Grassmann bundle

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In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X:

p : G d ( E ) → X {\displaystyle p:G_{d}(E)\to X} {\displaystyle p:G_{d}(E)\to X}

such that the fiber p − 1 ( x ) = G d ( E x ) {\displaystyle p^{-1}(x)=G_{d}(E_{x})} {\displaystyle p^{-1}(x)=G_{d}(E_{x})} is the Grassmannian of the d-dimensional vector subspaces of E x {\displaystyle E_{x}} {\displaystyle E_{x}}. For example, G 1 ( E ) = P ( E ) {\displaystyle G_{1}(E)=\mathbb {P} (E)} {\displaystyle G_{1}(E)=\mathbb {P} (E)} is the projective bundle of E. In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme.

The Grassmann bundle of the tangent bundle is the contact bundle.

Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle S and universal quotient bundle Q that fit into

0 → S → p ∗ E → Q → 0 {\displaystyle 0\to S\to p^{*}E\to Q\to 0} {\displaystyle 0\to S\to p^{*}E\to Q\to 0}.

Specifically, if V is in the fiber p−1(x), then the fiber of S over V is V itself; thus, S has rank r = d = dim(V) and ∧ d S {\displaystyle \wedge ^{d}S} {\displaystyle \wedge ^{d}S} is the determinant line bundle. Now, by the universal property of a projective bundle, the injection ∧ r S → p ∗ ( ∧ r E ) {\displaystyle \wedge ^{r}S\to p^{*}(\wedge ^{r}E)} {\displaystyle \wedge ^{r}S\to p^{*}(\wedge ^{r}E)} corresponds to the morphism over X:

G d ( E ) → P ( ∧ r E ) {\displaystyle G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)} {\displaystyle G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)},

which is nothing but a family of Plücker embeddings.

The relative tangent bundle TGd(E)/X of Gd(E) is given by[1]

T G d ( E ) / X = Hom ⁡ ( S , Q ) = S ∨ ⊗ Q , {\displaystyle T_{G_{d}(E)/X}=\operatorname {Hom} (S,Q)=S^{\vee }\otimes Q,} {\displaystyle T_{G_{d}(E)/X}=\operatorname {Hom} (S,Q)=S^{\vee }\otimes Q,}

which morally is given by the second fundamental form. In the case d = 1, it is given as follows: if V is a finite-dimensional vector space, then for each line l {\displaystyle l} {\displaystyle l} in V passing through the origin (a point of P ( V ) {\displaystyle \mathbb {P} (V)} {\displaystyle \mathbb {P} (V)}), there is the natural identification (see Chern class#Complex projective space for example):

Hom ⁡ ( l , V / l ) = T l P ( V ) {\displaystyle \operatorname {Hom} (l,V/l)=T_{l}\mathbb {P} (V)} {\displaystyle \operatorname {Hom} (l,V/l)=T_{l}\mathbb {P} (V)}

and the above is the family-version of this identification. (The general care is a generalization of this.)

In the case d = 1, the early exact sequence tensored with the dual of S = O(-1) gives:

0 → O P ( E ) → p ∗ E ⊗ O P ( E ) ( 1 ) → T P ( E ) / X → 0 {\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} (E)}\to p^{*}E\otimes {\mathcal {O}}_{\mathbb {P} (E)}(1)\to T_{\mathbb {P} (E)/X}\to 0} {\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} (E)}\to p^{*}E\otimes {\mathcal {O}}_{\mathbb {P} (E)}(1)\to T_{\mathbb {P} (E)/X}\to 0},

which is the relative version of the Euler sequence.

References

  1. Fulton 1998, Appendix B.5.8