\[
  \Omega : \mathcal{V}^{\circ} \longrightarrow (\mathrm{Ens})
\]

\[
  \Omega(\operatorname{int} X) \longrightarrow \Omega(\partial X),
\]

\[
  \Omega(e) \simeq \{+1, -1\} \subset \mathbb{Z} \quad\text{par}\quad \omega^{+} \mapsto +1,\ \omega^{-} \mapsto -1 :
\]

\[
  \Omega_{X} \mid Y \longrightarrow \underline{\operatorname{Hom}}(\rho_{Y/X}, \Omega_{Y})
\]

\[
  \Omega_{X} \times \rho_{Y/X} \longrightarrow \Omega_{Y}
\]

\[
  \Omega_{X} \mid Y \longrightarrow \underline{\operatorname{Hom}}(\rho_{Y/X}, \Omega_{Y})
\]

\[
  \rho_{Y/X} \longrightarrow \Omega_{Y}
\]

\[
  g = \exp f \qquad f' = \varphi = \psi^{1/2}
\]

\[
  g' = (\exp f) f' \quad \text{\struck{$= (\exp f)\varphi$}}
\]

\[
  g'' = (\exp f) f'^{2} + (\exp f) f'' = (\exp f)(f'^{2} + f'')
\]

\[
  \begin{aligned}
  g''' &= (\exp f) f'(f'^{2} + f'') + (\exp f)(2f'f'' + f''') \\
       &= (\exp f)(f'^{3} + 3f'f'' + f''')
  \end{aligned}
\]

\[
  \frac{g'''}{g'} = f'^{2} + 3f'' + \frac{f'''}{f'}
\]

\[
  \frac{g''}{g'} = f' + \frac{f''}{f'} \qquad \Bigl(\frac{g''}{g'}\Bigr)^{2} = f'^{2} + \Bigl(\frac{f''}{f'}\Bigr)^{2} + 2f''
\]

\[
  \begin{aligned}
  -2\frac{g'''}{g'} + 3\Bigl(\frac{g''}{g'}\Bigr)^{2}
    &= -2f'^{2} \text{\struck{$- 6f''$}} - 2\frac{f'''}{f'} \\
    &\quad + 3f'^{2} + 3\Bigl(\frac{f''}{f'}\Bigr)^{2} \text{\struck{$+ 6f''$}}
  \end{aligned}
\]

\[
  C(t, \exp f) = C(t, f) + 3(df)^{2}
\]

\[
  -c_{t} + c_{\exp f} = -c_{t} + c_{f} + 3(df)^{2}
\]

\[
  \varphi = \exp t \qquad \varphi' = \exp t \qquad \varphi'' = \exp t \qquad \varphi''' = \exp t
\]

\[
  \Bigl(-2\frac{\varphi'''}{\varphi'} + 3\Bigl(\frac{\varphi''}{\varphi'}\Bigr)^{2}\Bigr) dt^{2} = dt^{2}
\]

\[
  \text{\struck{$\ell$}}\ C(z, f) = \Bigl(-2\frac{f'''}{f'} + 3\Bigl(\frac{f''}{f'}\Bigr)^{2}\Bigr) dz^{2}
\]

\[
  C(f, z)\ \text{\struck{$df^{2}$}} = 0 \qquad \frac{d^{3}z}{df^{3}} \ldots
\]

\[
  C(f, z) = \ldots \qquad C(\gamma z, f) = \ldots
\]

\[
  \gamma.\,C(z, f) = C(\gamma.z, \gamma.f) = C(\gamma z, f) = C(z, f) \ldots
\]

\[
  \underline{D}^{*} : \mathrm{BT}(S)^{\circ} \longrightarrow \mathrm{Crisloclib}(S),
\]

\[
  \mathrm{Crisloclib}(S) \xrightarrow{\ \approx\ } \mathrm{Crisloclib}(S_{0})
\]

\[
  \underline{D}^{*} : \mathrm{BT}(S_{0})^{\circ} \longrightarrow \mathrm{Crisloclib}(S_{0}),
\]

\[
  V_{M} F_{M} = p\, \mathrm{id}_{M}, \qquad F_{M} V_{M} = p\, \mathrm{id}_{M^{(p)}} .
\]

\[
  \underline{D}^{*} : \mathrm{BT}(S_{0}) \longrightarrow \text{F-V-Cris}(S_{0}) .
\]

\[
  (*) \qquad 0 \longrightarrow \underline{\omega}_{G} \longrightarrow \underline{D}^{*}(G) \longrightarrow \underline{t}_{G^{*}} \longrightarrow 0 ,
\]

\[
  \mathrm{BT}(S) \longrightarrow \text{F-V-cris fil}\,(S) ,
\]

\[
  \mathrm{BT}(S') \Longrightarrow
\]

\[
  \phi : S_{0\,\mathrm{cris}/\underline{F}_p} \overset{\varepsilon}{\longrightarrow} S_0 .
\]

\[
  M_0^{(p)} \xrightarrow{F_{M_0}} M_0 \xrightarrow{V_{M_0}} M_0^{(p)} \xrightarrow{F_{M_0}} M_0
\]

\[
  \operatorname{Ker} V_{M_0} = \operatorname{Im} F_{M_0} \subset M_0 , \qquad
  \operatorname{Ker} F_{M_0} = \operatorname{Im} V_{M_0} \subset M_0^{(p)} .
\]

\[
  \phi^*(M_{S_0}) \simeq M_0^{(p)}
\]

\[
  \phi^*(\mathrm{Fil}^1(M_{S_0})) \subset M_0^{(p)} .
\]

\[
  \phi^*(\mathrm{Fil}^1(M_{S_0})) = \operatorname{Ker} F_{M_0} \;(= \operatorname{Im} V_{M_0}) \subset M_0^{(p)} .
\]

\[
  \mathrm{Schémab}(S) \longrightarrow \mathrm{Crisloclib}_{\mathrm{nilp}}(S)
\]

\[
  (*) \qquad \mathbb{D}^* : A \longmapsto R^1 f_{A\mathrm{cris}*}(\underline{O}_{A_{\mathrm{cris}}})
\]

\[
  f_{\mathrm{cris}} : A_{\mathrm{crisnilp}} \longrightarrow S_{\mathrm{crisnilp}}
\]

\[
  \mathbb{D}^*(A)_S = \text{\struck{\ill{}}}\; \underline{H}^1_{\mathrm{DR}}(A/S) \overset{\mathrm{dfn}}{=} R^1 f_{A*}(\Omega^{\bullet}_{A/S}) .
\]

\[
  \mathbb{D}^*(A)_{S'} = \text{\struck{\ill{}}}\; \underline{H}^1_{\mathrm{DR}}(A'/S') .
\]

\[
  0 \longrightarrow R^0 f_*(\underline{\Omega}^1_{A/S}) \longrightarrow \underline{H}^1_{\mathrm{DR}}(A/S) \longrightarrow R^1 f_*(\underline{O}_S) \longrightarrow 0 ,
\]

\[
  (**) \qquad 0 \longrightarrow \underset{\substack{\parallel \\ \underline{\omega}_A}}{\underline{\check{t}}_A} \longrightarrow \mathbb{D}^*(A)_S \longrightarrow \underline{t}_{A^*} \longrightarrow 0 .
\]

