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tautological line bundle

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Bundles

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Idea

The canonical line bundle over a projective space is sometimes called its “tautological line bundle”. For more see at classifying space.

Definition

As a topological line bundle

We discuss the tautological line bundle as a topological vector bundle. Hence let kk be the topological field either

equipped with their Euclidean metric topology.

Definition

(topological projective space)

Let nn \in \mathbb{N}. Consider the Euclidean space k n+1k^{n+1} equipped with its metric topology, let k n+1{0}k n+1k^{n+1} \setminus \{0\} \subset k^{n+1} be the topological subspace which is the complement of the origin, and consider on its underlying set the equivalence relation which identifies two points if they differ by multiplication with some ckc \in k (necessarily non-zero):

(x 1x 2)(ck(x 2=cx 1)). (\vec x_1 \sim \vec x_2) \;\Leftrightarrow\; \left( \underset{c \in k}{\exists} ( \vec x_2 = c \vec x_1 ) \right) \,.

The equivalence class [x][\vec x] is traditionally denoted

[x 1:x 2::x n+1]. [x_1 : x_2 : \cdots : x_{n+1}] \,.

Then the projective space kP nk P^n is the corresponding quotient topological space

kP n(k n+1{0})/. k P^n \;\coloneqq\; \left(k^{n+1} \setminus \{0\}\right) / \sim \,.
Definition

(standard open cover of topological projective space)

For nn \in \mathbb{N} the standard open cover of the projective space kP nk P^n (def. 1) is

{U ikP n} i{1,,n+1} \left\{ U_i \subset k P^n \right\}_{i \in \{1, \cdots, n+1\}}

with

U i{[x 1::x n+1]kP n|x i0}. U_i \coloneqq \left\{ [x_1 : \cdots : x_{n+1}] \in k P^n \;\vert\; x_i \neq 0 \right\} \,.

To see that this is an open cover:

  1. This is a cover because with the orgin removed in k n{0}k^n \setminus \{0\} at every point [x 1::x n+1][x_1: \cdots : x_{n+1}] at least one of the x ix_i has to be non-vanishing.

  2. These subsets are open in the quotient topology kP n=(k n{0})/k P^n = (k^n \setminus \{0\})/\sim, since their pre-image under the quotient co-projection k n+1{0}kP nk^{n+1} \setminus \{0\} \to k P^n coincides with the pre-image (pr iι) 1(k{0})(pr_i\circ\iota)^{-1}( k \setminus \{0\} ) under the projection onto the iith coordinate in the product topological space k n+1=i{1,,n}kk^{n+1} = \underset{i \in \{1,\cdots, n\}}{\prod} k (where we write k n{0}ιk npr ikk^n \setminus \{0\} \overset{\iota}{\hookrightarrow} k^n \overset{pr_i}{\to} k).

Definition

(tautological topological line bundle)

For kk a topological field and nn \in \mathbb{N}, the tautological line bundle over the projective space kP nk P^n is topological kk-line bundle whose total space is the following subspace of the product space of the projective space kP nk P^n with k nk^n:

T{([x 1::x n+1],v)kP n×k n+1|vx k}, T \coloneqq \left\{ ( [x_1: \cdots : x_{n+1}], \vec v) \in k P^n \times k^{n+1} \;\vert\; \vec v \in \langle \vec x\rangle_k \right\} \,,

where x kk n+1\langle \vec x\rangle_k \subset k^{n+1} is the kk-linear span of x\vec x.

(The space TT is the space of pairs consisting of the “name” of a kk-line in k n+1k^{n+1} together with an element of that kk-line)

This is a bundle over projective space by the projection function

T π kP n ([x 1::x n+1],v) [x 1::x n+1]. \array{ T &\overset{\pi}{\longrightarrow}& k P^n \\ ([x_1: \cdots : x_{n+1}], \vec v) &\mapsto& [x_1: \cdots : x_{n+1}] } \,.
Proposition

(tautological topological line bundle is well defined)

The tautological line bundle in def. 3 is well defined in that it indeed admits a local trivialization.

Proof

We claim that there is a local trivialization over the canonical cover of def. 2. This is given for i{1,,n}i \in \{1, \cdots, n\} by

U i×k T| U i ([x 1:x i1:1:x i+1::x n+1],c) ([x 1:x i1:1:x i+1::x n+1],(cx 1,cx 2,,cx n+1)). \array{ U_i \times k &\overset{}{\longrightarrow}& T\vert_{U_i} \\ ( [x_1 : \cdots x_{i-1}: 1 : x_{i+1} : \cdots : x_{n+1}] , c ) &\mapsto& ( [x_1 : \cdots x_{i-1} : 1 : x_{i+1} : \cdots : x_{n+1} ], (c x_1, c x_2, \cdots , c x_{n+1}) ) } \,.

This is clearly a bijection of underlying sets.

To see that this function and its inverse function are continuous, hence that this is a homeomorphism notice that this map is the extension to the quotient topological space of the analogous map

((x 1,,x i1,x i+1,,x n+1),c) ((x 1,,x i1,x i+1,,x n+1),(cx 1,cx i1,c,cx i+1,,cx n+1)). \array{ ( (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1}) , c) &\mapsto& ( (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1}) , (c x_1, \cdots c x_{i-1}, c, c x_{i+1}, \cdots, c x_{n+1}) ) } \,.

This is a polynomial function on Euclidean space and since polynomials are continuous, this is continuous. Similarly the inverse function lifts to a rational function on a subspace of Euclidean space, and since rational functions are continuous on their domain of definition, also this lift is continuous.

Therefore by the universal property of the quotient topology, also the original functions are continuous.

Examples

References

Lecture notes include

See also

Last revised on June 11, 2017 at 09:58:43. See the history of this page for a list of all contributions to it.