\[
  \boxed{\langle i, \varepsilon \rangle > 0}
\]

\[
  \Gamma' = \Gamma/W
  = \bigl[\mathrm{Hom}(G_{m}, T)/(\mathrm{Norm}_{G}(T)/T)\bigr](\mathbb{C})
  = \bigl[\mathrm{Hom}(G_{m}, G)/G\bigr](\mathbb{C})
\]

\[
  I(G) \longrightarrow \mathrm{Eff}(G)
\]

\[
  I^{*}(G) \longrightarrow \mathrm{Pol}(G) \times \mathrm{Eff}(G)
\]

\[
  \mathbb{Z}(M) \to \mathbb{Z}(\mathbb{Z})
  \quad \text{d'où} \quad
  G_{m,\mathbb{Z}} \xrightarrow{\ \alpha\ } S
\]

\[
  z = (\mathrm{id}_{A} \times i)_{*}(\Theta)
\]

\[
  z^{2} = i'_{*}(\Theta)^{2} = i'_{*}(\Theta^{2} c')
\]

\[
  K(z^{2}) = K(\Theta^{2} c') = K(\Theta^{2} \cdot 1_{A} \times c)
  = K(p_{*}(\Theta^{2}) \cdot c)
\]

\[
  \Theta = \Theta_{0} \times 1_{P} \quad \text{d'où} \quad
  \Theta^{2} = \Theta_{0}^{2} \times 1_{P} \quad \text{d'où}
\]

\[
  p_{*}(\Theta^{2}) = p'_{*}(\Theta_{0}^{2}) \times
  \mathrm{id}_{P*}(1_{P}) = p'_{*}(\Theta_{0}^{2}) \times 1_{P} .
\]

\[
  \Theta_{0}^{2} = \Delta^{2} + 0 + 0 - 2(0,0) - 2(0,0) + 2(0,0)
  = -2(0,0)
\]

\[
  p_{*}(\Theta^{2}) = -2(0 \times 0 \times P)
\]

\[
  K(s_{*}(z^{2})) = -2\, c(0,0)
\]

\[
  \mathrm{Cyc}(X)/N(X) \simeq A^{*}(X)/N'^{*}(X).
\]

\[
  f^{*} : A^{*}(Y) \to A^{*}(X)
\]

\[
  A'^{*}(X) \to A'^{*}(Y)
\]

\[
  S_{T \times T'}(\mathfrak{t} \times a') = S_{T}(\mathfrak{t}) = 0, \qquad
  S_{T \times T'}(a \times \mathfrak{t}') = S_{T'}(\mathfrak{t}') = 0
\]

\[
  Z_{1}(\mathfrak{t}_{1}) = Z(\mathfrak{t}), \qquad Z'(\mathfrak{t}'_{1}) =
  Z'(\mathfrak{t}'),
\]

\[
  Z_{1}(\mathfrak{t}'_{1}) = Z(\mathrm{pr}_{1*}(\mathfrak{t}'_{1})) = 0
\]

\[
  Z_{2} = Z_{1} + Z'_{1}, \qquad \mathfrak{t}_{2} = \mathfrak{t}_{1} +
  \mathfrak{t}'_{1},
\]

\[
  Z_{2}(\mathfrak{t}_{2}) = \mathfrak{z} + \mathfrak{z}' .
\]

\[
  f^{*}(Z'(\mathfrak{t}')) = Z_{1}(\mathfrak{t}'),
\]

\[
  \mathfrak{t} = \bigl((x) - (y)\bigr) \times \bigl((x') - (y')\bigr) =
  (x, x') - (x, y') - (y, x') + (y, y')
\]

\[
  u^{*}_{Z} : A^{*}_{Y}(k) \longrightarrow A^{i}_{\mathrm{alb}}(X),
\]

\[
  Z(\mathfrak{t}) = u_{Z}(S(\mathfrak{t}))
\]

\[
  u_{Z} : A_{Y}(k) \longrightarrow A^{i}_{\mathrm{alb}}(X) ,
\]

\[
  \varphi : A' \to A_{Y}
\]

\[
  u_{Z} \circ \varphi(k) = v_{Z'}
\]

\[
  Z(U(a)) = Z'(a)
\]

\[
  Z(\mathfrak{t}) = Z'(S(\mathfrak{t}))
\]

\[
  Z(U(a)) = Z'(S(U(a))) = Z'(a) \qquad \text{O.K.}
\]

\[
  S_{Y}(U(a)) = na
\]

\[
  Z'(S_{Y}(\mathfrak{t})) = n\, Z(\mathfrak{t}) \qquad \text{$\mathfrak{t}$ de
  degré $0$ sur $Y$.}
\]

\[
  S_{Y}(U(a)) = na = S_{Y}(n\mathfrak{t}),
\]

\[
  U(a) = n\mathfrak{t}
\]

