\[
  \mathrm{Aff}(r) \;=\; \mathrm{GL}(r)\cdot E^{r}
\]

\[
  {\scriptstyle r+1}
  \left\{
  \begin{pmatrix} A & C \\ 0 & 1 \end{pmatrix}
  \right\}
  {\scriptstyle r}
\]

\[
  0 \longrightarrow \Omega \longrightarrow E \longrightarrow
  \underline{O}_S \longrightarrow 0
\]

\[
  0 \longrightarrow \Omega \longrightarrow E \xrightarrow{\;p\;}
  \underline{O}_S \longrightarrow 0
\]

\[
  \widehat{S} = \underline{O}_{\widehat{P}}
\]

\[
  E^{\circ} \simeq \Omega \times \underline{O}_S
\]

\[
  \widehat{P}(S') \;\simeq\; \text{Quotients inversibles } L' \text{ de } E,
\]

\[
  (x,\lambda) \longmapsto \varphi(x) + \lambda, \qquad
  \varphi : \Omega \longrightarrow \underline{O}_S
\]

\[
  \widehat{P}^{\circ} = V(\Omega)^{\wedge} \quad
  \text{(complété formel le long de la section nulle)}
\]

\[
  = \mathrm{Spf}\;\widehat{\mathrm{Sym}}(\Omega)
  \;=\; \widehat{S}^{\circ}
\]

\[
  u_a(x,\lambda) = (x + \lambda a,\; \lambda)
\]

\[
  \widetilde{\varphi} \circ u_a : (x,\lambda) \longrightarrow
  \varphi(x + \lambda a) + \lambda
  = \varphi(x) + (1 + \varphi(a))\lambda
  = (1 + \varphi(a))\left( \frac{\varphi}{1 + \varphi(a)}(x) + \lambda \right),
\]

\[
  \varphi^a = u_a^{*}(\varphi) = \frac{\varphi}{1 + \varphi(a)}
\]

\[
  \varphi^a(\omega) = \frac{\varphi(\omega)}{1 + \varphi(a)}
  = \varphi\Big( \sum_{n \geqslant 0} (-1)^n \omega a^n \Big),
\]

\[
  \varphi^a(\omega) = \varphi(u_a(\omega))
\]

\[
  u_a(\omega) = \struck{\ill{}} \frac{\omega}{1+a}
  = \sum_{n \geqslant 0} (-1)^n \omega a^n
\]

\[
  E \otimes_{\underline{O}_S} \widehat{S} \longrightarrow \widehat{L}
  \simeq \widehat{S}
\]

\[
  \pi(\omega \otimes f,\, g) = \omega f + g,
  \qquad \omega \in \Omega(S),\; f,g \in \widehat{S}(S).
\]

\[
  \omega \otimes f + e \otimes g,
  \qquad \omega \in \Omega(S),\;\; f,g \in \widehat{S}(S),
\]

\[
  u_{a*}^{\widehat{E}}(\omega \otimes f + e \otimes g)
  = u_a(\omega) \otimes u_{a*}(f) + u_a(e) \otimes u_{a*}(g)
\]

\[
  = \omega \otimes u_{a*}(f) + (e + a) \otimes u_{a*}(g)
\]

\[
  = \big( \omega \otimes u_{a*}(f) + a \otimes u_{a*}(g) \big)
  + e \otimes u_{a*}(g)
\]

\[
  \pi\big( u_{a*}^{\widehat{E}}(\omega \otimes f + e \otimes g) \big)
  = \omega\, u_{a*}(f) + a\, u_{a*}(g) + u_{a*}(g)
\]

\[
  = \omega\, u_{a*}(f) + (a+1)\, u_{a*}(g)
\]

\[
  \pi(\omega \otimes f + e \otimes g) = \omega f + g .
\]

\[
  u_{a*}^{\widehat{L}}(g) = (a+1)\, u_{a*}(g)
  \qquad \Big( = \frac{g}{(1+a)^{\,n-1}} \;\text{ si } \deg g = n \Big)
\]

\[
  u_{a*}^{\widehat{L}}\big( \pi(\omega \otimes f + e \otimes g) \big)
  = (a+1)\, u_{a*}(\omega f + g)
\]

\[
  = (a+1) \big( u_{a*}(\omega)\, u_{a*}(f) + u_{a*}(g) \big)
\]

