\[
  \bigl|\widetilde{K}\bigr| - \bigl|\widetilde{L}\bigr| \xrightarrow{\ \sim\ } |K| - |L| ,
  \qquad
  (\widetilde{K}, \widetilde{L}) \rightsquigarrow
\]

\[
  \widetilde{K} \xrightarrow{\ \pi\ } K ,
  \qquad
  \widetilde{L} = \pi^{-1}(L) \longrightarrow L ,
  \qquad
  \widetilde{L} \subset \widetilde{K}
\]

\[
  \widetilde{K} = \varinjlim_{x \in K \setminus L} K_{x} ,
  \qquad
  K_{x} = \{\, y \in K \mid y \leqslant x \,\}
\]

\[
  \Sigma(X_{i}, \partial X_{i}) = \bigl|\widetilde{K}_{i}\bigr| ,
  \qquad \widetilde{K}_{i}
\]

\[
  N = \Bigl( \sum_{\substack{f \in \pi_{0}(X \setminus K) = F \\ f \ \text{pas un disque}}}
  \bigl( 6g(f) + 4\nu(f) - 6 \bigr) \Bigr)
\]

\[
  2 - 2g = \chi = \sum_{f \in F} \bigl( 2 - 2g(f) - \nu(f) \bigr) + 1 - \lambda
\]

\[
  2g - 2 = \sum_{f \in F} \bigl( 2g(f) - 2 + \nu(f) \bigr) + (\lambda - 1)
\]

\[
  g - 1 = \frac{1}{2} \sum_{f \in F} \bigl[ 2g(f) - 2 + \nu(f) \bigr] + \frac{3}{2}(\lambda - 1)
\]

\[
  3(g - 1) = \frac{3}{2} \sum_{f \in F} \bigl[ 2g(f) - 2 + \nu(f) \bigr] + \frac{3}{2}(\lambda - 1)
  = \sum_{f \in F} \Bigl[ 3\bigl(g(f) - 1\bigr) + \frac{3}{2}\nu(f) \Bigr] + \frac{3}{2}(\lambda - 1)
\]

\[
  N - (3g - 3) = \sum_{\substack{f \in F \\ f \ \text{pas un disque}}}
  \Bigl( 3g(f) + \frac{5}{2}\nu(f) - 3 \Bigr) + \frac{3}{2}(1 - \lambda)
  \ \overset{?}{\geqslant}\ 0
\]

\[
  N - 1 = \sum_{f \ \text{pas un disque}} \bigl( \underbrace{4\nu(f) - 6}_{\geqslant 2} \bigr) - 1
\]

\[
  \text{donc } \{\Gamma\} \simeq X/\Gamma_{0} \simeq X/\mathbb{R}
  \simeq \Delta / T_{\mathbb{C}} \cdot Z
  \simeq \text{cercles de } P
\]

\[
  \{\Gamma\} \simeq \mathrm{or}(\Gamma_{0}) \simeq \text{gén}(Z)
  \simeq \text{gén}(Z) \simeq \mathrm{Inv}(T)
\]

\[
  \Delta_{i} \subset T_{s_{i}}
\]

\[
  |z| \leqslant r_{i} ,
\]

\[
  r = (r_{i})_{i\in I} \in R^{\wedge} = \prod_{i\in I} R_{i}^{\wedge} .
\]

\[
  r_{i} < \rho_{i} ,
\]

\[
  \rho = (\rho_{i})_{i\in I} \in R = \prod_{i\in I} R_{i}
\]

\[
  0 < \lambda_{i} < 1 .
\]

\[
  \widetilde{MB} \simeq \widetilde{M} \times [0, 1[^{I} ,
\]

\[
  u_{+} \qquad \tfrac{1}{2}(u_{+} + u_{-})
\]

\[
  \underbrace{\left(x + \tfrac{1}{2} y\right)}_{m} u_{+} +
  \underbrace{\tfrac{1}{2} y}_{n} u_{-}
\]

\[
  0 \to \Pi \to V \to E \to 0
\]

