\[ (1.\ill{}) \qquad \pi_{i} \in \mathcal{E}^{*} \quad \text{pour tout } i . \]

\[ \operatorname{End}^{(\nu)}(M) = \textstyle\sum \operatorname{Hom}(M^{i}, M^{i+\nu}) \]

\[ \operatorname{End}^{(\nu)}(M) = \textstyle\sum_{i} \pi_{i+\nu} \operatorname{End}(M^{*})\, \pi_{i} , \]

\[ \struck{\mathrm{gr}}\ \mathrm{pr}_{\nu}(u) = \textstyle\sum_{i} \pi_{i+\nu}\, u\, \pi_{i} . \]

\[ (1.3) \qquad \text{a)}\ \ \mathcal{E}^{(\nu)} = \sum_{i} \pi_{i+\nu}\, \mathcal{E}^{*}\, \pi_{i} , \qquad \text{b)}\ \ \mathrm{pr}_{\nu}\, u = \sum_{i} \pi_{i+\nu}\, u\, \pi_{i} , \]

\[ (1.4) \qquad u \in \mathcal{E}^{(\nu)} \iff u\, \pi_{i} = \pi_{i+\nu}\, u \quad \text{pour tout } i . \]

\[ (1.1) \qquad \begin{cases} \pi_{i}^{2} = \pi_{i} \\ \pi_{i} \pi_{j} = 0 \ \text{ si } i \neq j \\ \sum_{i} \pi_{i} = 1 \end{cases} \]

\[ \begin{cases} \pi_{i}^{2} = \pi_{i} \\ \pi_{i} \pi_{j} = \pi_{j} \pi_{i} = 0 \ \text{ si } i \neq j \\ \sum_{i} \pi_{i} = 1 \end{cases} \]

\[ \begin{cases} \mathcal{E} = \bigoplus_{i,j} \pi_{j}\, \mathcal{E}\, \pi_{i} \\ \mathrm{pr}_{i,j}(u) = \pi_{j}\, u\, \pi_{i} \end{cases} \qquad \text{où } u = \sum_{\alpha,\beta} \pi_{\beta}\, u\, \pi_{\alpha} \]

\[ \pi_{\beta}\, u\, \pi_{i} = 0 \quad \text{si } \beta \neq i + \nu . \]

\[ (3.0) \qquad \pi_{i} = 0 \ \text{ si } i \notin [0, 2n], \qquad n \ \text{entier \add{fixé}} . \]

\[ u \in \mathcal{E}^{0} \iff \bigl( u \pi_{i} = \pi_{i} u \ \ \forall i \bigr) , \]

\[ \varphi_{i} = 1 - \sum_{\alpha < i} \pi_{\alpha} = \sum_{\alpha \geqslant i} \pi_{\alpha} , \qquad \psi_{2n-i} = 1 - \sum_{\alpha > 2n-i} \pi_{\alpha} = \sum_{\alpha \leqslant 2n-i} \pi_{\alpha} , \]

\[ \psi_{2n-i}\, \mathcal{E}^{2n-2i}\, \varphi_{i} = \sum_{\alpha} \psi_{2n-i}\, \pi_{\alpha + (2n-2i)}\, \mathcal{E}\, \pi_{\alpha}\, \varphi_{i} . \]

\[ \pi_{\alpha}\, \varphi_{i} = \begin{cases} 0 & \text{si } \alpha < i \\ \pi_{\alpha} & \text{si } \alpha \geqslant i \end{cases} \qquad \text{et} \qquad \psi_{2n-i}\, \pi_{\alpha+(2n-2i)} = \begin{cases} 0 & \text{si } \alpha > i \\ \pi_{\alpha+(2n-2i)} & \text{si } \alpha \leqslant i \end{cases} \]

\[ \pi_{i+(2n-2i)}\, \mathcal{E}\, \pi_{i} = \pi_{2n-i}\, \mathcal{E}\, \pi_{i} = \mathcal{E}_{2n-i,\,i} . \]

\[ (3.1)_{i} \qquad \bigl( u \in \mathfrak{z}\mathcal{E}^{0}, \ \mathcal{E}^{2n-i}_{i}\, u = 0 \bigr) \implies \pi_{i}\, u = 0 \]