\[
  = \omega\, u_{a*}(f) + (a+1)\, u_{a*}(g) \qquad \text{OK}
\]

\[
  \widehat{S} = \struck{\ill{}}\;
  T \wedge^{(\Omega,\, u^{\widehat{L}}_{\Omega*})}
  \widehat{\mathrm{Sym}}(\Omega)
\]

\[
  \widehat{L} =
  T \wedge^{(\Omega,\, u^{\widehat{L}}_{\Omega*})}
  \widehat{\mathrm{Sym}}(\Omega)
\]

\[
  \mathrm{gr}^1_{\widehat{J}}(\widehat{S}) \simeq \Omega
\]

\[
  \big( \mathrm{gr}_{\widehat{J}}(\widehat{S})
  \simeq \widehat{\mathrm{Sym}}(\Omega) \big)
\]

\[
  f + g \;\longrightarrow\; \frac{f}{(1+a)^{n}} + \frac{g}{(1+a)^{n+1}}
\]

\[
  f \;\longmapsto\; naf + g + \text{termes de degré } \geqslant n + 1
\]

\[
  \Omega \;\longrightarrow\; \mathrm{Hom}\big( \mathrm{Sym}^n \Omega,\;
  \mathrm{Sym}^{n+1} \Omega \big)
\]

\[
  v(a)(f) = af
\]

\[
  T \times_{\Omega} \big[ \mathrm{Hom}\big( \mathrm{Sym}^n \Omega,\;
  \mathrm{Sym}^{n+m} \Omega \big) \big]
\]

\[
    0 \longrightarrow \Omega \longrightarrow E \longrightarrow
    \underline{O}_X \longrightarrow 0
  \]

\[
    P^n_{X/S} \;\simeq\; \underline{O}_{\widehat{P}}
    \big/ \widehat{J}^{\,n+1}
    \qquad \big( \simeq \underline{O}_X \otimes_{\Omega}
    \mathrm{Sym}^{\bullet}(\Omega) \big)
  \]

\[
  (*) \qquad \Omega \;\simeq\; \Omega^1_{X/S}
\]

\[
  \Omega \xrightarrow{\;-u_1\;}
  \mathrm{Hom}\big( \Omega,\ \mathrm{Sym}^2(\Omega) \big)
\]

\[
  \mathrm{Hom}\big( \Omega^1_{X/S},\ \mathrm{Sym}^2 \Omega^1_{X/S} \big)
\]

\[
  P^{\infty}_{X/S} \simeq \underline{O}_S[[\delta f_1, \ldots]]
\]

\[
  \Delta : P^{\infty}_{X/S} \longrightarrow
  P^{\infty}_{X/S} \otimes_S P^{\infty}_{X/S}
  = P^{\infty}_{X/S}\big( P^{\infty}_{X/S} \big)
  \simeq P^{\infty}_{X/X}[[\ldots
\]

\[
  \Delta(\delta f_i) = \delta(f_i) \otimes 1 + 1 \otimes \delta(f_i)
  = \delta(f_i) + \delta f_i
\]

\[
  M \otimes P^n_{X/S} \longrightarrow P^n_{X/S} \otimes M
\]

\[
  M \xrightarrow{\;\Delta\;} P^n_{X/S} \otimes M
\]

\[
  \Delta(m) = m + \Sigma
\]

\[
  F_i \in \Gamma\, P^{\infty}_{X/S}
  = \Gamma\, \underline{O}_S[[\delta f_1, \ldots, \delta f_n]],
  \qquad 1 \leqslant i \leqslant n
\]

\[
  F_i = \delta f_i + G_i, \qquad G_i \text{ d'ordre } \geqslant 2
\]

\[
  \Phi_i \in \Gamma\, \widehat{\mathrm{Sym}}\;
  \underline{O}_S[[T_1, \ldots, T_n]],
  \qquad 1 \leqslant i \leqslant n
\]

\[
  \Phi_i = T_i + \Psi_i \qquad (\Psi_i \text{ d'ordre } \geqslant 2)
\]

\[
  \underline{O}_S[[T_1, \struck{\delta f}]] = \widehat{S}
  \xrightarrow{\;\Delta'\;} P^{\infty}_{X/S} \widehat{\otimes} \widehat{S}
  = P^{\infty}_{X/S}(\widehat{S}) \simeq P^{\infty}_{X/S}[[T \ldots
\]

