## On the Projective Geometry of Rational Homogeneous Varieties (2000)

Venue: | Comment. Math. Helv |

Citations: | 24 - 11 self |

### BibTeX

@ARTICLE{Landsberg00onthe,

author = {J. M. Landsberg and Laurent Manivel},

title = {On the Projective Geometry of Rational Homogeneous Varieties},

journal = {Comment. Math. Helv},

year = {2000},

volume = {78},

pages = {65--100}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper we determine the varieties of linear spaces on rational homogeneous varieties, provide explicit geometric models for these spaces, and establish basic facts about the local dierential geometry of rational homogeneous varieties. Let G be a complex simple Lie group, P a maximal parabolic subgroup. The space of lines on G=P in its minimal homogeneous embedding was determined in [4] in terms of Lie incidence systems. There is a dichotomy between the cases for which the simple root associated to P is short or not: for non-short roots, the space of lines on G=P is G-homogeneous and can be described using ideas of Tits; for short roots, it is not G-homogeneous. We present a renement of their result, due to [1] (for a parabolic subgroup P which does not need to be maximal), that each connected component of the space of lines consists of exactly two G-orbits

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Citation Context ...this H-module. For an ordinary Grassmannian G(k; n) = G(k; V ), T = E Q, where E is the tautological subbundle and Q = V=E the quotient bundle. Its symmetric powers are given by the Cauchy formula ([18], p. 33) S j T = M jj=j S E S Q; the sum is over all partitions with the sum of its parts jj equal to j. We have Res G H k V = k (E Q) = M h0 h E h Q = M h0 W ! k h +! k+h sinc... |

164 |
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Citation Context ...# j ) # # G/P S G/P S\# j #N(# j ) Indeed, the shadow on G/P S of a point in G/P S\# j #N(# j ) is a rational curve, on which a line bundle L # defined by a weight # # P (S) has degree #l,s# j # (see =-=[5]-=-, Lemme 2 p. 58). 14 J.M. LANDSBERG AND LAURENT MANIVEL In particular, suppose that G/P S is embedded in some PV # by a very ample line bundle L # , where # = P i#S l i # i , and contains a line of PV... |

163 | Lie algebra cohomology and the generalized Borel-Weil theorem - Kostant - 1961 |

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Citation Context ...representation, and N 2 is the vector representation V . The half-spin representations S+ and S can be constructed as the even and odd parts of the exterior algebra of a null 5-plane E in V (see e.g. =-=[1-=-0]): S+ ; S are dual to one another, the wedge product giving a perfect pairing S+ S ! 5 E = C . Moreover, the full exterior algebra of E is a module over the Cliord algebra of V . If F is a compleme... |

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Citation Context ...j ! N j+1 (recall that T = N 1 ). If N j is replaced by the space of sections (X;OX (j)) for a subvariety X PT , the homology of the corresponding Koszul complexes compute the syzygies of X [8]. For a classical minuscule variety X , there is a strange relation between the complexes constructed from their normal spaces, and the Koszul complexes computing the syzygies of another minuscule var... |

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Citation Context ...ice of H . The proposition holds for any such choice. Proof. Letsbe such that m j () = p; m i () = 0; i 2 Snfjg. Let Xs2 gs. Let v 2 V be a highest weight vector. Then Xsv 2 ^ Y jp T jp[v] X . By [6]=-=-=-, lemme 7.2.5, X p+1 v = 0 so the rational curve exp(tXs)v is contained in X and is of degree at most p. These propositions stress the importance of minuscule varieties in our study. The next section ... |

60 |
Algebraic geometry and local differential geometry
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Citation Context ...ctive second fundamental form FF 2 X,x = ∑ 1≤α,β≤n, n+1≤µ≤n+a q µ αβdxα ◦ dx β ⊗ ∂ ∂x µ ∈ S2T ∗ xX ⊗NxX. If x is a general point, FF2 X,x even contains information about the global geometry of X, see =-=[9]-=-, [12]. It is useful to consider the second fundamental form as a system of quadrics |FF2 X,x | := P(FF2 X,x (N ∗ xX)) ⊆ PS2T ∗ xX parametrized by N ∗ xX, and Base |FF2 X,x | ⊂ PTxX, their common zero... |