\[ (3.2)_{i} \qquad \bigl( u \in \mathfrak{z}\mathcal{E}^{0}, \ u\, \mathcal{E}^{2n-i}_{i} = 0 \bigr) \implies \pi_{2n-i}\, u = 0 \]

\[ (4.1)_{i} \qquad \text{l'intersection des noyaux des } u \colon M^{i} \to M^{2n-i}, \ u \in \mathcal{E}^{2n-i}_{i}, \ \text{est réduite à } 0 . \]

\[ (4.2)_{i} \qquad \text{la somme des images des } u \colon M^{i} \to M^{2n-i}, \ u \in \mathcal{E}^{2n-i}_{i}, \ \text{est égale à } M^{2n-i} . \]

\[ (4.3)_{i} \qquad \text{il existe } u \in \mathcal{E}^{2n-i}_{i} \ \text{qui soit un isomorphisme} \ \ill{}, \ \text{ou, de} \]

\[ (4.4)_{i} \qquad \begin{cases} \exists\, v_{i} \in \mathcal{E}^{(2n-2i)},\ w_{i} \in \mathcal{E}^{-(2n-2i)} \ \text{tels que l'on ait} \\[2pt] (w_{i} v_{i} - 1)\, \pi_{i} = 0 , \qquad (v_{i} w_{i} - 1)\, \pi_{2n-i} = 0 \end{cases} \]

\[ (6.0) \qquad L \in \mathcal{E}^{(2)} \]

\[ (5.1) \qquad \pi_{i} \in \mathcal{E} \quad \text{pour tout } i . \]

\[ v'_{i} = \psi_{2n-i}\, v_{i}\, \varphi_{i} = \pi_{2n-i}\, v_{i}\, \pi_{i} \in \mathcal{E} , \]

\[ w'_{i} = \varphi_{i}\, w_{i}\, \psi_{2n-i} = \pi_{i}\, w_{i}\, \pi_{2n-i} \in \mathcal{E} , \]

\[ w'_{i} v'_{i} = \pi_{i}\, w_{i}\, \pi_{2n-i}\, v_{i}\, \pi_{i} = \pi_{i}\, \pi_{i}\, w_{i}\, v_{i}\, \pi_{i} = \pi_{i}^{3} = \pi_{i} \]

\[ (6.1)_{i} \qquad L^{n-i} \,|\, M^{i} \colon M^{i} \to M^{2n-i} \ \text{ est un isom.} \]

\[ (6.2)_{i} \qquad \ill{} \ \text{est élément de } \mathcal{E}^{-(2n-2i)} . \]

\[ (6.3)_{i} \qquad \begin{cases} \exists\, w_{i} \in \mathcal{E}^{-(2n-2i)} \ \text{tel qu'on ait} \\[2pt] (w_{i} L^{n-i} - 1)\, \pi_{i} = 0 , \qquad (L^{n-i} w_{i} - 1)\, \pi_{2n-i} = 0 \end{cases} \]

\[ (6.4) \qquad \begin{cases} M^{i} \simeq P^{i} + L P^{i-2} + L^{2} P^{i-4} + \ldots & (i \leqslant n) \\[2pt] M^{2n-i} \simeq L^{n-i} P^{i} + L^{n-i+1} P^{i-2} + \ldots \end{cases} \]

\[ (6.7) \qquad \begin{cases} L \Lambda = 1 - \sum_{0 \leqslant i \leqslant n} \pi_{i,0} \\[2pt] \Lambda L = 1 - \sum_{n \leqslant i \leqslant 2n} \pi_{i,0} \end{cases} \]

\[ (6.7) \qquad \pi_{j,\alpha} \in \mathcal{E} \quad \text{pour } j \notin [i, 2n-i] \]

\[ (6.7') \qquad \pi_{j,\alpha} \in \mathcal{E} . \]

\[ (6.8) \qquad \pi_{i,\alpha'} \in \mathcal{E}, \quad \pi_{2n-i,\alpha'} \in \mathcal{E} \quad \text{si } \alpha' > \alpha \]

\[ (6.8') \qquad \pi_{i,\alpha} \in \mathcal{E}, \quad \pi_{2n-i,\alpha} \in \mathcal{E} . \]

\[ (6.9) \qquad \sum_{\alpha} \pi_{i,\alpha} = \pi_{i} , \]

