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1.1 Cell structure

We begin by recalling the cell decomposition of a quadric hypersurface in a form convenient for us. We will proceed by induction on dimension. The two smallest dimensions are :
n = 1:
Q is $ \mbox{$\bf P$}$1 $ \hookrightarrow$ $ \mbox{$\bf P$}$2 as a conic and has a natural cell decomposition Q = $ \mbox{$\bf C$}$ $ \cup$ {$ \infty$}. We have for d $ \in$ $ \mbox{$\bf N$}$ maps Fd : Q$ \to$Q ( z $ \mapsto$ zd) preserving the cell structure.
n = 2:
Q is $ \mbox{$\bf P$}$1×$ \mbox{$\bf P$}$1 $ \hookrightarrow$ $ \mbox{$\bf P$}$2 via the Segre embedding and has a natural cell decomposition

Q = ($\displaystyle \mbox{$\bf C$}$×$\displaystyle \mbox{$\bf C$}$) $\displaystyle \cup$ ($\displaystyle \mbox{$\bf C$}$×{$\displaystyle \infty$}) $\displaystyle \cup$ ({$\displaystyle \infty$$\displaystyle \mbox{$\bf C$}$) $\displaystyle \cup$ {($\displaystyle \infty$,$\displaystyle \infty$)}.

For all d $ \in$ $ \mbox{$\bf N$}$, we have maps Fd : Q$ \to$Q ( (z, w) $ \mapsto$ (zd, wd)) which preserve the cell structure.
We henceforth assume that n $ \geq$ 3.

Scholium 1   Q has a cell decomposition with cells in each even (real) dimension (the Bruhat cell decomposition) such that we have:
  1. For n $ \geq$ 3, Q(2n - 2) = C is the projective cone over Q' $ \subset$ $ \mbox{$\bf P$}$n - 1, a smooth quadric hypersurface of complex dimension n - 2.
  2. There is exactly one cell in each even (real) dimension except in dimension n for n = 2k.
  3. Q(n) may be explicitly described as follows:
    n = 2k:
    Q(n) = L' $ \cup$ L'', where L', L'' $ \cong$ $ \mbox{$\bf P$}$k are linear subspaces of $ \mbox{$\bf P$}$n + 1 and L' $ \cap$ L'' = L $ \cong$ $ \mbox{$\bf P$}$k - 1.
    n = 2k + 1:
    Q(n) = L $ \cong$ $ \mbox{$\bf P$}$k is a linear subspace of $ \mbox{$\bf P$}$n + 1.
  4. For n $ \geq$ 3, Q(n + 2) $ \subset$ C is the Thom space over Q'(n) of the complex line bundle $ \mbox{${\cal O}_{Q'^{(n)}}$}$(1).

Proof:     Let p be any point of Q and let H $ \subset$ $ \mbox{$\bf P$}$n + 1 be a hyperplane tangent to Q at p. Then Q $ \cap$ H = C is the projective cone over Q' $ \subset$ $ \mbox{$\bf P$}$n - 1, a smooth quadric hypersurface of complex dimension n - 2 (i.e. C is the Thom space of the complex line bundle $ \mbox{${\cal O}_{Q'}$}$(1) = $ \mbox{${\cal O}_{\P^{n-1}}$}$(1)$ \mid_{Q'}^{}$). By induction on dimension we get a cell decomposition of Q'. If Q'(m) is the m-skeleton of Q', and C(Q'(m)) is the Thom space of $ \mbox{${\cal O}_{Q'^{(m)}}$}$(1), then C(Q'(m)) - C(Q'(m - 1)) is a union of cells of dimension m + 2. Thus, we obtain a cell decomposition of C. Since Q - C = $ \mbox{$\bf C$}$n, we obtain the desired cell structure. $ \Box$
We shall use the following construction. Construction 1 Let X, Y be compact topological spaces, L and M complex line bundles on X and Y respectively. Let f : X$ \to$Y be a continuous map, such that there is an isomorphism $ \varphi$ : L$\scriptstyle \otimes$ d$ \to$f*(M), for some positive integer d. Then there exists a map $ \Phi$ : L$ \to$M giving a commutative diagram