\[
  \Delta(\struck{\ill{}}\; \delta f_i + G_i) \in \Gamma\, P^{\infty}_{X/S}
\]

\[
  c_{\infty} : P^{\infty}_{X/S} \longrightarrow \widehat{S}.
\]

\[
  \Omega^1_{X/S} = \omega, \qquad f_*(\omega) = \Omega, \qquad
  R^1 f_*\big( \Omega^{\bullet}_{X/k} \big) = DR
\]

\[
  R^1 f_*(\underline{O}_X) \simeq \check{\Omega} \simeq \tau
\]

\[
  0 \longrightarrow \Omega \longrightarrow DR \longrightarrow
  \check{\Omega} \longrightarrow 0
\]

\[
  DR \times DR \xrightarrow{\;q\;} \underline{O}_S \qquad \text{alternée}
\]

\[
  b = R^2 f_*\big( \Omega^{\bullet}_{X/S} \big)
  \simeq R^1 f_*\big( \Omega^1_{X/S} \big)
\]

\[
  0 \longrightarrow \check{\Omega} \otimes \check{\Omega} \longrightarrow
  \Sigma \longrightarrow \underline{O}_S \longrightarrow 0
\]

\[
  \Lambda^2 DR \xrightarrow{\;q\;} \underline{O}_S
\]

\[
  0 \longrightarrow \mathrm{Fil}^2 \longrightarrow \mathrm{Fil}^1
  \longrightarrow \Lambda^2 DR
\]

\[
  \begin{pmatrix} 0 & \pm\,\mathrm{id} \\ \mathrm{id} & \varphi_0 \end{pmatrix}
\]

\[
  q'(t) = q(t) + u(t) \qquad \big( u : \check{\Omega} \to \Omega \big)
\]

\[
  \varphi'_0(t,t') = \varphi\big( q(t) + u(t),\; q(t') + u(t') \big)
\]

\[
  = \varphi_0(t,t') + \big( t,\, u(t') \big) - \big( t',\, u(t) \big)
\]

\[
  0 \longrightarrow \Lambda^2 \Omega \longrightarrow \Sigma \longrightarrow
  \underline{O}_S \longrightarrow 0
\]

\[
  DR \longrightarrow DR \otimes \Omega^1_{S/T}
\]

\[
  \Omega \longrightarrow \check{\Omega} \otimes \Omega^1_{S/T}
\]

\[
  \Omega^1_{S/T} \longrightarrow \check{\Omega} \otimes \check{\Omega}
\]

\[
  \Omega^1_{S/T} \longrightarrow \Lambda^2 \check{\Omega}
\]

\[
  H^0(X, \Omega_X)
\]

\[
  0 \longrightarrow \Omega^1_X \xrightarrow{\;\omega\;} \Omega_X
  \xrightarrow{\;\det\;} \Omega^1_X \longrightarrow (DR)_X
\]

\[
  0 \longrightarrow \Gamma^2 \Omega \longrightarrow \Omega \otimes \Omega
  \longrightarrow \Lambda^2 \Omega \longrightarrow 0
\]

\[
  f_*(\omega^2) \;\simeq\; \Omega^1_{S/T}
\]

\[
  \frac{g(g+1)}{2} - (3g-3) \;=\; \frac{1}{2}\big( g^2 - 5g + 6 \big)
  \;=\; \frac{1}{2}(g-2)(g-3)
\]

\[
  \longrightarrow \underline{O}_X \longrightarrow P^1 \longrightarrow \omega
  \longrightarrow 0
\]

\[
  E \otimes_{\underline{O}} P \longrightarrow E
\]

\[
  2 - 2g
\]

\[
  P^{\infty}_{X/S}, \qquad \Omega\quad \Omega^2\quad \Omega^3
\]

\[
  \omega^2 \longrightarrow P^{3+} \longrightarrow \omega \longrightarrow 0
\]

\[
  \longrightarrow f_*(\omega^2) \longrightarrow f_*\big( P^{3+}_{X\varphi}
  \big) \longrightarrow f_*(\omega) \longrightarrow 0
\]

\[
  P^{\infty}\big( \widehat{\mathrm{Sym}}(\Omega) \big), \qquad
  P^{\infty}(\omega^n), \qquad \omega \otimes P^{\infty},
  \qquad P^{\infty} \otimes \omega
\]