\[
  H^{1}(\gamma, E) \xrightarrow{\ \sim\ } H^{2}(\gamma, \Pi) \simeq
  \Pi^{\gamma} / (1 + \sigma)(\Pi)
\]

\[
  z \longmapsto \exp 2i\pi z
\]

\[
  X = E(\mathbb{C}) / \text{conj.\ complexe} .
\]

\[
  H \subset \mathrm{Aut}_{\mathrm{conf}}(X) = G
\]

\[
  (H^{2}(\gamma, T) \simeq T^{\gamma} / N_{\gamma}(T) = T / T^{2} \simeq 0)
\]

\[
  H^{2}(\gamma, T) \simeq T^{\gamma} / N_{\gamma}(T) \simeq
  \underbrace{{}_{2}T}_{\mathbb{Z}/2\mathbb{Z}} / 0 \simeq
  \mathbb{Z}/2\mathbb{Z} .
\]

\[
  T = \{ z \mapsto az \mid a \in \mathbb{U} \text{ i.e. } a \in \mathbb{C},\
  |a| = 1 \text{ i.e. } a\bar{a} = 1 \}
\]

\[
  z \longmapsto a\bar{z} \quad (a \in \mathbb{C}^{*}) \quad
  \text{ceux qui fixent } 0, \infty
\]

\[
  z \longmapsto b/\bar{z} \quad (b \in \mathbb{C}^{*}) \quad
  \text{— intervertissant } 0, \infty
\]

\[
  \sigma(z) = b/\bar{z} , \quad \text{on a} \quad
  \sigma^{2}(z) = \tfrac{b}{\bar{b}}\, z ,
\]

\[
  z\bar{z} = b
\]

\[
  \underbrace{\underbrace{(\sigma' z)}_{(-b/\bar{z})}\,
  \underbrace{\overline{(\sigma' z)}}_{(-b/z)}}_{b^{2}/z\bar{z}} = b
\]

\[
  \sigma z = a\bar{z} , \quad (a \in \mathbb{C}^{*}) .
\]

\[
  \sigma^{2} z = a(\overline{a z}) = a\bar{a}\, z
\]

\[
  z \in \mathbb{C}^{\sigma} \iff z = \alpha \frac{\bar{z}}{\bar{\alpha}}
  \iff z/\alpha = \overline{(z/\alpha)} \quad \text{i.e.} \quad
  z \in \alpha \mathbb{R}
\]

\[
  (X \smallsetminus X^{H})/H = (X \smallsetminus X^{\sigma})/\sigma
\]

\[
  \text{surfaces conformes à bord} \;\approx\;
  \begin{array}{l}
    \text{surfaces holomorphes sans bord,} \\
    \text{munies d'une anti-involution}
  \end{array}
\]

\[
  X \longmapsto (X' = X^{\mathrm{or}}, \sigma)
\]

\[
  X = X'/\sigma \longleftarrow (X', \sigma)
\]

\[
  \chi(X^{\mathrm{or}}) = 2\chi(X)
\]

\[
  X \text{ hyperbolique} \iff \chi(X) < 0
\]

\[
  z \longmapsto a\bar{z} + b \qquad (a \in \mathbb{C}^{*},\ b \in \mathbb{C})
\]

\[
  \sigma z = a\bar{z}
\]

\[
  \sigma z = \alpha^{2}\bar{z} = \alpha/\bar{\alpha}\,\bar{z}
\]

\[
  \mathbb{C}^{\sigma} = \alpha.\mathbb{R}
\]

\[
  \overline{\mathbb{H}} = \{z \in \mathbb{C} \mid \Im z \geq 0\}.
\]

\[
  \pi_1(X) \longrightarrow \pi_1([Y,N])
\]

\[
  \pi_1(X,x) \longrightarrow \pi_1([Y,N],y)
\]

\[
  (r_1 r_2)^{2} = (r_2 r_3)^{3}, \qquad (r_1 r_2)^{4}\ \bigl(= (r_2 r_3)^{6}\bigr) = 1 .
\]

\[
  \ell_0\,\ell_1\,\ell_2\,\ell_3 = 1, \qquad \ell_0\,\ell_1\,\ell_\infty = 1, \qquad \ell'_0\,\ell'_2\,\ell'_3 = 1, \qquad \ell_\infty = \ell'_0
\]