\[
  Z'(S_{Y}(U(a))) = n\, Z(U(a)), \qquad Z'(S_{Y}(U(a))) = Z'(na), \quad
  n\,Z(U(a)) = n\,Z'(a) .
\]

\[
  Z'(S_{Y}(\mathfrak{t})) = n\, Z(\mathfrak{t}) .
\]

\[
  \varphi : Y(k) \longrightarrow A^{i}_{\mathrm{alb}}(X)
\]

\[
  A^{i}_{\mathrm{alb}}(X) \times \mathrm{Homalg}(A_{Y},
  A^{i}_{\mathrm{alb}}(X)) ,
\]

\[
  A_{0\,\mathrm{alb}}(A) = A^{\alpha}_{\mathrm{alb}}(A)^{(\text{alg.\ équiv.\
  }0)} \qquad (\alpha = \dim A)
\]

\[
  \varphi : P^{j}(X') \longrightarrow P^{i}(X) .
\]

\[
  u_{Z} : A(k) \longrightarrow A^{i}_{\mathrm{alb}}(X)
\]

\[
  a({}^{t}Z) : H^{2(n-i)+1}(X)(n-i) \longrightarrow H^{1}(A)
\]

\[
  u_{Z} = 0 \Longrightarrow a({}^{t}Z) : H^{2(n-i)+1} \to H^{1} \text{ est
  nul}
\]

\[
  u_{Z'} : C(k) \longrightarrow A^{i}_{\mathrm{alb}}(X)
\]

\[
  a({}^{t}Z') : H^{2(n-i)+1}(n-i) \longrightarrow H^{1}(C)
\]

\[
  u_{Z} = 0 \Longleftrightarrow u_{Z'} = 0, \qquad a({}^{t}Z) = 0
  \Longleftrightarrow a({}^{t}Z') = 0 .
\]

\[
  H^{N}(X \times C) \simeq H^{N}(X) \otimes H^{0}(C) + H^{N-1}(X) \otimes
  H^{1}(C) + H^{N-2}(X) \otimes H^{2}(C)
\]

\[
  \mathbb{T}_{K}(A) = \mathbb{T}_{\ell}(A)
\]

\[
  H^{2}(A, \mathbb{Q}_{\ell}(1))^{\mathrm{alg}} = H^{2}(A,
  \mathbb{Q}_{\ell}(1))^{\mathfrak{g}}
\]

\[
  D^{2i}(A, \mathbb{Q}_{\ell}(i)) = H^{2i}(A, \mathbb{Q}_{\ell}(i))^{\mathrm{alg}} = H^{2i}(A, \mathbb{Q}_{\ell}(i))^{U} \,]
\]

\[
  \begin{cases}
    H^{1}(A_T, \mathbb{Q}_{\ell}) \xrightarrow{\ \sim\ } H^{1}(T, \mathbb{Q}_{\ell}) \\
    H^{1}(B_T, \mathbb{Q}_{\ell}) \xleftarrow{\ \sim\ } H^{2n-1}(T, \mathbb{Q}_{\ell})(n-1)
  \end{cases}
\]

\[
  u_Z : A_Y \longrightarrow A_X
\]

\[
  S_X(Z(\mathfrak{A})) = u(S_Y(\mathfrak{A})) .
\]

\[
  \varphi_X : X \longrightarrow A^{1}_{X} = X'
\]

\[
  Z' = (\varphi_X \times \mathrm{id}_Y)_{*}(Z) \in \mathrm{Cyc}_{m}(X' \times Y)
\]

\[
  B = B_X = \underline{\mathrm{Pic}}^{00}_{A^{1}_{X}/k} \simeq \underline{\mathrm{Pic}}^{0}_{A_X/k}
\]

\[
  \pi : B \times X' \times Y \longrightarrow B \times Y
\]

\[
  c = \pi_{*}(\mathfrak{z}'' . \theta)
\]

\[
  c' \in \mathrm{Cl}(B, \struck{A} Y) .
\]

\[
  u_Z : A_Y \longrightarrow A_X ,
\]

\[
  S_X({}^{t}Z(\mathfrak{A})) = u_Z(S_Y(\mathfrak{A}))
\]

\[
  \Theta(S_X({}^{t}Z(\mathfrak{A}))) = \Theta(u_Z(S_Y(\mathfrak{A})))
\]

\[
  c'(\mathfrak{A}) = \Theta(u_Z(S_Y(\mathfrak{A}))) \qquad \text{dans } \mathrm{Pic}^{0}(B)
\]

\[
  D(\varphi_{X*}({}^{t}Z(\mathfrak{A}))) = C(\mathfrak{A}) \qquad \text{dans } \mathrm{Cyc}^{1}(B)
\]

\[
  a(\gamma^{n}(Z)) : H^{1}(X) \longrightarrow H^{1}(Y)
\]