\[
  M \otimes P^{\infty}, \qquad R^1 f_*\big( M \otimes \omega^n \big) = 0
  \quad \text{si } n \gg 1
\]

\[
  S^i = f_*\, P^{\infty}(\omega^n) = f_*\big( \omega^n \otimes P \big)
\]

\[
  \mathrm{gr}\, S^{\bullet} \;\simeq\; \sum_{n \geqslant \ill{}}
  f_*\big( \omega \otimes P \big).
\]

\[
  M_g^{\mathfrak{z}}(S) = \text{courbes algébriques}
\]

\[
    0 \longrightarrow \underline{O}_X \longrightarrow E \longrightarrow
    \underline{O}_X \longrightarrow 0
  \]

\[
  (*) \qquad H^0_{DR}(X, \mathfrak{g}) \simeq
  H^2_{DR}(X, \mathfrak{g})^{\vee}
\]

\[
  (**) \qquad \chi_{DR}(X, \mathfrak{g})
  \overset{\text{déf}}{=} \sum (-1)^i \dim H^i_{DR}(X, \mathfrak{g})
  \;=\; 3\, \chi_{DR}(X, \underline{O}_X)
\]

\[
  = \struck{6g} \; 6 - 6g
\]

\[
  (\!*\!*\!*) \qquad \dim H^1_{DR}(X, \mathfrak{g})
  = 2 \dim H^0_{DR}(X, \mathfrak{g}) + (6g - 6)
\]

\[
  H^0_{DR}(X, \mathfrak{g}) \simeq \mathrm{Lie}\;
  \mathrm{Aut}^{\mathfrak{z}}_{X/k}(P)
\]

\[
  \dim H^1_{DR}(X, \mathfrak{g}) = 2 \dim
  \mathrm{Aut}^{\mathfrak{z}}_{X/k}(P) + 6g - 6
\]

\[
  = 2 \Big( \dim \mathrm{Aut}^{\mathfrak{z}}_{X/k}(P) + 3g - 3 \Big)
\]

\[
  \pi \xrightarrow{\;\rho\;} \mathrm{PGL}(1, \mathbf{C}) = H,
  \qquad \pi = \pi_1(X, x)
\]

\[
  M_g^{\mathfrak{z}}(X) \;\simeq\;
  \mathrm{Hom}_{\mathrm{gr.\,ana}}(\pi, H) \,\big/\, H
\]

\[
  \rho \cdot u = \rho \circ u ;
\]

\[
  \rho \circ u = \mathrm{int}(\rho(g)) \circ u,
\]

\[
  \pi_1(Y) \;\simeq\; \pi = \pi_1(X, x)
\]

\[
  \rho_Y : \pi_1(Y) \longrightarrow H
\]

\[
  M_g(X) \hookrightarrow M_g^{\mathfrak{z}}(X)
\]

\[
  H^0\big( X, \omega^{\otimes 2}_{X/\mathbf{C}} \big), \ldots
\]

\[
  M_{\mathrm{ann}}(X) \longrightarrow M_g^{\mathfrak{z}}(X)
\]

\[
  \lambda c = c \quad\text{i.e.}\quad (\lambda - 1)\, c = 0,
\]

\[
  \mathrm{Aut}_X(P_X) \;\simeq\; H^0(X, \omega_X)
\]

\[
  \mathcal{U} \simeq \omega + \underline{O}_X + \omega^{-1}
\]

\[
  c_{D_t} : \mathcal{U} \longrightarrow \mathcal{U} \otimes \omega
  \simeq \omega^2 + \omega + \underline{O}_X
\]

\[
  \begin{cases}
    c_{D_t}(dt) = -2\, dt \\
    c_{D_t}(1) = 1 \\
    c_{D_t}\big( (dt)^{-1} \big) = 0
  \end{cases}
\]

\[
  c_{D_t}\big( \lambda\, dt + \mu \cdot 1 + \nu\, (dt)^{-1} \big)
  = \lambda'\, dt^2 + (\mu' - 2\lambda)\, dt + (\nu' \ldots
\]

\[
  u_\omega = \exp \Theta_\omega : \mathcal{U} \longrightarrow \mathcal{U}
\]

\[
  \exp \Theta_{-\omega}(1) = 1 + [-\omega,\, 1] = 1 + \omega
\]