\[
  \mathbb{D}^* : \mathrm{Schémab}(S) \longrightarrow \mathrm{Crisloclib}_{\mathrm{nilp}}\,\mathrm{fil}(S)
\]

\[
  \mathrm{Schémab}(S') \longrightarrow
\]

\[
  0 \longrightarrow \underline{\check{t}}_{A^*} \longrightarrow E(A) \longrightarrow A \longrightarrow 0 .
\]

\[
  E(A) \simeq \underline{E}(A)_S .
\]

\[
  \mathrm{Schémab}(S) \longrightarrow \mathrm{Cris\ Groupes\ lisses}_{\mathrm{nilp}}(S)
\]

\[
  \underline{E}(A)_{S'} = E(A') ,
\]

\[
  \underline{E}(A^*) = R^1 f_{A\mathrm{cris}*}(\underline{G}_{m\,A_{\mathrm{cris}}}) ,
\]

\[
  \text{\struck{\ill{}}}\; \mathbb{D}^*(A) = \underline{\mathrm{Lie}}(\underline{E}(A^*)) ,
\]

\[
  \underline{E}(A^*)_{S'}/L
\]

\[
  \mathbb{D}_*(A) = \underline{\mathrm{Lie}}\,\underline{E}(A) \xrightarrow[\sim]{\text{isom can}} \mathbb{D}^*(A^*)
\]

\[
  \mathbb{D}^*(A) \otimes \mathbb{D}^*(A^*) \longrightarrow \underline{O}_{S_{\mathrm{crisnilp}}}
\]

\[
  \mathbb{D}^*(A(\infty)) \simeq R^1 f_{\mathrm{cris}*}(\underline{O}_{A_{\mathrm{cris}}}) ,
\]

\[
  E(A(\infty)) \simeq E(A)(\infty)
\]

\[
  \underline{E}(A(\infty)) \simeq \underline{E}(A)(\infty) \quad \ldots
\]

\[
  \mathrm{Fil}^{1\,(p)} = \operatorname{Ker} F_M = \operatorname{Im} V_M ,
\]

\[
  \mathrm{IsoBT}(S) \qquad \mathrm{Iso\,F\text{-}V\text{-}cris}(S)
\]

\[
  \mathrm{IsoBT}(S') \longrightarrow \mathrm{IsoBT}(S)
\]

\[
  \left\{
  \begin{aligned}
    &\textstyle\sum_1^i \lambda'_j \leq \sum_1^i \lambda_j \\
    &\textstyle\sum_1^n \lambda'_j = \sum \lambda_j
  \end{aligned}
  \right.
  \qquad \textstyle\sum i h'_i
\]

\[
  \mathrm{BT}(S) \longrightarrow \mathrm{BT}(\eta)
\]

\[
  \mathrm{Cris\ loc\ lib}(S) \longrightarrow \mathrm{Cris\ loc\ lib}(\eta)
\]

\[
  \chi_!(X) = \chi(X) .
\]

\[
  \chi_!(X) = \chi_!(Y) + \chi_!(X-Y)
\]

\[
  \chi(X) = \chi(X') + \chi(X'') - \chi(X' \cap X'')
\]

\[
  \chi(X) = \chi(X')
\]

\[
  \chi(\mathrm{pt}) = 1
\]

\[
  \left\{
  \begin{aligned}
    &\chi(\mathbb{R}^n) = \chi(\overline{\mathbb{R}}^n) = \chi(B^n) = 1 \\
    &\phantom{\chi(\mathbb{R}^n) = {}} \chi_!(\overline{\mathbb{R}}^n) = \chi_!(B^n) \\
    &\chi_!(\mathbb{R}^n) = (-1)^n
  \end{aligned}
  \right.
  \qquad
  \chi(S^n) = \chi_!(S^n) = 1 + (-1)^n
\]

\[
  \chi(S^n) = \chi(S^n_+) + \chi(S^n_-) - \chi(S^{n-1}) = 2 - \chi(S^{n-1}) ,
\]

\[
  \chi(S^n) = 1 + (-1)^n .
\]

\[
  \chi(\mathbb{R}^n) = \chi(S^n) - \chi(\mathrm{pt}) = (1 + (-1)^n) - 1 = (-1)^n ,
\]

\[
  \chi_!(X) = \sum (-1)^i f_i
\]

\[
  \chi_!(X) = \chi_!(Y)\chi_!(F) .
\]

\[
  \chi(X) = \chi(Y)\chi(F)
\]

\[
  H^*_!(X) \Longleftarrow E_2^{pq} = H^p_!(Y, \underbrace{R^q f_!(k_X)}_{\substack{\text{syst.\ local sur } Y \\ \text{à fibres} \sim H^q_!(F)}})
\]

\[
\begin{aligned}
  \chi(X) &= \sum (-1)^{p+q} \operatorname{rg}_k H^p_!(Y, R^q f_!(k_X)) \\
  &= \sum_q (-1)^q \underbrace{\sum_p (-1)^p \operatorname{rg}_k H^p_!(Y, R^q f_!(k_X))}_{b^q_!(F)\chi_!(Y)} \\
  &= \chi_!(Y) \sum_q (-1)^q b^q_!(F) = \chi_!(Y)\chi_!(F)
\end{aligned}
\]

\[
  \chi_!(Y, \mathcal{R}) \overset{\mathrm{déf}}{=} \sum_p (-1)^p H^p(Y, \mathcal{R})
\]

\[
  \chi_!(Y, \mathcal{R}) = b\,\chi_!(Y)
\]

\[
  \chi_!(Y, \mathcal{R}) = \sum_{ij} \chi_!(Z_{ij}, \mathcal{R}) \underset{(\text{cellules } Z_{ij})}{=} \sum_{ij} b\,\chi_!(Z_{ij}) = b \sum \chi_!(Z_{ij}) = b\,\chi_!(Y) .
\]

\[
\chi(Y, R) \overset{\mathrm{def}}{=} \sum_p (-1)^p H_p^{\flat}(Y, R)
\]

\[
\chi(Y, R) = b\,\chi(Y)
\]

\[
\chi(X) = \chi(\hat{X} - \mathring{T}) = \chi(\hat{X}) - \chi(T) + \chi(\partial T)
\]

\[
\chi(X) = \chi(\hat{X}) - 1 + \chi(\partial T) = \chi_!(X) + \chi(\partial T), \qquad \partial T = S_\omega,
\]

\[
\boxed{\chi(X) - \chi_!(X) = \chi(S_\omega)}
\]

\[
\begin{cases} \chi_!(X) = \chi_!(Y)\,\chi_!(F) \\ \chi(X) = \chi(Y)\,\chi(F) \end{cases}
\]

\[
\begin{cases} \chi_!(X) = \chi(\hat{X}) - 1 \\ \chi(X) - \chi_!(X) = \chi(S) \end{cases}
\]

\[
\chi_!(X) = \sum (-1)^i f_i \qquad \text{(formule d'Euler)}
\]

\[
G : \mathcal{E}_X \times \mathcal{E}_X \longrightarrow \underline{\omega}_{X/\mathbb{C}}^{\otimes 2} \qquad (\underline{\omega}_{X/\mathbb{C}} = \underline{\Omega}^1_{X/\mathbb{C}})
\]

\[
G(f,g) = R(f, f', \dots, f^{(n)}, g, g', \dots, g^{(n)})\, dt^2
\]