\[
  u : B_X \longrightarrow A_Y , \qquad B_X = \underline{\mathrm{Pic}}^{00}_{X/k} ,
\]

\[
  u(\mathrm{cl}(D)) = S_Y({}^{t}Z(D))
\]

\[
  \mathfrak{z}(\Theta(b)) \simeq C(b)
\]

\[
  C = \pi_{*}\bigl((1_Y \times \Theta) . (\mathfrak{z} \times 1_{B_X})\bigr)
\]

\[
  S_Y\bigl(C((b) - (0))\bigr) = u(b) \qquad \text{i.e.} \qquad S_Y(\mathfrak{z}(\Theta(b))) = u(b) .
\]

\[
  \Theta_Y \circ Z \circ {}^{t}\Theta_X \quad \text{sur } B_X \times B_Y ,
\]

\[
  S_{X}(Z(D)) = {}^{t}(u_{Z})(D)
\]

\[
  {}^{t}Z(D) = {}^{t}(u_{Z})(D)
\]

\[
  \underline{\mathrm{Pic}}_{X/k} \longrightarrow \underline{\mathrm{Pic}}_{Y/k} \ !
\]

\[
  a(\gamma(Z)) : H^{1}(Y, \mathbb{Q}_{\ell}) \longrightarrow H^{2n-1}(X, \mathbb{Q}_{\ell})(n-1)
\]

\[
  H^{1}(Y, \mathbb{Z}_{\ell}) \longrightarrow H^{2n-1}(X, \mathbb{Z}_{\ell})(n-1) \longrightarrow
  \underline{H^{2n-1}(X, \mathbb{Z}_{\ell})(n-1)}
\]

\[
  H^{1}(X, \mathbb{Q}_{\ell}) \longrightarrow H^{2n-1}(X, \mathbb{Q}_{\ell})(n-1)
\]

\[
  u : B_{X} \longrightarrow A_{X}
\]

\[
  B_{X} \xrightarrow{\ \rho\ } B_{Y} \xrightarrow{\ \lambda\ } A_{Y} \xrightarrow{\ \sigma\ } A_{X}
\]

\[
  v : C \longrightarrow X \times Y ,
\]

\[
  v_{1} : C \longrightarrow X , \qquad v_{2} : C \longrightarrow Y .
\]

\[
  B_{X} \xrightarrow{\ v_{1}^{*}\ } B_{C} \xrightarrow{\ \text{dualité}\ } A_{C}
  \xrightarrow{\ v_{2*}\ } A_{Y} .
\]

\[
  \gamma(c) : H^{1}(A_{X}, \mathbb{Q}_{\ell}) \simeq H^{1}(X, \mathbb{Q}_{\ell})
  \longrightarrow H^{1}(B_{X}, \mathbb{Q}_{\ell}) \xleftarrow{\ \sim\ }
  H^{2n-1}(X, \mathbb{Q}_{\ell})(n-1) .
\]

\[
  u_{D} : A_{Y} \longrightarrow B_{X}
\]

\[
  a(\gamma(D)) : H^{2n-1}(X, \mathbb{Q}_{\ell})(n-1) \longrightarrow H^{1}(Y, \mathbb{Q}_{\ell}) .
\]

\[
  a(\gamma(Z)) : H^{1}(Y, \mathbb{Q}_{\ell}) \longrightarrow
  H^{2n-1}(X, \mathbb{Q}_{\ell})(n-1)
\]

\[
  u_{Z} : B_{X} \longrightarrow A_{Y} \quad (\text{une isog.})
\]

\[
  H^{2n-1}(X, \mathbb{Q}_{\ell})(n-1) \longrightarrow H^{1}(B_{X}, \mathbb{Q}_{\ell})
\]

\[
  \varphi_{X}^{-1} = \mu^{-1}\Psi_{X}\lambda
\]

\[
  H^{1}(X, \mu_{N}) \simeq {}_{N}\mathrm{Pic}(X)
\]

\[
  0 \to {}_{N}\mathrm{Pic}^{0}(X) \to H^{1}(X, \mu_{N}) \longrightarrow {}_{N}\mathrm{NS}(X) \longrightarrow 0
\]

\[
  H^{1}(X, \mathbb{Z}_{\ell}(1)) \xleftarrow{\ \simeq\ }
  T_{\ell}(\underline{\mathrm{Pic}}^{0}_{X/k})
\]

\[
  H^{1}(X, \mathbb{Z}_{\ell}) \xleftarrow{\ \varphi_{X}^{*}\ } H^{1}(A_{X}, \mathbb{Z}_{\ell}) .
\]

\[
  H^{2n-1}(X, \mathbb{Z}/N\mathbb{Z}) \simeq H^{1}(X, \mathbb{Z}/N\mathbb{Z})^{\vee}(-n)
\]