L $\displaystyle \;\stackrel{\Phi}{\longrightarrow}\;$ M
$\displaystyle \downarrow$   $\displaystyle \downarrow$
X $\displaystyle \;\stackrel{f}{\longrightarrow}\;$ Y

where $ \Phi$ is the d-th power map on fibres of the vertical arrows. The restriction of $ \Phi$ to the S1-bundles $ \widetilde{X}$, $ \widetilde{Y}$ of L, M respectively induces a map $ \widetilde{f}$ : $ \widetilde{X}$$ \to$$ \widetilde{Y}$, which has degree d along the fibres. If C(X, L) and C(Y, M) denote the Thom spaces of L and M respectively, we have a map C(f )= C(f,$ \varphi$) : C(X, L)$ \to$C(Y, M) induced by $ \Phi$.

In particular, if X $ \subset$ $ \mbox{$\bf P$}$N is a projective variety, f : X$ \to$X an algebraic self-map, L = M = $ \mbox{${\cal O}_{X}$}$(1), and $ \varphi$ is an isomorphism of algebraic line bundles, then C(X, L) is the projective cone (in $ \mbox{$\bf P$}$N + 1) of X and C(f ) can be regarded as an algebraic self-map of C(X, L). Note that $ \varphi$ is unique upto a scalar multiple. We had noted the existence of morphisms Fd : Q$ \to$Q for each d > 0, for a smooth quadric Q of dimension one or two. These maps satisfy Fd1oFd2 = Fd1d2 for all d1, d2 > 0. By repeatedly applying the above constructions, we obtain

Lemma 2   For each d > 0, there is an algebraic morphism

Fd, n : Q(n + 2)$\displaystyle \to$Q(n + 2)

and an algebraic isomorphism

$\displaystyle \varphi_{d,n}^{}$ : $\displaystyle \mbox{${\cal O}_{Q^{(n+2)}}$}$(d )$\displaystyle \to$Fd, n*($\displaystyle \mbox{${\cal O}_{Q^{(n+2)}}$}$(1))

such that
under the identification (by Scholium 1)

Q(n + 2) = C(Q'(n),$\displaystyle \mbox{${\cal O}_{Q'^{(n)}}$}$(1))

we have

Fd, n = C(Fd, n - 2,$\displaystyle \varphi_{d,n-2}^{}$)

Fd1, noFd2, n = Fd1d2, n.

$ \Box$
If Fd, n - 2 : Q'(n)$ \to$Q'(n) extends to a continuous map f' : Q'$ \to$Q', then the isomorphism $ \varphi_{d,n-2}^{}$ can be extended to an isomorphism (of topological complex line bundles)

$\displaystyle \varphi{^\prime}$ : $\displaystyle \mbox{${\cal O}_{Q'}$}$(d )$\displaystyle \to$f'*($\displaystyle \mbox{${\cal O}_{Q'}$}$(1)).

Then the induced continuous map C(f') = C(f',$ \varphi{^\prime}$) : C$ \to$C restricts to Fd, n on Q(n + 2) $ \subset$ C. Since Q is obtained from C by attaching $ \mbox{$\bf C$}$n via a map a : S2n - 1$ \to$C, the map C(f') : C$ \to$C extends to a map f : Q$ \to$Q if and only if C(f')*([a]) = m[a] $ \in$ $ \pi_{2n-1}^{}$(C), for some m $ \in$ $ \mbox{$\bf Z$}$. Thus, we need to compute $ \pi_{2n-1}^{}$(C), and the action of C(f')* on it.
next up previous
Next: 1.2 Computation of Homotopy Up: 1 Preliminaries Previous: 1 Preliminaries
Kapil Hari Paranjape 2002-11-21