\[
G(f,g) = \frac{1}{f'^{\,?} g'^{\,?}}\, P(f, f', \dots, f^{(n)}; g, g', \dots, g^{(n)}) \;]
\]

\[
\begin{array}{ll}
G(f, ag) = G(f,g) & a \in \mathbb{C}^* \\
G(f, g+b) = G(f,g) & b \in \mathbb{C} \\
G(f, 1/g) = G(f,g) & \text{si } g \text{ sans zéros}
\end{array}
\]

\[
G' = \varphi\, G, \quad \text{avec } \varphi \in \Gamma(X, \underline{\mathcal{O}}_X^*)
\]

\[
G' = c\, G \qquad c \in \mathbb{C}^*
\]

\[
R \in \mathbb{C}(F_0, F_1, \dots, F_n; G_0, \dots, G_n)
\]

\[
R(\ ) = \underbrace{P(F_0, \dots; G_0, \dots)}_{\in\, \mathbb{C}[F_0, \dots, G_n]} / F_1^{\,?}\, G_1^{\,?} \;]
\]

\[
G(f,g) = \left( \left[ -2\, \frac{g'''}{g'} + 3 \left( \frac{g''}{g'} \right)^2 \right] - \left[ -2\, \frac{f'''}{f'} + 3 \left( \frac{f''}{f'} \right)^2 \right] \right) dt^2
\]

\[
c^X : \mathcal{P}_X \longrightarrow \mathcal{T}_X
\]

\[
\begin{cases}
G(f,g) = c^X(g) - c^X(f) \\
c^X \text{ se factorise en } {}_H\backslash \mathcal{P}_X \to \mathcal{T}_X \\
\text{ce dernier morphisme est un iso}
\end{cases}
\]

\[
\mathcal{E}_X \times \mathcal{E}_X \xrightarrow{\ G\ } \omega^{2}_{X/S}
\]

\[
f,\ f' = D_t^{(1)} f,\ D_t^{(2)}(f), \dots, D_t^{(n)} f,\ g,\ g' = D_t^{(1)} g,\ D_t^{(2)} g, \dots, D_t^{(n)} g.
\]

\[
G_0(f,g) = \left[ \left( -\frac{D_t^{(3)} g}{D_t^{(1)} g} + \left( \frac{D_t^{(2)} g}{D_t^{(1)} g} \right)^2 \right) - \left( -\frac{D_t^{(3)} f}{D_t^{(1)} f} + \left( \frac{D_t^{(2)} f}{D_t^{(1)} f} \right)^2 \right) \right] dt^2
\]

\[
G_0(t, t^2) = \text{\struck{$\frac{1}{2t}$}}\ \frac{1}{2t}\, dt^2 \qquad \left(\text{au lieu de } -\frac{1}{2t}\, dt^2\right)
\]

\[
G_0(t, t^3) = \underbrace{\left( -\frac{1}{3t^2} + \left( \frac{3t}{3t^2} \right)^2 \right)}_{\frac{1}{t^2}\left(-\frac{1}{3} + 1\right)} dt^2 = \text{\struck{$\frac{1}{3t^2}$}}\ \frac{2}{3}\, \frac{1}{t^2}\, dt^2
\]

\[
\mathcal{T}^G \xrightarrow{\ \sim\ } \mathcal{T}^{\varphi G}
\quad \text{au-dessus de} \quad
\underline{\omega}_X^{\otimes 2} \xrightarrow{\ \varphi \cdot \mathrm{id}\ } \underline{\omega}_X^{\otimes 2}
\]

\[
d : \underline{\mathcal{O}}_X \longrightarrow \underline{\Omega}^1_{X/\mathbb{C}} = \underline{\omega}_X
\]

\[
A(f,g) = dg - df, \qquad A : \underline{\mathcal{O}}_X \times \underline{\mathcal{O}}_X \longrightarrow \underline{\omega}_X
\]

\[
A(f, g+c) = A(f,g) \qquad c \in \mathbb{C}
\]

\[
A(f,g) = A(f,h),\ g(x) = h(x) \Longrightarrow g = h \text{ au vois.\ de } x
\]

\[
g \longmapsto A(f,g) \qquad \underline{\mathcal{O}}_X \to \underline{\omega}_X \text{ épi}
\]

\[
L^* \xrightarrow{\ D\ } \underline{\mathrm{Conn}}_{X/\mathbb{C}}(L) \qquad (f \longmapsto D_f)
\]

\[
L^*/\mathbb{C}^* \xrightarrow{\ \sim\ } \underline{\mathrm{Conn}}_{X/\mathbb{C}}(L)
\]

\[
B(f,g) = \frac{d\, g/f}{g/f} \in \Gamma(\underline{\omega}_{X/\mathbb{C}})
\]

\[
g \longmapsto \frac{dg}{g} \qquad \underline{\mathcal{O}}_X^* \longrightarrow \underline{\omega}_{X/\mathbb{C}}
\]

\[
g \longmapsto \frac{g}{dg} \qquad \mathcal{E}_X \longrightarrow \underline{\omega}_{X/\mathbb{C}}^{\otimes(-1)}
\]

\[
C^!(f,g) = \text{\struck{$\frac{dg}{g} - \frac{df}{f}$}}\ g(dg)^{-1} - f(df)^{-1}
\]

\[
\mathcal{E}_X^* = \underline{\mathrm{Et}}_{\mathrm{an}}(X, \mathbb{C}^*)
\]

\[
C^{\natural}(f,g) = \frac{dg}{g} : \frac{df}{f} = \frac{dg}{df}\, \frac{f}{g} \in \Gamma(X, \underline{\mathcal{O}}_X^*)
\]

\[
{}_{\mathrm{Aff}(1,\mathbb{C})}\backslash \mathcal{E}_X \xrightarrow{\ \sim\ } \underline{\mathrm{Conn}}(\underline{\omega}_X)
\]

\[
{}_{\mathbb{G}_a(\mathbb{C})}\backslash \mathcal{E}_X \xrightarrow[\ \sim\ ]{d} \underline{\omega}_X^*
\]

\[
{}_{\mathbb{G}_m(\mathbb{C})}\backslash \underline{\omega}_X^* \xrightarrow{\ \sim\ } \underline{\mathrm{Conn}}(\underline{\omega}_X)
\]

\[
D(f,g) = \frac{d\,\frac{dg}{df}}{\frac{dg}{df}} \in \Gamma\, \underline{\omega}_X
\]

\[
D(f,g) = \frac{d\,\frac{g'}{f'}}{\frac{g'}{f'}} = \frac{dg'}{g'} - \frac{df'}{f'} = \left( \frac{g''}{g'} - \frac{f''}{f'} \right) dt
\]

\[
f \longmapsto \frac{1}{f} \quad \text{et} \quad f \longmapsto cf \quad (c \in \mathbb{C}^*)
\]

\[
\mathcal{E}_X^* \longrightarrow (\underline{\omega}_X^{\otimes 2})^*, \qquad
f \longmapsto \frac{df}{f} \in \underline{\omega}_X^*, \quad \omega \longmapsto \omega^2
\]

\[
{}^{c}\!\!\int \omega^{1/n} = {}^{c}\!\!\int f^{1/n}\, dt \quad \text{\struck{$= c_t +$}}
\]