\[
  0 \to ({}_{N}\mathrm{NS}(X))^{\vee} \to H^{2n-1}(X, \mathbb{Z}/N\mathbb{Z})
  \to {}_{N}A_{X}(k)(-n) \to 0
\]

\[
  0 \to (\mathrm{NS}(X)[\ell])^{\vee} \to H^{2n-1}(X, \mathbb{Z}_{\ell})(n-1)
  \to H^{1}(B_{X}, \mathbb{Z}_{\ell}) \to 0
\]

\begin{align*}
  &\pi_{1}^{\ell}(\underline{\mathrm{Alb}}^{0}_{X}) \xleftarrow[\ \sim\ ]{(t)} \pi_{1}^{\ell}(X)^{\mathrm{ab}}(\ell)
  && \text{i.e. \emph{torsion près}} \quad [\text{c'est toujours surjectif ?}] \\
  &\pi_{1}(\underline{\mathrm{Pic}}^{00}_{X}) \simeq H^{1}(\underline{\mathrm{Alb}}^{0}_{X})(1)
  \xrightarrow[\ \sim\ ]{t} H^{1}(X)(1)
  && \text{i.e. \emph{torsion près}} \quad [\text{c'est toujours injectif}] \\
  &H^{1}(\underline{\mathrm{Alb}}^{0}_{X}) \xrightarrow[\ \sim\ ]{t} H^{1}(X)
  && \text{i.e. \emph{torsion près}} \quad [\text{c'est toujours injectif}] \\
  &H^{1}(\underline{\mathrm{Pic}}^{00}_{X}) \simeq \pi_{1}(\underline{\mathrm{Alb}}_{X})(-1)
  \xleftarrow[\ \sim\ ]{(t)} H^{2n-1}(X)(n-1)
  && \text{toujours}
\end{align*}

\[
  \left\{
  \begin{array}{lll}
    \underline{\mathrm{Alb}}_{X} & \text{correspond :} & H^{1}(X) \\
    \underline{\mathrm{Pic}}_{X} & \text{------------} & H^{2n-1}(X)(n-1)
  \end{array}
  \right.
\]

\[
  \mathcal{VA}^{\circ} \xrightarrow{\ *\ } \mathcal{VA} \qquad A \rightsquigarrow A^{*}
\]

\[
  \mathcal{VA}^{\circ} \xrightarrow{\ C^{*}\ }
  \begin{pmatrix}
    \text{Anneaux gradués comm.,} \\
    \text{à degrés} \geqslant 0, \\
    C^{0} \simeq \mathbb{Z}
  \end{pmatrix}
  \qquad A \rightsquigarrow C^{*} \quad \text{cycles mod équiv. alg.}
\]

\[
  C^{1}(A \times B) \longrightarrow \mathrm{Hom}(A, B^{*})
\]

\[
  \mathcal{VA} \xrightarrow{\ T_{\mathbb{Q}_{\ell}}\ } \mathrm{Mod}(\mathbb{Q}_{\ell})
  \qquad A \rightsquigarrow T_{\ell}(A) \otimes_{\mathbb{Z}_{\ell}} \mathbb{Q}_{\ell}
\]

\[
  \mathcal{VA}^{\circ} \xrightarrow{\ H^{1}\ } \mathrm{Mod}(\mathbb{Q}_{\ell})
  \qquad A \rightsquigarrow H^{1}(A, \mathbb{Q}_{\ell}) =
  \mathrm{Hom}_{\mathbb{Q}_{\ell}}(T_{\mathbb{Q}_{\ell}}(A), \mathbb{Q}_{\ell})
\]

\[
  H^{*}(A) = {\textstyle\bigwedge^{*}} H^{1}(A)
\]

\[
  \mathcal{VA} \xrightarrow{\ \mathcal{C}^{*}\ } (\mathrm{Ab}) \qquad
  A \rightsquigarrow \mathcal{C}^{*}(A) \quad
  \text{anneau de Chow (des cycles mod \emph{équiv. alg})}
\]

\[
  \mathcal{C}^{i} \longrightarrow H^{2i} \otimes \mathbb{Q}_{\ell}(i) =
  {\textstyle\bigwedge^{2i}} H^{1} \otimes \mathbb{Q}_{\ell}(i) \qquad
  \mathcal{C}^{i}(A) \xrightarrow{\ \gamma_{A}^{2i}\ } H^{2i}(A, \mathbb{Q}_{\ell})(i)
\]