\[
  1 \to \Pi_{g,\nu-1} \to \mathbb{T}_{g,\nu} \to \mathbb{T}_{g,\nu-1} \to 1
\]

\[
  \bigl(\simeq \mathrm{Isom}_{\mathbb{U}}(\partial D'^{-1}, \partial D'') \simeq \mathrm{Isom}_{\mathbb{U}}(\partial D''^{-1}, \partial D')
\]

\[
  \simeq \mathrm{Isom}(T'^{-1}, T'') \simeq \mathrm{Isom}(T''^{-1}, T') \simeq \mathrm{Bi}\ldots(T' \otimes T'')
\]

\[
  \simeq T'^{\times} \wedge_{\mathbb{C}^{\times}} T''^{\times} \bigr) \simeq
\]

\[
  a_n z^{n}, \qquad n > 0,\ a_n \neq 0 \qquad (\text{OPS } x = 0)
\]

\[
  f : H \cap \mathring{D} \to \mathbb{C},
\]

\[
    \partial X \subset K \subset X,
  \]

\[
  D \subset T_{X',a}
\]

\[
  D \longmapsto o(D) \qquad (\text{« ombre de } D \text{ »})
\]

\[
  o(D_i) \subset T_{X',a_i} .
\]

\[
  \mathrm{TB}_{g,\nu} \longrightarrow T_{g,\nu}
\]

\[
  (D_i) \longmapsto (o(D_i))
\]

\[
  \mathrm{TB}_{g,I} \longrightarrow T_{g,I} \qquad (\operatorname{card} I = \nu),
\]

\[
  \varphi : \beta = \widehat{(1,\zeta)} \xrightarrow{\ \sim\ }
  \bar\beta = \widehat{(1,\bar\zeta)}
\]

\[
  f = f_{\varphi} : \mathbb{D} \longrightarrow \mathbb{D}
\]

\[
  \psi : [0,1] \xrightarrow{\ \sim\ } [0,1]
\]

\[
  f = \sum_{i \geq 0} a_i z^i \qquad (a_i \in \mathbb{C})
\]

\[
  T_{X'',x_1} \otimes_{\mathbb{C}} T_{X'',x_2}
\]

\[
  T^{*}_{X'',x_1} \wedge_{\mathbb{C}^{*}} T^{*}_{X'',x_2} .
\]

\[
  \mathrm{MDB}_{g,\nu} \longrightarrow \mathrm{M_{st}ND}_{g,\nu}
\]

\[
  u_i \in T^{*}_{x'_i} \wedge_{\mathbb{C}^{*}} T^{*}_{x''_i}
\]

\[
  T_{x'_i} \otimes T_{x''_i} \longrightarrow \mathbb{C}
\]

\[
  \mathrm{MDB} \longrightarrow \mathrm{M_{st}N}_{g,\nu} ,
\]

\[
  V(J) \hookrightarrow G_{X} = \operatorname{Aut}(X)
\]

\[
\begin{array}{lll}
  u(e_{0}) = (z \mapsto -z) = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}
    & & \text{pts fixes } 0, \infty \\[1ex]
  u(e_{1}) = (z \mapsto 1/z) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
    & & \text{pts fixes } \pm 1 \\[1ex]
  u(e_{0}+e_{1}) = (z \mapsto -1/z) = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}
    & & \text{pts fixes } \pm i
\end{array}
\]

\[
  (*) \qquad 1 \to V \to \Gamma \to \operatorname{Aut}(V) \to 1,
  \qquad \Gamma = \operatorname{Nor}_{G}(V), \quad
  \operatorname{Aut}(V) \simeq \mathfrak{S}_{J}
\]

\[
  \Gamma(V) \xrightarrow{\ \sim\ } \mathfrak{S}_{I(V)} ,
\]

\[
  \Gamma = \operatorname{Nor}_{G}(V)
\]

\[
  X(I)/V(I) \simeq X(J)
\]

\[
  X \simeq X(I(V)) \qquad \text{iso.\ canonique}
\]