\[
c_n(\omega) = c_t + \left[ -\frac{2}{n}\, \frac{f''}{f} + \frac{2n+1}{n^2} \left( \frac{f'}{f} \right)^2 \right] dt^2 \qquad \text{si } n \neq 0
\]

\[
\begin{aligned}
G_{m,n}(\omega_m, \varpi_n) &= c_n(\varpi_n) - c_m(\omega_m) \\
&= \left[ \left( -\frac{2}{n}\, \frac{g''}{g} + \frac{2n+1}{n^2} \left( \frac{g'}{g} \right)^2 \right) \right. \\
&\qquad \left. - \left( -\frac{2}{m}\, \frac{f''}{f} + \frac{2m+1}{m^2} \left( \frac{g'}{g} \right)^2 \right) \right] dt^2
\end{aligned}
\]

\[
\begin{cases}
G_{m,n}(\omega_m, \omega'_n) + G_{n,p}(\omega'_n, \omega''_p) = G_{m,p}(\omega_m, \omega''_p) \\
G_{m,n}(\omega_m, \omega_n) = G_{m,nr}(\omega_m, (\omega_n)^r) \quad \text{si } n \neq 0,\ r \neq 0 \\
G_{m,n}(\omega_m, ds) = G_{m,0}(\omega_m, s) \\
G_{0,0}(t, s) = G(t, s)
\end{cases}
\]

\[
G_{m,n}(\omega_m, (ds)^n) = G_{m,0}(\omega_m, s) \qquad c'_n(\omega) = n^2\, \frac{c_n(\omega)}{12} \in \mathcal{T}_X^{\otimes n^2}
\]

\[
G_{m,n}((dt)^m, (ds)^n) = G_{0,0}(t, s)
\]

\[
\omega_m \longmapsto \alpha^2 c'_m(\omega_m) = \alpha^2 m^2\, \frac{c_m(\omega_m)}{12} = \delta^2\, \frac{c_m(\omega_m)}{12} \in \Gamma\, \mathcal{T}_X^{\otimes \delta^2}
\]

\[
\omega'_n \longmapsto \beta^2 c'_n(\omega'_n) = \beta^2 n^2\, \frac{c_n(\omega'_n)}{12} = \delta^2\, \frac{c_n(\omega'_n)}{12} \in \Gamma\, \mathcal{T}_X^{\otimes \delta^2}
\]

\[
G'_{m,n}(\omega_m, \omega'_n) = \alpha^2 c'_m(\omega_m) - \beta^2 c'_n(\omega'_n) \in \Gamma\, \mathcal{T}_X^{\otimes \delta^2}
\]

\[
\begin{cases}
G(t, as) = G(t,s) & a \in \Gamma(S, \underline{\mathcal{O}}_S^*) \\
G(t, s+b) = G(t,s) & b \in \Gamma(S, \underline{\mathcal{O}}_S) \\
G(t, \tfrac{1}{s}) = G(t,s) & s \in \underline{\mathrm{Et}}_S(X, \mathbb{G}_{m,S})
\end{cases}
\]

\[
A_1(f,g) = dg/df = \frac{g'}{f'}, \qquad
A_n(f,g) = \Bigl(\frac{dg}{df}\Bigr)^n = \Bigl(\frac{g'}{f'}\Bigr)^n
\]

\[
A_{\mathrm{af}}(f,g) = D_{dg} - D_{df}
= \frac{d\,\frac{dg}{df}}{dg/df}
= \frac{d\,\frac{g'}{f'}}{g'/f'}
= \frac{dg'}{g'} - \frac{df'}{f'}
= \Bigl(\frac{g''}{g'} - \frac{f''}{f'}\Bigr)dt,
\qquad c_{\mathrm{af}}(f) = D_{df}
\]

\[
c_{D+\omega_1} = c_D + 2D\omega_1 + \omega_1^2 .
\]

\[
\exp(\pm f + b) = \exp b\,(\exp f)^{\pm 1}
\]

\[
\begin{aligned}
\text{Écad}(f,g) &= \frac{dg}{df} = \frac{g'}{f'} \in \Gamma\,\underline{O}^*_X\\
\text{Écaf}_n(f,g) &= \Bigl(\frac{dg}{df}\Bigr)^n = \Bigl(\frac{g'}{f'}\Bigr)^n \in \Gamma(\underline{O}^*_X)\\
\text{Écaf}(f,g) &= \frac{d(dg/df)}{dg/df} = \frac{d(g'/f')}{g'/f'} = \frac{dg'}{g'} - \frac{df'}{f'}\\
&= \Bigl(\frac{g''}{g'} - \frac{f''}{f'}\Bigr)dt \in \Gamma(\underline{\omega}^1_X)\\
\text{Écmu}(f,g) &= \frac{dg/g}{df/f} = \frac{g'/g}{f'/f} \in \Gamma(\underline{O}^*_X)\\
\text{Écmu}_2(f,g) &= \Bigl(\frac{dg/g}{df/f}\Bigr)^2 = \Bigl(\frac{g'/g}{f'/f}\Bigr)^2 \in \Gamma(\underline{O}^*_X)\\
\text{Écpr}(f,g) &= -2\,\frac{d^3g/df^3}{dg/df} + 3\Bigl(\frac{d^2g/df^2}{dg/df}\Bigr)^2\\
&= \Bigl[\Bigl(-2\frac{g'''}{g'} + 3\Bigl(\frac{g''}{g'}\Bigr)^2\Bigr) - \Bigl(-2\frac{f'''}{f'} + 3\Bigl(\frac{f''}{f'}\Bigr)^2\Bigr)\Bigr]dt^2\\
&\in \Gamma(\underline{\omega}^2_X)
\end{aligned}
\]

\[
\begin{aligned}
\alpha_n(\omega_1) &= \omega_1^n\\
\beta_n(\underset{\substack{\|\\ f\,dt^n}}{\omega_n}) &= \beta_1(\omega_n^{1/n}) = \beta_1(f^{1/n}\,dt) = D_{dt} + \frac{1}{n}\frac{df}{f} = D_{dt} + \frac{1}{n}\frac{f'}{f}\,dt\\
\gamma(D_{dt} + \underset{\substack{\|\\ f\,dt}}{\omega_1}) &= \underset{\substack{\|\\ c_{\mathrm{pr}}(t)}}{\gamma(D_{dt})} + 2D_{dt}(\omega_1) + \omega_1^2\\
&= c_{\mathrm{pr}}(t) + 2\frac{df}{dt}\,dt + f^2\,dt^2 = c_{\mathrm{pr}}(t) + (2f' + f^2)\,dt^2\\
\gamma\beta_n(\omega_n) &= c_{\mathrm{pr}}(t) + \Bigl(2\Bigl(\frac{1}{n}\frac{f'}{f}\Bigr)' + \Bigl(\frac{1}{n}\frac{f'}{f}\Bigr)^2\Bigr)dt^2\\
&= c_{\mathrm{pr}}(t) + \frac{1}{n^2}\Bigl(-2n\frac{f''}{f}\ \ldots\ \Bigl(\frac{f'}{f}\Bigr)^2\Bigr)dt^2\\
&= c_{\mathrm{pr}}(t) + \Bigl(-2\frac{f''}{f} + 3\Bigl(\frac{f'}{f}\Bigr)^2\Bigr)dt^2\\
\gamma\beta_1\,d(f) &= c_{\mathrm{pr}}(t) + \Bigl(-2\frac{f'''}{f'} + 3\Bigl(\frac{f''}{f'}\Bigr)^2\Bigr)dt^2
\end{aligned}
\]