\[
  \begin{array}{rcl}
  H^{2n}(A) \simeq {\textstyle\bigwedge^{2n}} H^{1}(A) & \xrightarrow[\ \sim\ ]{\eta_{A}} & \mathbb{Q}_{\ell}(-n) \\[2pt]
  H^{2n}(A^{*}) \simeq {\textstyle\bigwedge^{2n}} H^{1}(A^{*}) & \xrightarrow[\ \sim\ ]{\eta_{A^{*}}} & \mathbb{Q}_{\ell}(-n) \\[2pt]
  \wr & & \wr\ \text{(can.)} \\
  \mathbb{Q}_{\ell}(2n) & & \mathbb{Q}_{\ell}(-2n)
  \end{array}
\]

\[
  \begin{array}{ccc}
  u^{2n} : H^{2n}(A) & \longrightarrow & H^{2n}(A') \\
  \wr & & \wr \\
  \mathbb{Q}_{\ell}(-n) & & \mathbb{Q}_{\ell}(-n)
  \end{array}
\]

\[
  u_{*} : H^{*}(A) \longrightarrow H^{*}(A')
\]

\[
  u_{*}(1_{A}) = d \cdot 1_{A'}
\]

\[
  H^{i}(A) \longrightarrow H^{i+2(n'-n)}(A')(n'-n)
\]

\[
  u^{1} : H^{1}(A') \longrightarrow H^{1}(A) \qquad
  (\text{d'où } u^{*} : H^{*}(A') \to H^{*}(A) \text{ par } {\textstyle\bigwedge^{*}})
\]

\[
  u_{*}(1_{A}) \in H^{2(n'-n)}(A')(n'-n)
\]

\[
  u_{1} : T_{\mathbb{Q}_{\ell}}(A) \longrightarrow T_{\mathbb{Q}_{\ell}}(A'),
\]

\[
  0 \longrightarrow K \longrightarrow H^{1}(A'^{*}) \longrightarrow H^{1}(A)
  \longrightarrow 0
\]

\[
  \begin{array}{ccccc}
    \bigwedge^{2n'} H^{1}(A'^{*}) & \simeq & \bigwedge^{2n} H^{1}(A) & \otimes
    & \bigwedge^{2(n'-n)} K \\
    \wr & & \wr & & \\
    H^{2n'}(A') & & H^{2n}(A) & &
  \end{array}
\]

\[
  H^{i}(A) \times H^{2n-i}(A) \longrightarrow H^{2n}(A) \simeq
  \mathbb{Q}_{\ell}(-n)
\]

\[
  H^{i}(A) \simeq H^{2n-i}(A)'(-n) \simeq \bigwedge^{2n-i}
  T_{\mathbb{Q}_{\ell}}(A)\,(-n)
\]

\[
  \begin{array}{rccc}
    u_{*} : & H^{i}(A) & \longrightarrow & H^{i+2(n'-n)}(A')(n'-n) \\
    & \wr & & \\
    & \bigwedge^{2n-i} T_{\mathbb{Q}_{\ell}}(A)\,(-n) & \longrightarrow &
    \bigwedge^{2n-i} T_{\mathbb{Q}_{\ell}}(A')\,(-n)
  \end{array}
\]

\[
  \bigwedge^{2n} u_{1} \otimes \mathbb{Q}_{\ell}(-n) : \bigwedge^{2n}
  T_{\mathbb{Q}_{\ell}}(A)(-n) \longrightarrow \bigwedge^{2n}
  T_{\mathbb{Q}_{\ell}}(A')(-n)
\]

\[
  \bar{u} = (\mathrm{id}_{A}, u) \qquad A \longrightarrow A \times A^{*}
\]

\[
  H^{*}(A) \xrightarrow{\ \bar{u}_{*}\ } H^{*}(A \times A^{*})(\cdot\cdot)
  \qquad\qquad
  H^{*}(A) \xleftarrow{\ \bar{u}^{*}\ } H^{*}(A \times A^{*})
\]

\[
  \begin{array}{ccc}
    H^{0}(A) & \longrightarrow & H^{2n}(A \times A^{*})(n) \\
    \times & & \times \\
    H^{2n}(A) & \longleftarrow & H^{2n}(A \times A^{*}) \\
    \wr & & \wr \\
    \mathbb{Q}_{\ell}(-n) & & \mathbb{Q}_{\ell}(-n)
  \end{array}
\]

\[
  H^{1}(A) \xleftarrow{\ (\mathrm{id}_{H^{1}(A)},\, u^{1})\ }
  H^{1}(A \times A^{*}) = H^{1}(A) \times H^{1}(A^{*})
\]

\[
  \bar{u}_{*}(1)
\]

\[
  C^{1}(A \times B) / C^{1}(A) \times C^{1}(B) \simeq \mathrm{Hom}(A, B^{*})
\]

\[
  \begin{array}{c}
    \bigwedge^{2}\bigl(H^{1}(A) \times H^{1}(B)\bigr)(1) \big/
    \bigwedge^{2} H^{1}(A)(1) \times \bigwedge^{2} H^{1}(B)(1) \simeq
    \bigl(H^{1}(A) \otimes H^{1}(B)\bigr)(1) \\
    \wr \\
    \mathrm{Hom}\bigl(T_{\ell}(A), T_{\ell}(B^{*})\bigr)
  \end{array}
\]