\[ q_{i,\alpha} = 1 - \sum_{\alpha' > \alpha} \pi_{i,\alpha'} = \sum_{\alpha' \leqslant \alpha} \pi_{i,\alpha'} , \]

\[ L^{n-i+\alpha}\, q_{i,\alpha} \colon M^{i} \longrightarrow M^{2n-i+2\alpha} \]

\[ L^{n-i+2\alpha} \colon P^{i-2\alpha} \longrightarrow L^{n-i+2\alpha} P^{i-2\alpha} \subset M^{2n-i+2\alpha} . \]

\[ (6.9) \qquad \pi_{i,\alpha} = L^{\alpha}\, w_{i-2\alpha}\, L^{n-i+\alpha}\, q_{i,\alpha} = L^{\alpha}\, w_{i-2\alpha}\, L^{n-i+\alpha}\Bigl( 1 - \sum_{\alpha' > \alpha} \pi_{i,\alpha'} \Bigr) \]

\[ (6.10) \qquad \pi_{2n-i,\alpha} = L^{n-i}\, \pi_{i,\alpha}\, w_{i} \ \struck{\ill{}} \]

\[ (6.11) \qquad \Lambda = \sum_{j} \struck{\ill{}}\ \Lambda\, \pi_{j,\alpha}\ \struck{\ill{}} \]

\[ (6.12) \qquad \begin{cases} \Lambda\, \pi_{i,0} = 0 , \\[2pt] \Lambda\, \pi_{i,\alpha} = \struck{L}\ w_{i-2\alpha}\, L^{n-i+\alpha}\, \pi_{i,\alpha} \quad \text{si } \alpha \geqslant 1 , \end{cases} \]

\[ (6.13) \qquad \Lambda\, \pi_{2n-i,\alpha} = L^{n-i+\alpha-1}\, w_{i-2\alpha}\, \struck{\ill{}}\ L^{\alpha}\, \pi_{2n-i,\alpha} \qquad (\text{si } i \leqslant n-1) \]

\[ (6.15) \qquad \begin{cases} L \Lambda = 1 - \sum_{0 \leqslant i \leqslant n} \pi_{i,0} \\[2pt] \Lambda L = 1 - \sum_{n \leqslant j \leqslant 2n} \pi_{j,0} \end{cases} \]

\[ L_{0}\, x = \xi_{0}\, x , \qquad \Lambda_{0}\, x = \xi^{*}_{0} \lrcorner\, x \]

\[ \Phi_{0} = \mathbb{Z}[L_{0}, \Lambda_{0}] \subset \operatorname{End}(M^{*}_{0}) . \]

\[ 0 \leqslant i \leqslant n , \quad 0 \leqslant \alpha, \beta \leqslant 2n-i \]

\[ \begin{cases} \varphi^{\beta,\alpha}_{i} = L_{0}^{\beta-\alpha}\, \pi_{i+2\alpha,\alpha} & \text{si } \alpha \leqslant \beta \\[2pt] \varphi^{\beta,\alpha}_{i} = \Lambda_{0}^{\alpha-\beta}\, \pi_{i+2\alpha,\alpha} & \text{si } \alpha \geqslant \beta \end{cases} \]

\[ \begin{cases} = L_{0}^{\beta-\alpha}\, \pi_{i,\alpha} = \pi_{j,\beta}\, L_{0}^{\beta-\alpha} & \text{si } \alpha \leqslant \beta \\[2pt] = \Lambda_{0}^{\alpha-\beta}\, \pi_{i,\alpha} = \pi_{j,\beta}\, \Lambda_{0}^{\alpha-\beta} & \text{si } \alpha \geqslant \beta \end{cases} \]

\[ \bigl( \Lambda_{0}^{n-i} L_{0}^{n-i} - 1 \bigr)\, \pi_{i} = 0 , \qquad \bigl( L_{0}^{n-i} \Lambda_{0}^{n-i} - 1 \bigr)\, \pi_{2n-i} = 0 , \]

\[ \bigl( \Lambda^{n-i} L^{n-i} - 1 \bigr)\, \pi_{i} = 0 , \qquad \bigl( L^{n-i} \Lambda^{n-i} - 1 \bigr)\, \pi_{2n-i} = 0 , \]