\[
\begin{aligned}
\delta(\omega_2) &= \gamma\beta_2(\omega_2) + \omega_2\\
&= c_{\mathrm{pr}}(t) + \Bigl[\ldots\frac{f''}{f} + \frac{5}{4}\Bigl(\frac{f'}{f}\Bigr)^2\Bigr]dt^2\\
\delta\alpha'_2(\omega_1) &= \gamma\beta_1(\omega_1) + \omega_1^2\\
&= c_{\mathrm{pr}}(t) + \Bigl[f^2 - 2\frac{f''}{f} + 3\Bigl(\frac{f'}{f}\Bigr)^2\Bigr]dt^2
\end{aligned}
\]

\[
\begin{aligned}
&k_1 \in H^3(\pi_1(X), \pi_2(X)) \overset{?}{=} 0\\
&\qquad\uparrow \qquad\qquad \uparrow\\
\exists\ &k_1' \in H^4(K(\pi_1(X),2), \pi_2(X))\\
&\qquad\qquad \uparrow\\
&k_1'' \quad H^5(K(\pi_1(X),3), \pi_2(X))\\
&\qquad\qquad \simeq\\
&\qquad H^{5+i}(K(\pi_1(X),3+i), \pi_2(X))
\end{aligned}
\]

\[
\Bigl(\frac{df}{f}\Bigr)^2 = \frac{(df)^2}{f^2} = \Bigl(\frac{f'}{f}\Bigr)^2 dt^2
\]

\[
E(f,g) = \Bigl(\frac{dg}{g}\Bigr)^2 \Big/ \Bigl(\frac{df}{f}\Bigr)^2 = \frac{\bigl(\frac{dg}{df}\bigr)^2}{(g/f)^2}
\]

\[
E = f_*(L),
\]

\[
M = \check{E}\otimes E \simeq \underline{\mathrm{End}}(E) \simeq E\otimes E\otimes(\textstyle\bigwedge^2 E)^{-1}
\]

\[
M \overset{?}{\simeq} f_*(\mathrm{P}^1_{P/X}(\underline{t}_{P/X}))\quad \text{\uncertain{addition}\,?}
\]

\[
\simeq f_*(\underline{t}_{P/X})
\]

\[
0 \to \Omega \to \mathcal{L} \to \underline{O}_X \to 0,
\]

\[
\simeq f_*(\underline{L}/\mathcal{J}\underline{L}) \qquad (\mathcal{J} = \text{Idéal de } \sigma(X) \text{ dans } P)
\]

\[
\simeq \sigma^*(\mathrm{P}^1(\underline{L}))
\]

\[
\mathfrak{g} = M/\underline{O}_X \simeq \mathrm{Sym}^2(\mathcal{L})\otimes\Omega^{-1} \qquad (\textit{NB}\ \det\mathcal{L}\simeq\Omega)
\]

\[
\begin{array}{ccc}
\Omega & \underline{O}_X & \Omega^{-1}\\
\| & \| & \|\\
\mathrm{Fil}_1 & \mathrm{Fil}_2/\mathrm{Fil}_1 & \mathrm{Fil}_3/\mathrm{Fil}_2\\
\| & \| & \|\\
\mathrm{gr}_1 & \mathrm{gr}_2 & \mathrm{gr}_3
\end{array}
\]

\[
\underbrace{\Omega \subset \overbrace{\mathcal{L}}^{\underline{O}_X} \subset \mathfrak{g}}_{}
\qquad \text{(gradués : } \Omega,\ \underline{O}_X,\ \Omega^{-1})
\]

\[
(P, \sigma, c)
\]

\[
\begin{cases}
P \text{ fibré en droites projectives sur } X\\
\sigma \text{ section}\\
c \text{ connexion de } P \text{ rel. à } S
\end{cases}
\]

\[
0 \to \Omega \to \mathcal{L} \to \underline{O}_X \to 0
\]

\[
\mathfrak{g} \simeq \underset{\substack{\|\\ \mathcal{L}_{-}}}{\Omega} + \underset{\substack{\|\\ \underline{O}_X}}{\underline{O}^0_X} + \Omega^{-1},
\qquad \mathcal{L} = \Omega + \underline{O}^0_X
\]

\[
\begin{aligned}
[\omega_1, \lambda] &= -[\lambda, \omega_1] = -\lambda\omega_1\\
[\omega_{-1}, \lambda] &= -[\lambda, \omega_{-1}] = \lambda\omega_{-1}\\
[\omega_1, \omega_{-1}] &= -[\omega_{-1}, \omega_1] = 2\omega_1\omega_{-1}
\end{aligned}
\]

\[
-2\lambda\omega_1,\quad 2\lambda\omega_{-1},\quad \omega_1\omega_{-1}\ )
\]

\[
\mathfrak{g} \simeq \Omega + \underset{\substack{\|\\ \underline{O}_X}}{\Omega^0} + \Omega^{-1}
\]

\[
\mathfrak{g}\otimes\omega \simeq \Omega\otimes\omega + \omega + \Omega^{-1}\otimes\omega
\]

\[
\mathfrak{g} \xrightarrow{\ c_0\ } \mathfrak{g}\otimes\omega
\]

\[
c_0(\delta) = c_0(1) = c_0(\delta^{-1}) = 0
\]

\[
c_0(\underbrace{\lambda\delta + \mu\cdot 1 + \nu\delta^{-1}}_{\substack{\updownarrow\\ (\lambda,\mu,\nu)}}) = \underbrace{\delta\,d\lambda + 1\,d\mu + \delta^{-1}d\nu}_{\substack{\updownarrow\\ (d\lambda,\,d\mu,\,d\nu)}}
\]

\[
\mathrm{Der}(\mathfrak{g},\mathfrak{g})\otimes\omega \simeq \mathfrak{g}\otimes\omega
\]

\[
\begin{aligned}
\varpi_2 &\in \Gamma(\Omega\otimes\omega)\\
\varpi_1 &\in \Gamma(\omega)\\
\varpi_0 &\in \Gamma(\Omega^{-1}\otimes\omega)
\end{aligned}
\]

\[
c_{\varpi}(\underbrace{\lambda\delta + \mu\cdot 1 + \nu\delta^{-1}}_{(\lambda,\mu,\nu)}) \overset{?}{=} \text{\struck{\ill{}}}
\]

\[
\begin{aligned}
c_{\varpi}(\underset{\substack{\|\\ \omega_0}}{\lambda\delta}) &= \underbrace{\delta\,d\lambda}_{\text{poids } 2} + \lambda\bigl[\underset{\substack{\|\\ 0}}{[\varpi_2,\lambda\delta]} + [\varpi_1,\lambda\delta] + [\varpi_0,\lambda\delta]\bigr]\\
&\qquad + \underbrace{\varpi_1(\lambda\delta)}_{\text{poids } 2}\ \ 2\varpi_0(\lambda\delta)
\end{aligned}
\]

