Standard Embeddings of Smooth Schubert Varieties in Rational Homogeneous Manifolds of Picard Number 1
|
摘要
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 20-dimensional F_4- homogeneous manifold associated to a short simple root.
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 20-dimensional F_4- homogeneous manifold associated to a short simple root.
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 20-dimensional F_4- homogeneous manifold associated to a short simple root.
引文
[1]Billey,S.,Postnikov,A.:Smoothness of Schubert varieties via patterns in root subsystems.Adv.Appl.Math.,34,447–466(2005)
[2]Brion,M.,Polo,P.:Generic singularities of certain Schubert varieties.Math.Z.,231(2),301–324(1999)
[3]Choe,I.,Hong,J.:Integral varieties of the canonical cone structure on G/P.Math.Ann.,329(4),629–652(2004)
[4]Hong,J.:Classification of smooth Schubert varieties in the symplectic Grassmannians.J.Korean Math.Soc.,52(5),1109–1122(2015)
[5]Hong,J.:Smooth horospherical varieties of Picard number one as linear sections of rational homogeneous varieties.J.Korean Math.Soc.,53(5),433–446(2016)
[6]Hong,J.,Kim,S.Y.:Characterization of smooth pro jective horospherical varieties of Picard number one,preprint(2017)
[7]Hong,J.,Kwon,M.:Rigidity of smooth Schubert varieties in a rational homogeneous manifold associated to a short root,preprint(2016)
[8]Hong,J.,Mok,N.:Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogeneous manifolds.J.Differential Geom.,86(3),539–567(2010)
[9]Hong,J.,Mok,N.:Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1.J.Algebraic Geom.,22(2),333–362(2013)
[10]Hong,J.,Park,K.D.:Characterization of standard embeddings between rational homogeneous manifolds of Picard number 1.Int.Math.Res.Not.IMRN,2011(10),2351–2373(2011)
[11]Humphreys,J.E.:Introduction to Lie algebras and representation theory,Graduate Texts in Mathematics9,Springer-Verlag,New York,1972
[12]Hwang,J.M.:Geometry of minimal rational curves on Fano manifolds.In:Demailly,J.P.,Goettsche,L.,Lazarsfeld,R.(eds)School on Vanishing Theorems and Effective Results in Algebraic Geometry,Trieste(Italy),25 Apr–12 May 2000,ICTP Lect.Notes Series 6,335–393(2001)
[13]Hwang,J.M.,Mok,N.:Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kahler deformation.Invent.Math.,131(2),393–418(1998)
[14]Hwang,J.M.,Mok,N.:Varieties of minimal rational tangents on uniruled pro jective manifolds.In:Schneider,M.,Siu,Y.T.(eds)Several Complex Variables(Berkeley,CA,1995–1996),MSRI publications 37,Cambridge University Press,351–389(1999)
[15]Hwang,J.M.,Mok,N.:Deformation rigidity of the rational homogeneous space associated to a long simple root.Ann.Sci.Ecole.Norm.Sup.,35(2),173–184(2002)
[16]Hwang,J.M.,Mok,N.:Deformation rigidity of the 20-dimensional F4-homogeneous space associated to a short root.In:Popov,V.L.(ed)Algebraic transformation groups and algebraic varieties,Encyclopaedia of Mathematical Sciences 132,Springer,Berlin,37–58(2004)
[17]Hwang,J.M.,Mok,N.:Prolongations of infinitesimal linear automorphisms of pro jective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kahler deformation.Invent.Math.,160(3),591–645(2005)
[18]Kim,S.Y.:Geometric structures modeled on smooth pro jective horospherical varieties of Picard number one.Transform.Groups,22(2),361–386(2017)
[19]Kollar,J.:Rational Curves on Algebraic Varieties,Ergebnisse der Mathematik und ihrer Grenzgebiete 32,Springer,Berlin,1996
[20]Lakshmibai,V.,Weyman,J.:Multiplicities of points on a Schubert variety in a minuscule G/P.Adv.Math.,84(2),179–208(1990)
[21]Landsberg,J.M.,Manivel,L.:On the pro jective geometry of rational homogeneous varieties.Comment.Math.Helv.,78(1),65–100(2003)
[22]Mihai,I.A.:Odd symplectic flag manifolds.Transform.Groups,12(3),573–599(2007)
[23]Mok,N.:Characterization of standard embeddings between complex Grassmanninas by means of varieties of minimal rational tangents.Sci.China Ser.A,51(4),660–684(2008)
[24]Mok,N.:Geometric structures on uniruled pro jective manifolds defined by their varieties of minimal rational tangents,In:Hijazi,O.(ed)Geometrie Differentielle,Physique Mathematique,Mathematiques et Societe(II),Volume in honor of Jean Pierre Bourguignon,Asterisque,322,151–205(2008)
[25]Mok,N.:Geometric structures and substructures on uniruled pro jective manifolds,In:Cascini,P.,Mc Kernan,J.,Pereira,J.V.(eds)Foliation Theory in Algebraic Geometry,Simons Symposia,Springer,Cham,103–148(2016)
[26]Pasquier,B.:On some smooth pro jective two-orbit varieties with Picard number 1.Math.Ann.,344(4),963–987(2009)