\[
    \chi(F \otimes L^{\otimes n}) \longrightarrow +\infty \quad \text{si }
    n \longrightarrow +\infty
  \]

\[
  n \geqslant n_{0} \Longrightarrow H^{i}(X, L^{\otimes n}) = 0 \quad
  \text{pour } i \geqslant 2
\]

\[
  \text{pour } n \geqslant n_{0} \Longrightarrow X \dashrightarrow
  \mathbb{P}\, H^{0}(X, L^{\otimes n}) \text{ partout défini}
\]

\[
  \begin{array}{ccccc}
    \mathring{P}^{+} & \subset & P^{+} & \subset & \overline{P^{+}} \\
    \cup & & & & \cup \\
    \mathring{Q}^{+} & \subset & Q^{+} & = & \overline{Q^{+}} \\
    \cup & & & & \cup \\
    \mathring{R} & = & R & \subset & \overline{R}
  \end{array}
\]

\[
  \overline{P^{+}}^{\,\circ} = P^{+\circ} \subset Q^{+}
\]

\[
  P^{+} \ni x \iff \exists\, C > 0,\ n > 0, \text{ tel que }
  \gamma(C) = n x \iff \ell(nx) \geqslant 2 \text{ si } n \text{ grand}
\]

\[
  Q^{+} \ni x \iff \left\{\begin{array}{l} x^{2} \geqslant 0 \\ x \cdot a
  \geqslant 0 \end{array}\right. , \qquad \mathring{Q}^{+} \ni x \iff
  x^{2} > 0,\ x \cdot a \geqslant 0
\]

\[
  R \ni x \iff \exists\, C \text{ \textit{ample}},\ n > 0, \text{ tel que }
  \gamma(C) = n x
\]

\[
  \boxed{\ x \in R \iff x \cdot P > 0 \text{ et } x^{2} > 0\ }
\]

\[
  \overline{R} \subset \overline{P}^{\,\circ} \cap Q = P^{\circ} =
  \overline{P}^{\,\circ},
\]

\[
  \mathring{P}^{\circ} \subset \mathring{Q}
\]

\[
  \boxed{\begin{array}{l}
    R = \mathring{\overline{R}} = \mathring{\overline{P^{\circ}}} \subset
    \mathring{Q} \\
    \overline{R} = P^{\circ} = \overline{P}^{\,\circ} \subset Q
  \end{array}}
\]

\[
  H^{2}(L^{\otimes n} \otimes K^{\otimes(2n-1)})
\]

\[
  \boxed{\left\{\begin{array}{l} D^{d} > 0 \\ D \cdot C > 0 \text{ si }
  C > 0 \end{array}\right.}
\]

\[
  \Big\Updownarrow
\]

\[
  \boxed{\ D \in P_{1}^{+\circ},\quad \alpha_{d-1} D^{d-1} \in P_{1}^{+}\ }
\]

\[
  D \in P_{1}^{+\circ} \Longrightarrow \alpha D^{d-1} \in P_{d-1}^{+\circ}
\]

\[
  \boxed{\begin{array}{ccccccc}
    P_{i}^{+\circ} & \subset & \mathcal{A}^{i} & \xrightarrow{\ \alpha_{i}\ } &
    \mathcal{A}_{d-i} & \supset & P_{d-i}^{+} \\
    \Vert & & & & & & \\
    P^{i}_{+} & & & & & & \\[1ex]
    P_{d-i}^{+\circ} & \subset & \mathcal{A}^{2d-i} &
    \xrightarrow{\ \alpha_{d-i}\ } & \mathcal{A}_{i} & \supset & P_{i}^{+}
    \\[1ex]
    & & \mathrm{int}\bigl(P_{i}^{+\circ}\bigr) & \longrightarrow &
    P_{d-i}^{+} & &
  \end{array}}
\]

\[
  c_{i}(E)
\]

\[
  D \cdot C \geqslant 0 \text{ si } C > 0
\]

\[
  \Big\Downarrow ?
\]

\[
  D^{i} X_{i} \geqslant 0 \text{ si } X_{i} > 0 \quad ??
\]

\[
  D \in P_{1}^{+\circ} \Longrightarrow D^{i} \in P_{i}^{+\circ} \quad
  \text{pour } i \leqslant d-1
\]

\[
  \begin{array}{ccc}
    D^{d} = & D^{i} & D^{d-i} \\
    & \cap & \cap \\
    & P_{i}^{+\circ} & P_{d-i}^{+\circ}
  \end{array}
  \qquad (0 < i < d)
\]