\[
\begin{aligned}
c_{\varpi}(\underset{\substack{\|\\ \lambda\delta}}{\omega_1}) &= \overset{2}{\delta\,d\lambda} + \omega_1(\overset{2}{\varpi_1} - \overset{1}{2\varpi_0})\\
c_{\varpi}(\underset{\substack{\|\\ \mu\cdot 1}}{\omega_0}) &= \overset{1}{d\mu} + \omega_0(-\overset{2}{\varpi_2} + \overset{0}{\varpi_0})\\
c_{\varpi}(\underset{\substack{\|\\ \nu\delta^{-1}}}{\omega_{-1}}) &= \overset{0}{\delta^{-1}d\nu} + \omega_{-1}(\overset{1}{2\varpi_2} - \overset{0}{\varpi_1})
\end{aligned}
\]

\[
\mathfrak{g} : \Omega,\ \underset{\substack{\|\\ \underline{O}_X}}{\Omega^0},\ \Omega^{-1}
\qquad\qquad
\mathfrak{g}\otimes\underline{\omega} : \Omega\otimes\underline{\omega},\ \underline{\omega},\ \Omega^{-1}\otimes\underline{\omega}
\]

\[
c_{i,i+1} : \mathrm{gr}_i\,\mathfrak{g} \to \mathrm{gr}_{i+1}(\mathfrak{g}\otimes\omega) \simeq \mathrm{gr}_{i+1}(\mathfrak{g})\otimes\omega
\]

\[
\begin{aligned}
c_{1,2} &: \mathrm{gr}_1 \to \mathrm{gr}_2,\quad \Omega \to \omega && \text{i.e. } c_{12} \in \Gamma\,\Omega^{-1}\otimes\underline{\omega}\\
c_{2,3} &: \mathrm{gr}_2 \to \mathrm{gr}_3,\quad \underline{O}_X \to \Omega^{-1}\otimes\omega && \text{i.e. } c_{23} \in \Gamma\,\Omega^{-1}\otimes\underline{\omega}
\end{aligned}
\]

\[
\boxed{c_{1,2} = -2\,c_{2,3}}
\]

\[
c^{\Omega} : \underset{\substack{\wr\\ \mathcal{L}'\simeq\mathcal{L}\otimes\Omega^{-1}}}{\mathfrak{g}/\Omega} \longrightarrow \Omega^{-1}\otimes\underline{\omega}
\]

\[
\text{on a}\quad \mathfrak{g}\otimes\underline{\omega} \to \Omega^{-1}\otimes\underline{\omega}
\]

\[
\underline{\mathrm{Conn}} \xrightarrow{\ \varpi_0\ } \Omega^{-1}\otimes\omega
\]

\[
\begin{aligned}
\Omega = \mathrm{gr}_1(\mathfrak{g}) &\xrightarrow[-2\varpi_0]{} \omega \simeq \mathrm{gr}_2(\mathfrak{g}\otimes\underline{\omega})\\
\Omega^0 \simeq \underline{O}_X = \mathrm{gr}_2\,\mathfrak{g} &\xrightarrow[\varpi_0]{} \Omega^{-1}\otimes\underline{\omega} = \mathrm{gr}_3(\mathfrak{g}\otimes\underline{\omega})
\end{aligned}
\]

\[
\mathcal{L}\otimes\underline{\omega} \to \underline{\omega}
\]

\[
\Omega \xrightarrow{\ c_{\Omega}\ } \mathcal{L}\otimes\omega
\]

\[
(**)\quad c_{\Omega}(f\omega) - f\,c_{\Omega}(\omega) = \omega\otimes df \qquad \omega\in\Gamma\,\Omega,\ f\in\Gamma\,\underline{O}_X
\]

\[
(-2i) : \Omega \longrightarrow \mathcal{L}\otimes\underline{\omega} \to \underline{\omega}
\]

\[
(\mathfrak{g}, \mathcal{L}, c_{\Omega}) \quad (\text{où, ce qui revient au même,}\ (\mathcal{L}\supset\Omega,\ c_{\Omega}))
\]

\[
\mathcal{L} \simeq \mathrm{P}^1_{X/S}(\underline{\omega})\otimes\underline{\omega}^{-1}
\qquad (\text{d'où } \mathcal{L}\otimes\underline{\omega} \simeq \mathrm{P}^1_{X/S}(\underline{\omega}))
\]

\[
c_{\Omega} : \Omega \simeq \underline{\omega} \longrightarrow \mathcal{L}\otimes\underline{\omega} \simeq \mathrm{P}^1_{X/S}(\underline{\omega})
\]

\[
c_{2,3} : \Omega \longrightarrow \underline{\omega} = \Omega^1_{X/S}
\]

\[
\Omega \simeq \sigma^*(\underline{\Omega}^1_{P/X})
\]

\[
\Omega \xrightarrow[\ \sim\ ]{\ i\ } \underline{\omega} = \Omega^1_{X/S}
\]

\[
c|\underset{\substack{\|\\ \mathrm{Fil}_1\mathfrak{g}}}{\Omega} : \Omega \longrightarrow \underbrace{\mathcal{L}\otimes\omega}_{\Omega\otimes\omega,\ \omega} \ldots
\]

\[
\underline{\Omega} \xrightarrow[\substack{\text{op. diff.}\\ \text{con.}\\ \text{d'ordre } 1}]{\ d^1_{\underline{\omega}}\ } \mathrm{P}^1_{X/S}(\underline{\Omega}) \xrightarrow[\text{lin.}]{\ \gamma\ } \mathcal{L}\otimes\underline{\omega}
\]

\[
\underline{\Omega} \xrightarrow{\ c|\underline{\omega}\ } \mathcal{L}\otimes\underline{\omega} \to \underline{\omega}
\]

\[
\begin{array}{ccccccccc}
0 \to & \Omega\otimes\omega & \to & \mathrm{P}^1_{X/S}(\Omega) & \to & \Omega & \to & 0\\
& \downarrow{\scriptstyle \gamma_0 = ?} & & \downarrow{\scriptstyle \gamma} & & \downarrow{\scriptstyle -2i} & &\\
0 \to & \Omega\otimes\omega & \to & \mathcal{L}\otimes\omega & \to & \omega & \to & 0
\end{array}
\]

\[
\gamma(\underset{\substack{\cap\\ \Gamma\Omega}}{\omega_1}\otimes \underset{\substack{\cap\\ \Gamma\underline{O}_X}}{df}) = c(f\omega_1) - f\,c(\omega_1)
\]

\[
\begin{aligned}
\gamma(\lambda\delta\otimes df) &= \delta\,[d(f\lambda) - f\,d(\lambda)]\\
&= \delta\,[\lambda\,df] = \lambda\delta\otimes df
\end{aligned}
\]

\[
\mathcal{L} \xrightarrow[\gamma^{-1}\otimes\mathrm{id}_{\omega^{-1}}]{\ \sim\ } \mathrm{P}^1_{X/S}(\Omega)\otimes\omega^{-1} \simeq \mathrm{P}^1_{X/S}(\omega)\otimes\omega^{-1}
\]

\[
c\,|\,\underset{\substack{\|\\ \mathrm{Fil}_1\mathfrak{g}\\ \|\\ \mathcal{L}^{\perp}}}{\Omega \simeq \underline{\omega}}
\quad \underset{d^1_{\underline{\omega},X/S}}{=} \quad
\underset{\substack{\wr\\ \omega}}{\Omega} \xrightarrow{\ c\ } \mathcal{L}\otimes\omega \simeq \mathrm{P}^1(\omega)
\]

\[
0 \to \Omega \to \mathcal{L} \to \underline{O}_X \to 0
\]