\[ u \colon \Phi_{0} \longrightarrow \mathcal{E} \]

\[ \bigl( w_{i} L^{n-i} - 1 \bigr)\, \pi_{2n-i} \struck{\ill{}} = 0 , \]

\[ \mathcal{E}^{(2n)} \neq 0 \iff \pi_{2n} \neq 0 \implies L^{n} \neq 0 \implies \Lambda^{n} \neq 0 . \]

\[ (8.13) \qquad \mathrm{Pic}(U) = \underline{\mathrm{Pic}}_{U}(k) \]

\[ (8.14) \qquad 0 \longrightarrow \mathrm{Pic}(U)^{0} \longrightarrow \mathrm{Pic}(U) \longrightarrow \mathrm{NS}(U) \longrightarrow 0 , \]

\[ (8.15) \qquad \mathrm{Pic}(U)^{0} = \underline{\mathrm{Pic}}_{U}^{0}(k) , \qquad \mathrm{NS}(U) = \underline{\mathrm{NS}}_{U}(k) , \]

\[ (C_{i} . C_{j})_{1 \leqslant i, j \leqslant r} \]

\[ \underline{\mathbb{Z}}^{r} \longrightarrow \underline{\mathbb{Z}}^{r} \]

\[ (8.16) \qquad \mathrm{NS}(U) \times \mathrm{NS}(U) \longrightarrow \mathbb{Q}/\mathbb{Z} , \]

\[ n b_{0} + (n-1) b_{1} = \ldots , \qquad 2 b_{0} + b_{1} , \qquad \sum_{i} (n-i)\, b_{i} \]

\[ M^{0} \xrightarrow{\ L\ } M^{2n} \]

\[ L , \qquad \mathcal{E}^{(n-i)} , \qquad \Lambda , \qquad \Lambda L \]

\[ L \sim \lambda L , \qquad \Lambda = P(L \ , \qquad \Lambda = \Omega(L)\, \Delta(L) \]

\[ L , L' \quad | \quad L^{2}, L L' \quad | \quad L^{3} \]

\[ L'^{2} = L L' , \qquad L^{3} = L^{2} L' , \qquad L'^{3} = L' L'^{2} = L^{2} L' = L^{3} \]

\[ \bigl( L^{q},\ L'^{2} - L L',\ L^{2} L' - L^{3},\ L^{3} L' \bigr) \]

\[ \mathbb{Z}[L]/(L^{4}) \ \ill{}\ \mathbb{Z}[L'] / (L'^{2} - L'') \]

\[ (8.3) \qquad 0 \longrightarrow t^{i-1}(X, \ell) \longrightarrow H^{i}(X, \underline{\mathbb{Z}}_{\ell}[1]) \longrightarrow T_{\ell}\bigl( H^{i}(X, \mu_{\ell^{\infty}}) \bigr) \longrightarrow 0 , \]

\[ (8.4) \qquad t^{i-1}(X, \ell) = H^{i-1}(X, \mu_{\ell^{\infty}}) \big/ H^{i-1}(X, \mu_{\ell^{\infty}})^{0} \]

\[ (8.5) \qquad H^{i}(X, \underline{\mathbb{Z}}_{\ell}[1]) = \varprojlim H^{i}(X, \mu_{\ell^{\nu}}) , \]

\[
L^{n}_{0} = L^{n}_{0}\,\pi_{0} = L^{n}_{0}\,\Lambda'^{\,n} L'^{\,n}
 = L^{n}_{0}\,(\beta \Lambda^{n}_{0})(\alpha L^{n}_{0})
\]

\[
= L^{n}_{0}\,\alpha\beta\,\pi_{0} = \alpha\beta\,L^{n}_{0}
\]

\[
L^{4} = 0, \quad L'^{\,2} - LL' = 0, \quad L^{2}L' - L^{3} = 0, \quad L^{3}L' = 0,
\]

\[
1 \ \| \ L,\ L' \ \| \ L^{2},\ LL' \ \| \ L^{3}
\qquad (\deg : \ 0 \quad 2 \quad 4)
\]

\[
\begin{array}{ccccc}
H^{0} & H^{1} & H^{2} & H^{3} & H^{4} \\
k & V & k\xi + W & \check{V} & k\xi^{2}
\end{array}
\]