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Citation Context ...e l-th prolongation of A. Let Jac(A) := fvyP jv 2 V; P 2 Ag S d 1 V , the Jacobian ideal of A. Note that A (1) = fP 2 S d+1 V jJac(P ) Ag. A basic fact about fundamental forms, due to Cartan ([3], p 377) (and rediscovered in [9]), is that if x 2 X is a general point, then the prolongation property holds at x: jFF k X;x j jFF k 1 X;x j (1) : A geometric consequence is as follows. Dene the k... |

22 |
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Citation Context ... 7 # ImO = V # 1 = V . Let # # # 3 V # be a generic element and let # : GL(V ) # GL(# 3 V # ) be the induced representation. Here are some descriptions of G 2 # GL(V ) (see [10] pp. 114, 116, 278 and =-=[23]-=- chapter 2): G 2 = Aut(O) = {g # GL(V ) | #(g)# = #} = {g = (g + , g - , g 0 ) # Spin 8 (V ) | g + = g - = g 0 }. 22 J.M. LANDSBERG AND LAURENT MANIVEL The third line should be understood as follows: ... |

18 |
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Citation Context ...ven a mapping : Y ! PV , one denes its fundamental forms FF k in the same manner. FF 2 ;x quotiented by ker x is isomorphic to the second fundamental form of the image, FF 2 (Y );(x) . See [13] for details. In what follows, we will slightly abuse notation by ignoring twists, which will not matter as we study fundamental forms only at somesxed base point. We will use N k to denote both N ... |

16 | Representation theory and projective geometry’, math.AG/0203260
- Landsberg, Manivel
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Citation Context ...minimally homogeneously embedded Hermitian symmetric space of H). We study minuscule varieties in 3 and prove our main result on their infinitesimal geometry in 4. This is the first paper in a series =-=[15, 14, 16, 17, 18]-=- establishing new relations between the representation theory of complex simple Lie groups and the algebraic and di#erential geometry of their homogeneous varieties. The surprising connection between ... |

14 |
A system of quadrics describing the orbit of the highest vector
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Citation Context ... ideal is generated in degree d, and L a linear space osculating to order d at a smooth point x 2 X , then L X . The ideal of a projective homogeneous variety is generated in degree two (see e.g. [1=-=6-=-]), so if X PV is homogenous, then Base jFF 2 X;x j is the set of tangent directions to lines on X through x. If y 2 ~ T x X \ X then the line P 1 xy is contained in X . 2.2. Osculating spaces of hom... |

13 | Construction and classification of complex simple Lie algebras via projective geometry
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Citation Context ...nimally homogeneously embedded Hermitian symmetric space of H). We study minuscule varieties in §3 and prove our main result on their infinitesimal geometry in §4. This is the first paper in a series =-=[15, 14, 16, 17, 18]-=- establishing new relations between the representation theory of complex simple Lie groups and the algebraic and differential geometry of their homogeneous varieties. The surprising connection between... |

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Citation Context ...1 ; : : : ; m ; m ) . Finally, the case of quadrics is immediate since they are hypersurfaces. For exceptional minuscule varieties the same argument goes through, except that we use the LiE package [1=-=7-=-], or Littelmann paths, instead of the above classical decomposition formulas. 3.4. Algebraic structures induced by innitesimal geometry. We remark on some consequences of the strict prolongation prop... |

11 |
On Second Fundamental Forms of Projective Varieties
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Citation Context ...h is the projective second fundamental form FF 2 X;x = q sdx sdx @ @x 2 S 2 T x X N x X . If x is a general point, FF 2 X;x even contains information about the global geometry of X , see [9], [12]. It is useful to consider the system of quadrics jFF 2 X;x j := P(FF 2 X;x (N x X)) PS 2 T x X , and Base jFF 2 X;x j PT x X , their common zero locus. More generally, the k-th projective fun... |