\[
  u_\omega(c_{D_t}) = \struck{\ill{}}\; u \circ c_{D_t} \circ u^{-1}
  = \big( \exp \Theta_{-\omega} \big) \ldots
\]

\[
  u_\omega(c_{D_t}) - c_{D_t} \;\equiv\; \varphi
\]

\[
  \varphi(dt) \;\struck{=}\; c_{D_t}(dt)
  = \big( \mathrm{id} + \Theta_{-\omega} \big)
  \big( \struck{D_t}(dt) \big) = -2\, dt + \ldots
\]

\[
  \varphi(1) = -2\,\omega\, dt
\]

\[
  \varphi(1) + c_{D_t}(1) = \Big( \mathrm{id} + \Theta_{-\omega}
  + \tfrac{1}{2} \Theta_{-\omega}^{\,2} \Big)\,
  c_{D_t}\big( 1 + [\omega,\, 1] \big)
\]

\[
  = 1 + [-\omega,\, 1] + \big[ -\omega,\, [-\omega,\, 1] \big] + \ldots
\]

\[
  = 1 + (-2\omega) + 2\omega^2 - \lambda'\, dt^2 + 2\lambda\, dt + \ldots
\]

\[
  \varphi(1) = \big( -\lambda' + \struck{\ill{}} \big)\, dt^2
  \;\struck{\ill{}}
\]

\[
  u_\omega(c_{D_t}) = c_{D_t} + d^1_{\omega/S}(\omega)
\]

\[
  \boxed{\;u_\omega(c) = c + d^1_{\omega/S}(\omega)\;}
\]

\[
  \ill{} \longrightarrow M \longrightarrow F \longrightarrow
  \underline{O}_X \longrightarrow 0
\]

\[
  E \simeq F \otimes L,
\]

\[
  \underline{\omega} \hookrightarrow E \simeq F \otimes L \twoheadrightarrow L
\]

\[
  \pi_1(X^{\mathrm{an}}, x) \longrightarrow H(\mathbb{C})
\]

\[
  (P,\sigma) \quad\longleftrightarrow\quad 0 \to \Omega \to E \to \underline{O}_S \to 0
\]

\[
  J/J^2 = \omega \;\xrightarrow{\ \omega\ }\; \Omega^1_{X/S}
\]

\[
  \widehat{S} \;\underset{\sim}{\overset{u}{\longleftrightarrow}}\; P
\]

\[
    \widehat{S}/\widehat{J}^{\,4}_P \longrightarrow P/J^4
  \]

\[
  \Omega \longrightarrow \widehat{S} \otimes \Omega^1_{X/S},
  \qquad
  \prod_{i \geqslant 0} \Omega^{\otimes i} \otimes \Omega^1_{X/S},
  \qquad g \in G(c)
\]

\[
  S = \mathrm{Sym}(\Omega) \;\xrightarrow{\ c\ }\; \widehat{S} \,\widehat{\otimes}\, \Omega^1_{X/S},
  \qquad
  \Omega \;\xrightarrow{\ c\ }\; \widehat{S} \,\widehat{\otimes}\, \Omega^1_{X/S}
\]

\[
  C(\omega_0) = \sum_{i \geqslant 0} \omega_0^{\,i} \otimes \varpi_i,
  \qquad \varpi_i \in \Omega^1_{X/S}
\]

\[
  C(\lambda\omega_0) = \lambda\,C(\omega_0) + \omega_0\,d\lambda
\]

\[
  = 1 \otimes \lambda\varpi_0 + \omega_0 \otimes (\lambda\varpi_1 + d\lambda)
  + \omega_0^2 \otimes \lambda\varpi_2 + \cdots + \omega_0^{\,i} \otimes \lambda\varpi_i \cdots
\]

\[
  d(\lambda\omega) = \lambda\,d\omega + \struck{\omega\,d\lambda}
\]

\[
  J^i \longrightarrow J^{i-1} \otimes \Omega^1_{X/S}
\]

\[
  \lambda\omega^i \longrightarrow \struck{C(\ill{})}\ \omega^i \otimes d\lambda
  + \lambda\,\omega^{i-1} C(\omega)
\]

\[
  \struck{\Omega}\ \Omega^i \xrightarrow{\ \uncertain{h_i}\ } \Omega^{i-1} \otimes \Omega^1_{X/S}
\]