\[
\Lambda^{2}V \xrightarrow{\ \alpha\ } k\xi + W, \qquad
\alpha(x \wedge y) = \varphi(x,y)\,\xi + \psi(x,y)
\]

\[
\mathrm{Sym}^{2}(W) \xrightarrow{\ \beta\ } k\xi^{2}, \qquad
\beta(xy) = Q(x,y)\,\xi^{2}
\]

\[
\varphi(x,y)\varphi(t,z) + Q(\psi(x,y), \psi(t,z))
 = \varphi(t,x)\varphi(y,z) + Q(\psi(t,x), \psi(y,z))
\]

\[
x\,y\,z\,t = \chi(x,y,z,t)\,\xi^{2} \qquad (x,y,z,t \in V)
\]

\[
      \Lambda^{2}V \xrightarrow{\ \alpha\ } \tilde W
      \]

\begin{align}
\mathrm{Im}\ L_{X} &= \bigcap_{0 \le i \le n} \mathrm{Ker}\ p^{X}_{i}, &
\mathrm{Ker}\ L_{X} &= \sum_{n \le i \le 2n} \mathrm{Im}\ p^{X}_{i}, \\
\mathrm{Im}\ \Lambda_{X} &= \bigcap_{n \le i \le 2n} \mathrm{Ker}\ p^{X}_{i}, &
\mathrm{Ker}\ \Lambda_{X} &= \sum_{0 \le i \le n} \mathrm{Im}\ p^{X}_{i},
\end{align}

\begin{align}
p^{X}_{i} L_{X} &= 0 \ \text{ si } 0 \le i \le n, &
L_{X} p^{X}_{i} &= 0 \ \text{ si } n \le i \le 2n, \\
\uncertain{p^{X}_{i}}\, \Lambda_{X} &= 0 \ \text{ si } n \le i \le 2n, &
\Lambda_{X} p^{X}_{i} &= 0 \ \text{ si } 0 \le i \le n,
\end{align}

\begin{equation}
p^{X}_{n} L_{X} = L_{X} p^{X}_{n} = p^{X}_{n} \Lambda_{X} = \Lambda_{X} p^{X}_{n} = 0 .
\end{equation}

\begin{equation}
\Lambda_{X} L_{X} = \mathrm{id}_{X} - \sum_{n \le i \le 2n} p^{X}_{i}, \qquad
L_{X} \Lambda_{X} = \mathrm{id}_{X} - \sum_{0 \le i \le n} p^{X}_{i},
\end{equation}

\begin{equation}
\varphi_{*} \Lambda_{Y} \varphi^{*} = \mathrm{id}_{X} - \sum_{0 \le i \le 2n} p^{X}_{i}
\end{equation}

\begin{equation}
T_{X} = \mathrm{id}_{X} + \varphi_{*} \Lambda_{Y} \varphi^{*},
\end{equation}

\begin{equation}
L^{2}_{X} \Lambda_{X} + 2\,L_{X} \Lambda_{X} L_{X} + \Lambda_{X} L^{2}_{X}
 = \struck{L_{X}(\mathrm{id}_{X} + \varphi_{*}\Lambda_{Y}\varphi^{*}) + (\mathrm{id}_{X} + \varphi_{*}\Lambda_{Y}\varphi^{*})L_{X}}
 \ \add{L_{X} T_{X} + T_{X} L_{X}},
\end{equation}

\[
i \le n-2 \ \Longrightarrow \ 2(n-1) - i \ge n
\]

\[
H^{d}(X) = \struck{\ill{}} \coprod_{\substack{i+j \le n \\ i,j \ge 0}} L^{j}_{X} P^{i}(X)
 \ \oplus \coprod_{j \ge n} P^{j}(X)
\]

\[
H^{d}(Y) = \coprod_{\substack{i+j \le n-1 \\ i,j \ge 0}} L^{j}_{Y} P^{i}(Y)
\]

\[
\coprod_{\substack{i+j \le n-1 \\ i,j \ge 0}} L^{j}_{X} P^{i}(X)
\]

\[
P^{i}(X) \longrightarrow H^{d}(Y) = \coprod_{\substack{i+j \le n-1 \\ i,j \ge 0}} L^{j}_{Y} P^{i}(Y) \ ;
\]