\[
  \begin{array}{ccc}
    \boxed{P_{i}^{+\circ} \cdot P_{d-i}^{+\circ} \geqslant 0 \ ?} & &
    \\[1ex]
    \Big\Updownarrow & \Longleftarrow & P_{i}^{+\circ}\, P_{j}^{+\circ}
    \subset P_{i+j}^{+\circ} \\[1ex]
    \boxed{\alpha_{i}(P_{i}^{+\circ}) \subset P_{d-i}^{+}} & & \\[1ex]
    \boxed{\alpha_{d-i}(P_{d-i}^{+\circ}) \subset P_{i}^{+}} & &
  \end{array}
\]

\[
  \begin{array}{l}
    D \cdot C \geqslant 0 \text{ si } C \geqslant 0 \\
    D^{2} \cdot D' \geqslant 0 \text{ si } D' \geqslant 0 \\
    \quad \Big\Downarrow ? \\
    D^{3} \geqslant 0
  \end{array}
\]

\[
  P_{1}^{+\circ} \subset \mathcal{A}^{1} \longrightarrow \mathcal{A}_{2}
  \supset P_{2}^{+}
\]

\[
  (D' + nD)^{3} = D'^{3} + 3n D'^{2} D + 3n^{2} D' D^{2} + n^{3} D^{3}
\]

\[
  \boxed{D^{3} \geqslant 0}
\]

\[
B^{1}(X) \longrightarrow J^{1}(Y) \qquad [\,B^{1}(X) \text{ contenant } K\,]
\]

\[
0 \to K \xrightarrow[\text{mod gpes finis}]{\text{inj.}} M^{1}(X) \longrightarrow
\underbrace{N^{1}(Y)/\operatorname{Im} N^{1}(X)}_{\text{partie « mobile » de } N^{1}(Y)}
\]

\[
\begin{cases}
X = S - \text{pt fermé} \\
Y = X \cap V(f)
\end{cases}
\]

\[
\begin{array}{lll}
\text{bij.\ [exc.\ } p\text{] si} & i \leqslant \frac{N-2}{2} \\[2pt]
\text{inj.\ [exc.\ } p\text{] si} & i \leqslant \frac{N-1}{2} \\[2pt]
\text{nul si} & i \geqslant \frac{N}{2}
\end{array}
\]

\[
\begin{array}{lll}
\text{bij.\ [exc.\ } p\text{] si} & i \leqslant \frac{N-3}{2} \\[2pt]
\text{inj.\ [exc.\ } p\text{] si} & i \leqslant \frac{N-2}{2} \\[2pt]
\text{nul [mod.\ gpes finis] si} & i \geqslant \frac{N-1}{2}
\end{array}
\]

\[
J^{1}(X) \xrightarrow{\ \sim\ } J^{1}(Y)^{\pi}
\]

\[
N^{1}(X) \xrightarrow{\ \sim\ } N^{1}(Y)^{\pi}
\]

\[
\mathcal{A}^{m+1}(X) \equiv 0 \qquad \bigl(\text{et } \mathcal{A}^{i}(X)
\equiv 0 \text{ si } i \geqslant m+1\bigr)
\]

\[
\begin{cases}
\text{surjectif [exc.\ } p\text{]} \\
\text{induit } \mathcal{A}^{r}_{i}(Y)^{\tau} \to \mathcal{A}^{r}_{i}(X)^{\tau}
\quad \text{surj.\ [exc.\ } p\text{]} \\
\text{induit } \mathcal{A}^{a}_{i}(Y) \to \mathcal{A}^{a}_{i}(X)
\quad \text{bij.\ [exc.\ } p\text{]}
\end{cases}
\]

\[
\Longrightarrow \qquad \mathcal{A}^{i}_{r}(X) \xrightarrow{E^{p}}
\mathcal{A}^{i+p}_{r}(X)
\]

\[
\mathcal{A}^{i}_{r}(U) \simeq \mathcal{A}^{i}_{r}(X)/\operatorname{Im}
\mathcal{A}^{i}_{r}(Y), \quad \text{de même pour } \mathcal{A}_{a}(U) \ldots
\]

\[
\begin{cases}
\mathcal{A}^{i}_{a}(E) \simeq P^{i}_{a}(X) & \text{(mod gpes finis) si } 2i \leqslant n \\
\mathcal{A}^{i}_{a}(E) \simeq 0 & \text{mod gpes finis si } 2i \geqslant n+1
\end{cases}
\]

\[
\begin{cases}
\mathcal{A}^{i}_{r}(E)^{a} \simeq I^{i}(X) & \text{mod gpes finis si } 2i \leqslant n+1 \\
\mathcal{A}^{i}_{r}(E)^{a} \simeq 0 & \text{mod gpes finis si } 2i \geqslant n+2
\end{cases}
\]