\[
  X(T \wedge T') = X(T) \wedge^{V} T'
\]

\[
  \boxed{X = X(I(V)) = X(I) \wedge^{V} \underline{t}}
\]

\[
  \boxed{X/V \simeq X(I)/V \simeq \Sigma(J)}
\]

\[
  \underline{t} \longmapsto X(I) \wedge^{V} \underline{t}
\]

\[
  X/V \simeq \Sigma(J) \qquad \text{(iso.\ canonique)}
\]

\[
  X_{\xi} = X(I)_{\xi} \wedge \underline{t} \simeq I \qquad \text{i.e.} \qquad
  \underline{t} = I \wedge X(I)_{\xi}^{-1}
\]

\[
  X = X(I) \wedge^{V(J(I))} t(\xi, I)
\]

\[
  I \hookrightarrow X \text{ est la fibre de } X \to \xi, \text{ compte tenu
  que}
\]

\[
\begin{array}{c}
  X/V(I) \simeq X(I)/V(I) \simeq \Sigma(J) \\
  X_{\xi} \simeq X(I)_{\xi} \wedge t(\xi, I) .
\end{array}
\]

\[
  I \simeq X(I)_{\xi} .
\]

\[
\begin{array}{l}
  P = M_{+} = \operatorname{Ker}(\mathrm{id}_{M} - \sigma)
    = \{x \in M \mid \sigma x = x\} \\
  Q = M_{-} = \operatorname{Ker}(\mathrm{id}_{M} + \sigma)
    = \{x \in M \mid \sigma x = -x\}
\end{array}
\]

\[
  2M \subset M_{+} \oplus M_{-} \subset M
  \quad \text{car } \forall x \in M, \text{ on a }
  2x = \underbrace{(x + \sigma x)}_{M_{+}} + \underbrace{(x - \sigma x)}_{M_{-}}
\]

\[
  \underset{\substack{\Vert \\ 2M}}{M'} \subset M_{+} \oplus M_{-} = P \oplus Q
\]

\[
  2P \oplus 2Q \subset M' \subset P \oplus Q
\]

\[
  (P \oplus Q)/(2P \oplus 2Q) \simeq \underbrace{P/2P}_{p} \oplus
  \underbrace{Q/2Q}_{q} ,
\]

\[
\begin{cases}
  m \cap p = 0 & \text{i.e.\ }
    \underbrace{M' \cap \bigl(P + (2P + 2Q)\bigr)}_{M' \cap P + (2P + 2Q)}
    = (2P + 2Q) \\
  m \cap q = 0 & \text{i.e.\ } M' \cap \bigl(Q + (2P + 2Q)\bigr) = 2Q
\end{cases}
\]

\[
  P \oplus Q \supset M' \supset 2P \oplus 2Q
\]

\[
\begin{array}{l}
  \sigma(x, y) = x - y \\
  (1 + \sigma)(x, y) = 2x \\
  (1 - \sigma)(x, y) = 2y
\end{array}
\]

\[
\begin{cases}
  M' \cap P = 2P \\
  M' \cap Q = 2Q
\end{cases}
\qquad \text{i.e.\ } M' \cap (P + 2Q) = 2P + 2Q
\ \ \bigl(= (M' \cap P) + 2Q\bigr)
\]

\[
  m \subset \underbrace{P/2P}_{p} \times \underbrace{Q/2Q}_{q}
  \qquad \text{sous-module de } k/2k\text{-modules}
\]

\[
\begin{array}{ll}
  m \cap p = 0 & \qquad m \to p \text{ injectif} \\
  m \cap q = 0 & \qquad m \to q \text{ injectif}
\end{array}
\]

\[
\begin{array}{l}
  M_{+} = \{x \in M \mid \sigma x = x\} = \operatorname{Ker}(1 - \sigma) \\
  M_{-} = \{x \in M \mid \sigma x = -x\} = \operatorname{Ker}(1 + \sigma) \\
  M_{+} \cap M_{-} = 0 \\
  M/M_{+} + M_{-} \ \text{annulé par } 2 \\
  2x = \underbrace{(x + \sigma x)}_{\in M_{+}} + \underbrace{(x - \sigma x)}_{\in M_{-}}
\end{array}
\]