\[
L^{0}_{X} P^{n-1}(X) = P^{n-1}(X) \longrightarrow \struck{\ill{}}\ \varphi^{*} L^{0}_{Y} P^{n-1}(Y) = P^{n-1}(Y).
\]

\[
E^{n-1} = H^{n-1}(Y)/\mathrm{Im}\ H^{n-1}(X) = P^{n-1}(Y)/\mathrm{Im}\ P^{n-1}(X)
\]

\[
\simeq \mathrm{Ker}\bigl(\varphi_{*} : H^{n-1}(Y) \to H^{n+1}(X)\bigr)
\]

\[
\text{(b)} \qquad H^{d}(Y) \simeq
 \coprod_{\substack{i+j \le n-1 \\ i,j \ge 0}} L^{j}_{X} P^{i}(X) \ \oplus\ E^{n-1}
\]

\[
\varphi_{*} : H^{d}(Y) \longrightarrow H^{d+2}(X)
\]

\[
\mathrm{Ker}\ \varphi_{*} \ \struck{=\ \ill{}}\ \simeq\ \mathrm{Ker}\ E^{n-1}(Y)
\]

\[
\mathrm{Im}\ \varphi_{*} = \mathrm{Im}\ \varphi_{*}\varphi^{*} = \mathrm{Im}\ L_{X}
 = \bigcap_{0 \le i \le n} \mathrm{Ker}\ p^{X}_{i}
\]

\[
\varphi^{*} L_{X} = L_{Y} \varphi^{*}, \qquad L_{X} \varphi_{*} = \varphi_{*} L_{Y}
\]

\[
p^{X}_{i} \varphi_{*} = 0 \ \text{ si } 0 \le i \le n, \qquad
p^{X}_{j} \varphi_{*} = \varphi_{*} p^{Y}_{j-2} \ \text{ si } \struck{n+1} \le i \le 2n
\]

\[
\varphi_{*} \varphi^{*} = L_{X}, \qquad \varphi^{*} \varphi_{*} = L_{Y}
\]

\[
\coprod_{\substack{i+j = n-1 \\ i,j \ge 0}} L^{j}_{X} P^{j}(X) \ \oplus\ E^{(n-1)}
 \ = \ \coprod_{i \ge n+1} P^{j}(X) \ \oplus\ E^{(n-1)}
\]

\[
\coprod_{0 \le i \le n-1} P^{i}(X) + E^{n-1} \ \simeq\
\coprod_{0 \le i \le n-1} P^{i}(Y),
\]

\[
\longrightarrow \coprod_{\substack{i+j \le n-2 \\ i,j \ge 0}} L^{j-1}_{X} P^{i}(X) .
\]

\[
\Lambda_{Y} \varphi^{*} = \varphi^{*} \Lambda_{X}\Bigl(\mathrm{id}_{X} - \sum_{n+1 \le j \le 2n} p^{j}_{X}\Bigr)
 = \Bigl(\mathrm{id}_{Y} - \struck{\sum_{n+1 \le j \le 2(n-1)}} p^{j}_{X}\Bigr) \varphi^{*} \Lambda_{X}
\]

\[
\varphi^{*} \Lambda_{X} p^{j}_{X} = p^{j-2}_{Y} \varphi^{*} \Lambda_{X} \struck{\ill{}}
\quad \text{si } j \ge n+1
\]

\[
\varphi_{*} \Lambda_{Y} = \Bigl(\mathrm{id}_{X} - \sum_{0 \le i \le n-1} p^{i}_{X}\Bigr) \Lambda_{X} \varphi_{*}
 = \Lambda_{X} \varphi_{*} \Bigl(\mathrm{id}_{Y} - \sum_{0 \le i \le n-1} p^{i}_{Y}\Bigr)
\]

\[
p^{i}_{X} \Lambda_{X} \varphi^{*} = \Lambda_{X} \varphi^{*} p^{i}_{Y} = p^{i}_{X} \varphi_{*} \Lambda_{Y}
\quad \text{si } 0 \le i \le n-1
\]