\[
  \lambda\omega^i \longrightarrow \lambda\,\omega^{i-1}\,\underline{C(\omega)},
  \qquad
  \lambda\omega_0^{\,i} \longrightarrow \lambda\,\omega_0^{\,i-1}\,C(\omega_0)
\]

\[
  C(\omega_0^{\,u})^{u^{-1}} =
  \left(\varpi_0 + \frac{\omega_0}{1+\sum \alpha_i \omega_0^{\,i}}\,\varpi_1
  + \left(\frac{\omega_0}{1+\sum \alpha_i \omega_0^{\,i}}\right)^{\!2}\varpi_2\right)
  \left(1 + \sum (i+1)\alpha_i \left(\frac{\ill{}}{1 + \ill{}}\right)\right)
\]

\[
  + \sum \left(\frac{\omega_0}{\sum 1 + \alpha_j \omega_0^{\,j}}\right)^{\!\ill{}} \otimes\, d\alpha_i
\]

\[
  C(\omega_0^{\,u})^{u^{-1}} \equiv (\varpi_0 + \omega_0\varpi_1 + \omega_0^2\varpi_2)
  \left[1 + \struck{\ill{}}\ 4\alpha_3\omega_0^3\right]
\]

\[
  \sum_{i \geqslant 1} \Omega^i \otimes \Omega^1_{X/S}
  \qquad = \struck{\ill{}}\ C(\omega_0) + 4\alpha_3\,\omega_0^3\,\varpi_0
\]

\[
  C(\omega_0^{\,u})^{u^{-1}} = (\varpi_0 + \omega_0\varpi_1 + \omega_0^2\varpi_2)
  \bigl(1 + (n+1)\alpha_n\,\omega_0^{\,n}\bigr)
\]

\[
  = \struck{\varpi_0}\ C(\omega_0) + \varpi_0\ \struck{\ill{}}\ (n+1)\alpha_n\,\omega_0^{\,n}\,\varpi_0
\]

\[
  C(\omega_0^{\,u})^{u^{-1}} = (\varpi_0 + \omega_0\varpi_1 + \omega_0^2\varpi_2)
  \bigl(1 + 3\alpha_2\,\omega_0^2\bigr)
  = C(\omega_0) + 3\alpha_2\,\omega_0^2\,\varpi_0
\]

\[
  (c,u) \longmapsto \struck{\ill{}}\ u(\hat{c})
\]

\[
  \mathrm{Conn}\,P \;(\times\, G_0) \longrightarrow \mathrm{Conn}_{\mathrm{Alg}}(\widehat{S})
  \quad \struck{mod}
\]

\[
  G \times \mathrm{Conn}(P), \qquad (g,c) \longmapsto g\cdot\hat{c}.
\]

\[
  h \in G(c) \iff h\cdot\hat{c} \ \text{de la forme}\ \hat{c}
\]

\[
  \struck{G(c) \times{}}\quad G_0 \times G(c) \longrightarrow G,
  \qquad (g,h) \longmapsto g\cdot h
\]

\[
  C(\omega^{\,u}) = \struck{\ill{}}\ C\omega + \sum_{i \geqslant 2} C(\omega)u_i
  + \omega\,C(u_i)
  = C(\omega)\Bigl(1 + \sum u_i\Bigr) + \omega \sum C(u_i)
\]

\[
  C(\lambda\,\omega^{\otimes i}) = \omega^{\otimes i} \otimes d\lambda
  + \lambda\,i\,\omega^{\otimes i-1}\,\ill{}
\]

\[
  C\bigl((\lambda\omega_0)^{\,u}\bigr) =
  \bigl(\lambda\,\add{C(\omega_0)} + \omega_0 \otimes d\lambda\bigr)
  \Bigl(1 + \sum_{i \geqslant 2}\alpha_i\,\omega_0^{\,i}\Bigr)
  + \lambda\omega_0 \sum_{i \geqslant 2}
  \bigl(\omega_0^{\,i} \otimes d\alpha_i + i\,\alpha_i\,\omega_0^{\,i-1}\,C(\omega_0)\bigr)
\]

\[
  C(\omega_0) = \struck{\ill{}}\ \varpi_0 + \omega_0\varpi_1 + \omega_0^2\varpi_2,
  \qquad \varpi_i \in \struck{\ill{}}\ \widehat{S} \otimes \Omega^1_{X/S}
\]