\[
\begin{array}{l}
  M_{+} \oplus M_{-} \subset M \subset \tfrac{1}{2}(M_{+} \oplus M_{-}) \\
  2M_{+} \oplus 2M_{-} \subset \underbrace{2M}_{M'} \subset M_{+} \oplus M_{-}
\end{array}
\]

\[
  m \in P/2P \oplus Q/2Q
\]

\[
  (1 + \sigma) M' = 2M_{+}, \qquad (1 - \sigma) M' = 2M_{-}
\]

\[
  Q(e_{+}) = 1
\]

\[
  Q(x e_{+} + y e_{-}) = x^{2} + a y^{2}
\]

\[
  1 \longrightarrow \underset{\substack{\Vert \\ \operatorname{Aut}(X)}}{G^{+}}
  \longrightarrow G \longrightarrow \pm 1 \longrightarrow 1
\]

\[
  1 \longrightarrow G^{+} \longrightarrow G \longrightarrow \pm 1
  \longrightarrow 1
\]

\[
  a = a(\sigma) = \frac{Q(e_{-})}{Q(e_{+})}
\]

\[
  a(-\sigma) = 1/a(\sigma)
\]

\[
  a(-\sigma) = a(\sigma) \iff a(\sigma) = 1 \iff \text{cas \emph{carré}} .
\]

\[
  a' = \frac{Q(e_{0} - e_{1})}{Q(e_{0} + e_{1})} = 1
\]

\[
  a = \frac{Q(2v - u)}{Q(u)} = \sqrt{3}
\]

\[
  \boxed{\tau = \sqrt{a}\, i}
\]

\[
  \underbrace{e_{0}}_{1}, \ \underbrace{\tfrac{1}{\sqrt{a}}\, e_{1}}_{i}
  \qquad \text{base orthonormale}
\]

\[
  \boxed{\tau = \tfrac{1}{2}(1 + \sqrt{a}\, i)} = \frac{1}{2} + \frac{\sqrt{a}}{2}\, i
\]

\[
  \sigma x \equiv x \quad (\Pi)
\]

\[
  \sigma(x, y) - (x, y) = (0, -2y)
\]

\[
\begin{array}{l}
  P = k, \quad Q = k, \quad m = \text{diagonale de } k_{0} \times k_{0}
    \text{ i.e.} \\
  M' = (2k \oplus 2k) + \text{diag.\ de } k \oplus k
    = \{(x, y) \mid x - y \in 2k\}
\end{array}
\]

\[
  u_{0} = (2, 0), \quad v_{1} = (1, 1), \quad \text{d'où} \quad
  -u_{0} + 2 v_{1} = (0, 2) = \operatorname{sym}(u_{0})
\]

\[
  \sigma u_{0} = u_{0}, \quad \sigma v_{1} = (1, -1) = u_{0} - \operatorname{sym} v_{1}
  = u_{0} - v_{1}
\]

\[
  \begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix}
\]

\[
\begin{array}{l}
  \sigma(x, y) = \sigma(xu + yv) = xu + y(u - v) = (x + y) u - y v . \\
  \sigma(x, y) - (x, y) = yu - 2yv = y(u - 2v),
\end{array}
\]

\[
  0 \longrightarrow \Pi \longrightarrow V \longrightarrow
  E(\mathbb{C}) \longrightarrow 0
\]

\[
  H^{1}(\gamma, E(\mathbb{C})) \simeq H^{2}(\gamma, \Pi) \simeq
  \Pi^{\gamma}/(1 + \sigma)\Pi = \Pi_{+}/(1 + \sigma)\Pi
\]

\[
  H^{1}(\mathbb{R}, E_{\mathbb{R}}) \simeq H^{1}(\gamma, E(\mathbb{C}))
  \simeq H^{2}(\gamma, \Pi) \simeq
  \underbrace{H^{2}(\gamma, \Pi_{+})}_{\mathbb{Z}/2\mathbb{Z}} \oplus
  \underbrace{H^{2}(\gamma, \Pi_{-})}_{0}
\]