert Varieties in Rational Homogeneous Manifolds of Picard Number 1
Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20dimensional
F
_{4}
homogeneous manifold associated to a short simple root.
2018 Vol. 34 (3): 466487 [
Abstract
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13
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339KB] (
32
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488
Bernhard KRÖTZ, Eitan SAYAG, Henrik SCHLICHTKRULL
Geometric Counting on Wavefront Real Spherical Spaces
We provide
L
^{p}
versus
L
^{∞}
bounds for eigenfunctions on a real spherical space
Z
of wavefront type. It is shown that these bounds imply a nontrivial error term estimate for lattice counting on
Z
. The paper also serves as an introduction to geometric counting on spaces of the mentioned type.
2018 Vol. 34 (3): 488531 [
Abstract
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6
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511KB] (
23
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532
Duo LI
Twodimensional Irreducible Algebraic Semigroups
We study twodimensional irreducible projective smooth algebraic semigroups. Minimal surface semigroups with Kodaira dimension at most one are partially classified. We also calculate the local dimension of the moduli scheme parameterizing all algebraic semigroup laws on a fixed minimal surface.
2018 Vol. 34 (3): 532541 [
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3
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220KB] (
27
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542
Boris PASQUIER
The Log Minimal Model Program for Horospherical Varieties Via Moment Polytopes
In a previous work, we described the Minimal Model Program in the family of QGorenstein projective horospherical varieties, by studying certain continuous changes of moment polytopes of polarized horospherical varieties. Here, we summarize the results of the previous work and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs (
X
, Δ) when
X
is a projective horospherical variety.
2018 Vol. 34 (3): 542562 [
Abstract
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5
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301KB] (
25
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563
Kay PAULUS, Guido PEZZINI, Bart VAN STEIRTEGHEM
On Some Families of Smooth Affine Spherical Varieties of Full Rank
Let
G
be a complex connected reductive group. Losev has shown that a smooth affine spherical
G
variety
X
is uniquely determined by its weight monoid, which is the set of irreducible representations of
G
that occur in the coordinate ring of
X
. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify:(a) all such varieties for
G
=SL(2)×C
^{×}
and (b) all such varieties for
G
simple which have a
G
saturated weight monoid of full rank. We also use the characterization and Knop's classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward's result that every reflective Delzant polytope is the moment polytope of such a manifold.
2018 Vol. 34 (3): 563596 [
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3
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