\[
  C(\struck{\lambda}\,\omega_0^{\,u}) = \struck{C(\omega_0}\ (\varpi_0 + \varpi_1 + \varpi_2)
  \Bigl(1 + \sum_{i \geqslant 2}\alpha_i\,\omega_0^{\,i} + \ill{}\Bigr)
\]

\[
  C(\omega_0^{\,u}) = C\Bigl(\omega_0\bigl(1 + \sum_{i \geqslant 2}\alpha_i\,\omega_0^{\,i}\bigr)\Bigr)
  = C(\omega_0)\Bigl(1 + \sum \alpha_i\,\omega_0^{\,i}\Bigr)
  + \omega_0 \sum_{i \geqslant 2}\bigl(\omega_0^{\,i} \otimes d\alpha_i
  + i\,\alpha_i\,\omega_0^{\,i-1}\,C(\omega_0)\bigr)
\]

\[
  = C(\omega_0)\Bigl(1 + \sum_{i \geqslant 2}(i+1)\alpha_i\,\omega_0^{\,i}\Bigr)
  + \omega_0^2 \sum_{i \geqslant 2} \ill{}
\]

\[
  C(\omega_0^{\,u})^{u^{-1}} = \struck{\ill{}}
  \Biggl(1 + \sum_{i \geqslant 2}(i+1)\alpha_i
  \biggl(\frac{\omega_0}{1 + \sum_j \alpha_j\,\omega_0^{\,j}}\biggr)^{\!i}\Biggr)
  + \ill{} \sum_{i \geqslant 2}
  \biggl(\frac{\omega_0}{1 + \sum_{j \geqslant 2} \alpha_j\,\omega_0^{\,j}}\biggr)^{\!i+1}
  \otimes\, d\alpha_i
\]

\[
  G_0 \times C_0 \;\xrightarrow{\ \sim\ }\; C, \qquad (h,c) \longmapsto h\cdot c
\]

\[
  G(c) = \{\, g \in G \ \mid\ g\cdot c \in C_0 \,\}
\]

\[
  G_0 \times G(c) \;\xrightarrow{\ \sim\ }\; G, \qquad (h,g) \longmapsto h\,g
\]

\[
  C : P \longrightarrow P \,\widehat{\otimes}\, \Omega^1_{X/S}
\]

\[
  \begin{cases}
    C(fg) = f\,C(g) + g\,C(f), & f, g \ \text{sections de}\ P \\
    C(\lambda \cdot 1) = 1 \otimes d\lambda, & \lambda \ \text{section de}\ \underline{O}_X
  \end{cases}
\]

\[
  \Omega \longrightarrow \widehat{\mathrm{Sym}}(\Omega) \,\widehat{\otimes}\, \Omega^1_{X/S}
  = \prod_{i \geqslant 0} \mathrm{Sym}^i(\Omega) \otimes \ill{}
\]

\[
  \begin{cases}
    \varpi_i \in \struck{\ill{}}\ \Gamma\bigl(\Omega^{\otimes (i-1)} \otimes \Omega^1_{X/S}\bigr) & (i \neq 1) \\
    \varpi_1 \ \text{une connexion sur}\ \Omega
  \end{cases}
\]

\[
  \mathrm{gr}^i(S) = J^i/J^{i+1} \simeq \Omega^i
  \;\xrightarrow{\ \widetilde{c}\ }\;
  \struck{\ill{}}\ \bigl(P \,\widehat{\otimes}\, \Omega^1_{X/S}\bigr)
\]

\[
  \widetilde{c}(\omega_0) = \omega_i \otimes \varpi_0
\]

\[
  \varpi_0 \in \Gamma\bigl(\Omega^{-1} \otimes \Omega^1_{X/S}\bigr)
  = \mathrm{Hom}\bigl(\Omega, \Omega^1_{X/S}\bigr)
\]

\[
  \mathrm{Conn}(P) \longrightarrow \mathrm{Conn}(\widehat{S})
\]

\[
  \mathrm{Conn}^{*}(P) \times G_0 \longrightarrow \mathrm{Conn}^{*}(\widehat{S}),
  \qquad (c,g) \longmapsto g\cdot\hat{c}
\]

\[
  G(c) \;\xrightarrow{\ \sim\ }\; G/G_0 \simeq \Gamma(X, \Omega^{\otimes 2}).
\]