9 | Manivel, Classification of complex simple Lie algebra via projective geometry
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(Show Context)
Citation Context ...minimally homogeneously embedded Hermitian symmetric space of H). We study minuscule varieties in 3 and prove our main result on their infinitesimal geometry in 4. This is the first paper in a series =-=[15, 14, 16, 17, 18]-=- establishing new relations between the representation theory of complex simple Lie groups and the algebraic and di#erential geometry of their homogeneous varieties. The surprising connection between ... |

8 |
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Citation Context ...the sets of simple roots of G. Consider the diagram G/PS∪S ′ π ւ ց π ′ X = G/PS X ′ = G/PS ′ Let x ′ ∈ X ′ and consider Y := π(π ′−1 (x ′ )) ⊂ X. Then X is covered by such varieties Y . Tits shows in =-=[24]-=- that Y = H/Q where D(H) = D(G)/(S\S ′ ), and Q ⊂ H is the parabolic subgroup corresponding to S ′ \S. He calls such subvarieties Y of X, L-subvarieties, and Y the shadow of x ′ . Example 4.1. For X =... |

8 |
Spaces of constant curvature. Third edition. Publish or
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Citation Context ...tion of g=p by P-submodules. The quotients T k = M m i ()=k g are irreducible P-modules. Proof. The fact that each s k =s k 1 is a P-module is clear. The irreducibility of T k is a special case of [21], 8.13.3 (which is attributed to Kostant). The irreducibility of T k implies that the set f 2 + j m i () = kg has a unique minimal element which we denote by k when we consider the root as a weig... |

5 |
Lie incidence systems from projective varieties
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(Show Context)
Citation Context ...ntial geometry of rational homogeneous varieties. Let G be a complex simple Lie group, P a maximal parabolic subgroup. The space of lines on G=P in its minimal homogeneous embedding was determined in =-=[4]-=- in terms of Lie incidence systems. There is a dichotomy between the cases for which the simple root associated to P is short or not: for non-short roots, the space of lines on G=P is G-homogeneous an... |

5 |
The projective geometry of Freudenthal’s magic chart and the exceptional homogeneous spaces, in preparation
- Landsberg, Manivel
(Show Context)
Citation Context ... forms are simply the secant varieties of the base locus of the second fundamental forom. We exploit the suprising connection between secant varieties, prolongations and nilpotent orbit structures in =-=[14, 15]-=-. We explain how to determine the higher dimensional linear spaces associated to non-short roots using Tits methods. For short roots, we provide explicit descriptions of the spaces we study, especiall... |

3 |
Algebraic Geometry and Local Di#erential Geometry
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- 1979
(Show Context)
Citation Context ... which is the projective second fundamental form FF 2 X;x = q sdx sdx @ @x 2 S 2 T x X N x X . If x is a general point, FF 2 X;x even contains information about the global geometry of X , see [9], [12]. It is useful to consider the system of quadrics jFF 2 X;x j := P(FF 2 X;x (N x X)) PS 2 T x X , and Base jFF 2 X;x j PT x X , their common zero locus. More generally, the k-th projecti... |

3 |
Construction and classi of complex simple Lie algebras via projective geometry
- Landsberg, Manivel
(Show Context)
Citation Context ... forms are simply the secant varieties of the base locus of the second fundamental forom. We exploit the suprising connection between secant varieties, prolongations and nilpotent orbit structures in =-=[14, 15]-=-. We explain how to determine the higher dimensional linear spaces associated to non-short roots using Tits methods. For short roots, we provide explicit descriptions of the spaces we study, especiall... |

3 |
exceptional Lie algebras, and Deligne dimension formulas, arXiv:math.AG/0107032
- Landsberg, Manivel, et al.
(Show Context)
Citation Context ...minimally homogeneously embedded Hermitian symmetric space of H). We study minuscule varieties in 3 and prove our main result on their infinitesimal geometry in 4. This is the first paper in a series =-=[15, 14, 16, 17, 18]-=- establishing new relations between the representation theory of complex simple Lie groups and the algebraic and di#erential geometry of their homogeneous varieties. The surprising connection between ... |

2 | Dsingularisation des varits de Schubert gnralises - Demazure - 1974 |