\[
\Omega \xrightarrow{\ c_{\Omega}\ } \mathcal{L}\otimes\underline{\omega} \qquad \underline{\omega} = \Omega^1_{X/S}
\]

\[
c_{\Omega}(f\omega) - f\,c_{\Omega}(\omega) = \omega\otimes df
\]

\[
\underline{O}_{\hat{P}} \simeq \mathrm{P}^{\infty}_{X/S}
\]

\[
\mathfrak{g} \simeq \Omega \oplus \Omega^0 \oplus \Omega^{-1} \qquad (\Omega\simeq\omega)
\]

\[
-2\varpi_2 = 0
\]

\[
\begin{aligned}
c_0(\underset{\substack{\|\\ \lambda\delta}}{\omega_1}) &= \delta\,d\lambda + \omega_1(-2\varpi_0) \qquad\qquad \varpi_0\in\Gamma\,\Omega^{-1}\otimes\omega\\
c_0(\underset{\substack{\|\\ \mu\cdot 1}}{\omega_0}) &= 1\,d\mu + \omega_0(\varpi_0)\\
c_0(\underset{\substack{\|\\ \nu\delta^{-1}}}{\omega_{-1}}) &= \delta^{-1}d\nu \quad \text{\struck{\ill{}}}
\end{aligned}
\]

\[
s = s_0 + \omega_1 \qquad (s_0 = (0,1))
\]

\[
\begin{cases}
\text{\struck{$\ldots$}} : \underline{O}_X \to \mathcal{L}\\
v_s : \Omega^{-1} \to \mathfrak{g}
\end{cases}
\qquad \text{i.e. } v_s \in \Gamma(\mathfrak{g}\otimes\Omega)
\]

\[
\begin{aligned}
s &= \exp\Theta_{\omega_1} s_0 \quad \text{\struck{$= s_0 + [-\omega_1, s_0] + [-\omega_1 \ldots$}}\\
v_s &= \exp\Theta_{\omega_1} v_{s_0} = v_{s_0} + \underbrace{[-\omega_1, v_{s_0}]}_{-2\omega_1 v_{s_0}} + \tfrac{1}{2}\underbrace{[-\omega_1,[-\omega_1, v_{s_0}]]}_{\omega_1^2 v_{s_0}}
\end{aligned}
\]

\[
\mathfrak{g} \simeq \Omega + \text{\struck{$\exp\Theta_{\ldots}$}}\,\underline{O}_X\cdot s + v_s(\Omega^{-1})
\]

\[
c(v_s) = \underbrace{\varpi_2(s)}_{\Omega\otimes\omega} + \text{\struck{\ill{}}}\ s\otimes\varpi_1(\ldots) + \underbrace{(v_s\ldots)(\varpi_0)}_{\Omega^{-1}\otimes\omega}
\]

\[
= \text{composante de } \exp\Theta_{+\omega_1}(c(s))
\]

\[
s = s_0 + \omega_1 \longmapsto c_0 + \Bigl(\delta\,d\bigl(\tfrac{\omega_1}{\delta}\delta^{-1}\bigr) + \omega_1^2\Bigr)
\]

\[
s_0 + \lambda\,dt \longmapsto c_0 + \Bigl(\frac{d\lambda}{dt} + \lambda^2\Bigr)(dt)^2
\]

\[
\frac{d\lambda}{dt} + \lambda^2 = \varphi
\]

\[
(L, \sigma)
\]

\[
\sigma : \mathrm{P}^1_{X/S}(\omega) \to L \quad \text{épi. lin.}
\]

\[
D : \underline{\omega} \longrightarrow L,
\]

\[
\sigma_0 : \underline{\omega}^2 \to L \qquad \text{i.e. } \sigma_0 \in \Gamma(L\otimes\omega^{-2})
\]

\[
D(f\omega) - f\,D(\omega) = \sigma_0(\omega\cdot df)
\]

\[
\omega \longrightarrow \underbrace{L \simeq \underline{O}_X}_{\text{\uncertain{trivialisé}}}
\]

\[
D(f\,dt) = a\frac{df}{dt} + b
\]

\[
\lambda_t = \frac{2}{t}\,(+\ \text{pas de termes constants}) + \text{termes d'ordre} \geq 1
\]

\[
P = \mathbb{P}(\mathrm{P}^1_{X/S}(\underline{\omega})),\ \sigma,\ c
\longmapsto \sigma,\ c + \varpi_2 \qquad (\varpi_2 \neq 0)
\]

\[
\begin{aligned}
&= \text{splittings de } \mathrm{P}^1_{X/S}(\underline{\omega})\\
&= \text{connexions sur } \omega
\end{aligned}
\]

\[
D(f\omega) - f\,D(\omega) = \omega\otimes df
\]

\[
D(f\,dt) = f\,\underbrace{D(dt)}_{\lambda} + dt\,df
\]

\[
0 \to \underbrace{\omega}_{} \subset \mathcal{L} \subset \mathfrak{g} \to 0
\qquad
\begin{aligned}
&v_s : \omega^{-1} \to \mathfrak{g}\\
&v_s \in \Gamma\,\mathfrak{g}\otimes\underline{\omega} = \underline{\omega}^2\oplus\underline{\omega}\oplus\underline{\omega}^0
\end{aligned}
\]

\[
\underline{\omega} + \underline{O}_X\cdot s + v_s(\underline{\omega}^{-1}) \qquad c(s) \subset s\otimes\omega + v_s\cdot\underline{O}_X\ ?
\]

\[
\omega_a + \underline{O}_s + \omega' \qquad
\text{\struck{$\lambda$}}\ c(s) = \varphi_2 + s\otimes\varphi_1 + v_s\cdot\varphi_0
\quad \text{avec } \varpi_0 = 0\ ?
\]

\[
s = s_0 + a_1 \quad (a_1\in\Gamma\underline{\omega})
\]

\[
\begin{aligned}
(\exp(\Theta_{a_1})\otimes\mathrm{id}_{\omega})(c(s)) &= \varphi_2 + (\exp\Theta_{a_1} s)\otimes\varphi_1 + (\exp\Theta_{a_1} v_s)\,\varphi_0\\
&= \varphi_2 + s_0\otimes\varphi_1 + \varphi_0
\end{aligned}
\]

\[
\frac{dt}{ds}=\frac{1}{ds/dt}=\frac{1}{s'} \qquad ds=s'\,dt
\]

\[
\text{\struck{$d\frac{1}{s}$}}\quad d\,\frac{dt}{ds}=-\frac{s''}{s'^2}\,dt
\qquad ds\,d\,\frac{dt}{ds}=-\frac{s''}{s'}\,dt^2
\]

\[
c_{D_s}=c_{D_t}-2\Bigl(\frac{s''}{s'}\Bigr)'dt^2+\Bigl(\frac{s''}{s'}\Bigr)^2dt^2
\]

\[
\boxed{c_{D_s}=c_{D_t}+\Bigl[-2\Bigl(\frac{s''}{s'}\Bigr)'+\Bigl(\frac{s''}{s'}\Bigr)^2\Bigr]dt^2}
\]

\[
\Bigl[\;s'=\frac{ds}{dt},\quad s''=\frac{ds'}{dt},\quad \Bigl(\frac{s''}{s'}\Bigr)'=\frac{d}{dt}\Bigl(\frac{s''}{s'}\Bigr)\Bigr]
\]