\[
\varphi_{*} \Lambda_{Y} \varphi^{*}
 = \underbrace{\varphi_{*}\varphi^{*} \Lambda_{X}}_{L_{X}\Lambda_{X}}
   \Bigl(\mathrm{id}_{X} - \sum_{n+1 \le j \le 2n} p^{j}_{X}\Bigr)
 = \underbrace{\mathrm{id}_{X} - \sum_{0 \le j \le 2n} p^{j}_{X}}_{q}
\]

\[
\varphi_{*} \Lambda^{m}_{Y} \varphi^{*}
 = \struck{\ill{}} \Bigl(\mathrm{id} - \sum_{0 \le i \le n} p^{i}_{X}\Bigr)
   \Lambda^{m-1}_{X} \Bigl(\mathrm{id} - \sum_{n+1 \le j \le 2n} p^{j}_{X}\Bigr)
 = q\,\Lambda^{m-1}_{X}\,q
\]

\[
\varphi^{*} \Lambda_{X} \varphi_{*} = \mathrm{id}_{Y} - p_{E}
\]

\[
\Lambda_{Y} = \Lambda_{Y} \varphi^{*} \Lambda_{X} \varphi_{*}
 = \varphi^{*} \Lambda_{X} \Bigl(\mathrm{id}_{X} - \sum_{n+1 \le j \le 2n} p^{j}_{X}\Bigr) \Lambda_{X} \varphi_{*}
\]

\[
\boxed{\ \Lambda_{Y} = \varphi^{*} \Lambda^{2}_{X} \varphi_{*}\ }
\]

\[
\varphi_{*}\, p^{i,j}_{Y}\, \varphi^{*} = p^{i,j+1}_{X}
\qquad \text{pour } 0 \le i+j \le n-1,\ i,j \ge 0 \ \struck{\ill{}}
\]

\[
  \sum_{i} H^{2(n-1)-i}(Y) \otimes H^{i}(X) = H^{2(n-1)}(X \times Y)
  = H^{m-1}(X \times Y), \qquad m = \dim X \times Y .
\]

\[
  \Lambda_{X}\varphi_{*}L_{Y} + L_{X}\Lambda_{X}\varphi_{*}
  = (\Lambda_{X}L_{X} + L_{X}\Lambda_{X})\,\varphi_{*}
  = (\varphi_{*}\Lambda_{Y}\varphi^{*} + \mathrm{id}_{X})\,\varphi_{*}
\]

\[
  C(X) \Longleftrightarrow \bigl[\,C(Y) \text{ et } A^{\circ}(X \times Y)\,\bigr]
\]

\[
  C(X) \Longrightarrow \bigl(A^{\circ}(X \times Y) \text{ et }
  A^{\circ}(Y \times Z) \text{ et } A^{\circ}(Z \times T) \dots\bigr)
\]

\[
  H^{2n-i}(X) \longrightarrow H^{2n-i-2}(Y) \longrightarrow H^{i}(Y)
  \longrightarrow H^{i}(X)
\]

\[
  (\Lambda_{X}\varphi_{*})\,L_{Y} + L_{X}\,(\Lambda_{X}\varphi_{*})
  = (\varphi_{*}\Lambda_{Y}\varphi^{*} + \mathrm{id}_{X})\,\varphi_{*} .
\]

\[
  u^{b} - \sigma_{1}(w)\,u^{b-1} + \dots + \struck{(\pm 1)} (-1)^{b}\sigma_{b}(w) = 0
\]

\[
  H^{2m-2}(X) \xrightarrow{\ \varphi^{*}\ } H^{2m-2}(Y)
  \xrightarrow{\ \varphi_{*}\ } H^{2m}(X)
  \xrightarrow{\ L_{X}\ } H^{2m+2}(X)
\]

\[
  H^{2m}(X) \xrightarrow{\ \varphi^{*}\ } H^{2m}(Y)
  \xrightarrow{\ \varphi_{*}\ } H^{2m+2}(X)
\]

\[
  u_{i}\,\zeta^{n-i} : H^{i}(X) \longrightarrow H^{2n-i}(X) \longrightarrow H^{i}(X)
\]

\[
  T_{k} = \mathrm{id} - \sum_{i<k} p_{i} - \sum_{j>2n-k} p_{j} ;
\]

\[
  H^{k-2j} \xrightarrow{\ \zeta^{j}\ } \zeta^{j}H^{k-2j}
\]