[27]Perrin,N.:On the geometry of spherical varieties.Transform.Groups,19(1),171–223(2014)
[28]Richmond,E.,Slofstra,W.:Billey–Postnikov decompositions and the fibre bundle structure of Schubert varieties.Math.Ann.,366(1–2),31–55(2016)
[29]Springer,T.A.:Linear Algebraic Groups,Progress in Mathematics 9,Birkhauser,Boston,1998
[2]Brion,M.,Polo,P.:Generic singularities of certain Schubert varieties.Math.Z.,231(2),301–324(1999)
[3]Choe,I.,Hong,J.:Integral varieties of the canonical cone structure on G/P.Math.Ann.,329(4),629–652(2004)
[4]Hong,J.:Classification of smooth Schubert varieties in the symplectic Grassmannians.J.Korean Math.Soc.,52(5),1109–1122(2015)
[5]Hong,J.:Smooth horospherical varieties of Picard number one as linear sections of rational homogeneous varieties.J.Korean Math.Soc.,53(5),433–446(2016)
[6]Hong,J.,Kim,S.Y.:Characterization of smooth pro jective horospherical varieties of Picard number one,preprint(2017)
[7]Hong,J.,Kwon,M.:Rigidity of smooth Schubert varieties in a rational homogeneous manifold associated to a short root,preprint(2016)
[8]Hong,J.,Mok,N.:Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogeneous manifolds.J.Differential Geom.,86(3),539–567(2010)
[9]Hong,J.,Mok,N.:Characterization of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1.J.Algebraic Geom.,22(2),333–362(2013)
[10]Hong,J.,Park,K.D.:Characterization of standard embeddings between rational homogeneous manifolds of Picard number 1.Int.Math.Res.Not.IMRN,2011(10),2351–2373(2011)
[11]Humphreys,J.E.:Introduction to Lie algebras and representation theory,Graduate Texts in Mathematics9,Springer-Verlag,New York,1972
[12]Hwang,J.M.:Geometry of minimal rational curves on Fano manifolds.In:Demailly,J.P.,Goettsche,L.,Lazarsfeld,R.(eds)School on Vanishing Theorems and Effective Results in Algebraic Geometry,Trieste(Italy),25 Apr–12 May 2000,ICTP Lect.Notes Series 6,335–393(2001)
[13]Hwang,J.M.,Mok,N.:Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kahler deformation.Invent.Math.,131(2),393–418(1998)
[14]Hwang,J.M.,Mok,N.:Varieties of minimal rational tangents on uniruled pro jective manifolds.In:Schneider,M.,Siu,Y.T.(eds)Several Complex Variables(Berkeley,CA,1995–1996),MSRI publications 37,Cambridge University Press,351–389(1999)
[15]Hwang,J.M.,Mok,N.:Deformation rigidity of the rational homogeneous space associated to a long simple root.Ann.Sci.Ecole.Norm.Sup.,35(2),173–184(2002)
[16]Hwang,J.M.,Mok,N.:Deformation rigidity of the 20-dimensional F4-homogeneous space associated to a short root.In:Popov,V.L.(ed)Algebraic transformation groups and algebraic varieties,Encyclopaedia of Mathematical Sciences 132,Springer,Berlin,37–58(2004)
[17]Hwang,J.M.,Mok,N.:Prolongations of infinitesimal linear automorphisms of pro jective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kahler deformation.Invent.Math.,160(3),591–645(2005)
[18]Kim,S.Y.:Geometric structures modeled on smooth pro jective horospherical varieties of Picard number one.Transform.Groups,22(2),361–386(2017)
[19]Kollar,J.:Rational Curves on Algebraic Varieties,Ergebnisse der Mathematik und ihrer Grenzgebiete 32,Springer,Berlin,1996
[20]Lakshmibai,V.,Weyman,J.:Multiplicities of points on a Schubert variety in a minuscule G/P.Adv.Math.,84(2),179–208(1990)
[21]Landsberg,J.M.,Manivel,L.:On the pro jective geometry of rational homogeneous varieties.Comment.Math.Helv.,78(1),65–100(2003)
[22]Mihai,I.A.:Odd symplectic flag manifolds.Transform.Groups,12(3),573–599(2007)
[23]Mok,N.:Characterization of standard embeddings between complex Grassmanninas by means of varieties of minimal rational tangents.Sci.China Ser.A,51(4),660–684(2008)
[24]Mok,N.:Geometric structures on uniruled pro jective manifolds defined by their varieties of minimal rational tangents,In:Hijazi,O.(ed)Geometrie Differentielle,Physique Mathematique,Mathematiques et Societe(II),Volume in honor of Jean Pierre Bourguignon,Asterisque,322,151–205(2008)
[25]Mok,N.:Geometric structures and substructures on uniruled pro jective manifolds,In:Cascini,P.,Mc Kernan,J.,Pereira,J.V.(eds)Foliation Theory in Algebraic Geometry,Simons Symposia,Springer,Cham,103–148(2016)
[26]Pasquier,B.:On some smooth pro jective two-orbit varieties with Picard number 1.Math.Ann.,344(4),963–987(2009)
[27]Perrin,N.:On the geometry of spherical varieties.Transform.Groups,19(1),171–223(2014)
[28]Richmond,E.,Slofstra,W.:Billey–Postnikov decompositions and the fibre bundle structure of Schubert varieties.Math.Ann.,366(1–2),31–55(2016)
[29]Springer,T.A.:Linear Algebraic Groups,Progress in Mathematics 9,Birkhauser,Boston,1998