\[
\begin{aligned}
P_{a}(X) &\simeq \operatorname{Ker}\bigl(\mathcal{A}^{i}_{a}(X)
\xrightarrow{L^{n-2i}} \mathcal{A}^{n-i}_{a}(X)\bigr)
&& \text{partie « primitive » de la « coh.\ algébrique »} \\
I_{r}(X) &= \operatorname{Ker}\bigl(\mathcal{A}^{i}_{r}(X)^{a}
\xrightarrow{L^{n-2i+1}} \mathcal{A}^{n-i+1}_{r}(X)^{a}\bigr)
&& \text{partie « primitive » de la V.A.\ intermédiaire}
\end{aligned}
\]

\[
S \simeq S_{0} \times_{S'_{0}} S', \qquad S' \ [= \operatorname{Spec} A']
\text{ rev.\ fini normal de } S.
\]

\[
\begin{cases}
H^{*}(X) & \text{cohomologie } \ell\text{-adique} \\
P^{*}(X) & \text{partie primitive de la cohomologie}
\end{cases}
\]

\[
\begin{cases}
\mathcal{A}^{*}(X) = \mathcal{A}^{*}_{r}(X) & \text{anneau de Chow (équiv.\ rationnelle)} \\
N^{*}(X) = \mathcal{A}^{*}_{a}(X) & \text{id.\ pour équiv.\ algébrique} \\
J^{*}(X) = \mathcal{A}^{*}_{r}(X)^{a}
\end{cases}
\]

\[
0 \longrightarrow J^{*}(X) \longrightarrow \mathcal{A}^{*}(X) \longrightarrow
N^{*}(X) \longrightarrow 0
\]

\[
\begin{cases}
M^{*}(X) & \text{partie primitive de } N^{*}(X) \\
I^{*}(X) & \text{partie primitive de } J^{*}(X)
\end{cases}
\]

\[
\Longrightarrow
\begin{cases}
(\text{équiv.\ num.} = \tau\text{-équivalence}) \\
\text{les } \mathcal{A}^{i}_{a}(X) \text{ sont de type fini} \ldots
\end{cases}
\ \Downarrow
\]

\[
\begin{cases}
\mathcal{A}^{i}_{r}(X)^{a} \times \mathcal{A}^{j}_{r}(Y)^{a} \longrightarrow
\mathcal{A}^{i+j}_{r}(X \times Y)^{a} & \text{est une \ill{}} \\
\mathcal{A}^{i}_{r}(X)^{a} \times \mathcal{A}^{j}_{r}(X)^{a} \longrightarrow
\mathcal{A}^{i+j}_{r}(X)^{a} & \text{est une \ill{}}
\end{cases}
\]

\[
\xi = \gamma(F) - \gamma\bigl(\mathcal{O}(-n)^{N}\bigr) + \gamma(R)
\]

\[
\xi + \gamma\bigl(\mathcal{O}(-n)^{N}\bigr) = \gamma(F + R).
\]

\[
(\mathrm{ii}) \Longrightarrow (\mathrm{i}) \Longleftrightarrow (\mathrm{iv}) \Longrightarrow (\mathrm{iii}) \Longrightarrow (\mathrm{v})
\]

\[
(x+y) - (x) - (y) + (0) \sim 0 ,
\]

\[
\mathcal{A}_{0}(A)^{a} \longrightarrow A
\]

\[
\begin{array}{cc} X & Y \\ a & b \end{array}
\qquad X \times Y = Z
\qquad\qquad A \times A \to A
\]

\[
(\underset{\sim\, 0}{z} \times b) \times (\underset{\sim\, 0}{a \times z'}) \quad \text{dans} \quad Z \times Z = X \times Y \times X \times Y
\]

\[
(z \times z') \times (b \times a)
\]

\[
\mathcal{A}_{0}(X \times Y)^{\tau} \longrightarrow \mathcal{A}_{0}(X)^{\tau} \times \mathcal{A}_{0}(Y)^{\tau}
\]

\[
y_{y} = i_{y}^{*}(y')
\]

\[
f_{*}(y') = x
\]

\[
i_{y}^{*}(x) = i'^{*}_{y}(f^{*}f_{*}(y')) \qquad f^{*}f_{*}y' = y' + \alpha_{*}(u)
\]

\[
\begin{aligned}
i_{y}^{*}(x) &= i'^{*}_{y}(y') + i'^{*}_{y}(\alpha_{*}(u)) \\ &= y' + \alpha_{y*}(u_{y})
\end{aligned}
\qquad \text{Or } u_{y} = u, \ u \in A(C)
\]

\[
\boxed{A(Y(y)) = \mathrm{Im}\, A(X) + \mathrm{Im}\, A(C)}\,.
\]

\[
A^{i}(Y(y)) = \mathrm{Im}\, A^{i}(X) + \mathrm{Im}\, A^{i-j}(C)
\]