\[
[\;\;]=-2\,\frac{s'''s'-s''^2}{s'^2}+\frac{s''^2}{s'^2}=\frac{3s''^2-2s's'''}{s'^2}
\]

\[
\boxed{c_{D_s}=c_{D_t}+\frac{3s''^2-2s's'''}{s'^2}\,dt^2}
\]

\[
3s''^2=2s's'''\quad\text{i.e.}\quad 2\,\frac{s'''}{s''}=3\,\frac{s''}{s'}
\]

\[
\text{i.e.}\quad (2\log s'')'=(3\log s')'
\]

\[
2\log s''=3\log s'+\mathrm{c}^{\mathrm{te}},\qquad \log\frac{s''^2}{s'^3}=\mathrm{c}^{\mathrm{te}}
\]

\[
\frac{s''^2}{s'^3}=\mathrm{c}^{\mathrm{te}}\quad\text{i.e.}\quad s''^2=Cs'^3,\qquad s''=Cs'^{3/2}
\]

\[
s=\frac{at+b}{ct+d}=u_M(t)\qquad M\ \text{la matrice}\ \begin{pmatrix}a&b\\c&d\end{pmatrix}
\]

\[
D_{u_M(t)}=D_t
\]

\[
s=\frac1t,\quad s'=-\frac{1}{t^2},\quad s''=+\frac{2}{t^3},\quad s'''=-\frac{6}{t^4}
\]

\[
3s''^2-2s's'''=\frac{12}{t^6}-\frac{12}{t^6}=0
\]

\[
\underline{\omega}+\mathcal{O}_X\xi_0+\underline{\omega}^{-1}v_{s_0}
\]

\[
s=s_0+a_1=(\exp\theta_{-a_1})s_0=s_0+[-a_1,s_0]=\dots
\]

\[
\Bigl[\;v_s=(\exp\theta_{-a_1}\otimes\mathrm{id}_{\underline{\omega}})v_{s_0}
=v_{s_0}+\underbrace{[-a_1,v_{s_0}]}_{-2a_1 s_0}+\tfrac12\bigl[-a_1,\underbrace{[-a_1,v_{s_0}]}_{-2a_1}\bigr]
\]

\[
v_s=\underbrace{-a_1^2}_{2}-\underbrace{2a_1}_{1}+\underbrace{1}_{0}\;\Bigr]
\]

\[
(\exp(\theta_{a_1})\otimes\mathrm{id}_{\underline{\omega}})(c(s))
\]

\[
c(s_0)=\text{\struck{\ill{}}}\;\varpi_2+v_{s_0}
\]

\[
(\exp\theta_{a_1}\otimes\mathrm{id}_{\underline{\omega}})(c(s))
=(\exp\theta_{a_1}\otimes\mathrm{id}_{\underline{\omega}})(c(a_1))+\dots
\]

\[
c(s_0)+c(a_1)=-\varpi_2+c(a_1)+v_{s_0}
\]

\[
c(a_1)=\text{\struck{\ill{}}}\;\underbrace{\mathcal{D}_1(a_1)}_{2}-\underbrace{2a_1}_{1}\;\text{\struck{$v_{s_0}$}}\;(\text{\struck{\ill{}}}\dots
\]

\[
(\exp(\theta_{a_1})\otimes\mathrm{id}_{\underline{\omega}})(c(s))\|=\varpi_2+\underbrace{\mathcal{D}_1(a_1)}_{2}-\underbrace{2a_1}_{1}-\underbrace{2[a_1,a_1]}_{+2a_1a_1}
\]

\[
\text{\struck{$+2a$}}\;\underbrace{-a_1^2}_{2}+2a_1+1
\]

\[
\boxed{\mathcal{D}_1(a_1)+a_1^2=\varpi_2}
\]

\[
(d\lambda)\delta+\lambda^2\delta^2=0
\]

\[
\boxed{\frac{d\lambda}{dt}+\lambda^2=A}
\]

\[
\text{Sections de }P-\sigma(X)=\text{connexions sur }\underline{\omega}
\]

\[
\boxed{c_{D_0+\omega_1}=c_{D_0}+2D_0(\omega_1)+\omega_1^2}
\]

\[
c_{D_0+\omega_1+\varpi_1}=c_{D_0+\omega_1}+2(D_0+\omega_1)(\varpi_1)+\varpi_1^2
\]

\[
=\underbrace{c_{D_0}+2D_0\omega_1+\omega_1^2}+2D_0(\varpi_1)+2\omega_1\varpi_1+\varpi_1^2
\]

\[
=c_{D_0}+2D_0(\omega_1+\varpi_1)+\omega_1^2+\varpi_1^2+2\omega_1\varpi_1
=c_{D_0}+2D_0(\omega_1+\varpi_1)+(\omega_1+\varpi_1)^2
\]

\[
c_{D_0}+\varpi_2=c_{D_0+\omega_1}\quad\text{i.e.}\quad
\boxed{2D_0(\omega_1)+\omega_1^2=\varpi_2}
\]

\[
D_t(\lambda\,dt)=\frac{d\lambda}{dt}\,dt^2\quad\text{i.e.}\quad D_t(\omega)=\Bigl(\frac{d}{dt}\Bigl(\frac{\omega}{dt}\Bigr)\Bigr)dt^2
\]

\[
\begin{array}{l}
\alpha_1+\alpha_{-1}=a\\
\rho_a\,\omega_1=\text{\struck{$\omega_1\alpha_{-1}$}}\;\dots\;+\lambda\alpha_{-1}\;\dots\\
\rho_a\,\lambda=\lambda\alpha_1+\lambda\alpha_{-1}\qquad
\rho_a\,\omega_{-1}=\alpha_1\omega_{-1}\;\dots\\
(\rho_a)^2\omega_1=(\alpha_{-1}\alpha_1)\omega_1+(\alpha_{-1})^2\omega_2\;\dots\\
(\rho_a)^2\lambda=2\alpha_1\alpha_{-1}\lambda\\
(\rho_a)^2\omega_{-1}=(\alpha_1^2)\omega_{-1}+(\alpha_1\alpha_{-1})\omega_{-1}\\
(\rho_a)^3\omega_1=2\alpha_{-1}^2\alpha_1\omega_1\\
(\rho_a)^3\lambda=2\alpha_1^2\alpha_{-1}\lambda+2\alpha_1\alpha_{-1}^2\lambda
\end{array}
\]

\[
D_t(f\omega)=(\text{\struck{\ill{}}}f)D_t\omega+f\omega\,df
\]

\[
\left[\;
\begin{aligned}
&d\Bigl(\frac{f\omega}{dt}\Bigr)dt &&\qquad f\Bigl(d\frac{\omega}{dt}\Bigr)dt+\omega\,df\\
&\Bigl(f\,d\frac{\omega}{dt}+df\cdot\frac{\omega}{dt}\Bigr)dt\\
&f\,dt\,d\frac{\omega}{dt}+df\,\omega
\end{aligned}
\;\right]
\]

\[
\text{\struck{$2\lambda'+\lambda^2$}}\qquad
\boxed{2\frac{d\lambda}{dt}+\lambda^2=\mu}
\]

\[
\text{\struck{$D_s(\omega)=ds\,d\frac{\omega}{ds}=\dots=\varpi_1\frac{dt}{dt}$}}
\qquad
\text{\struck{$D_t(\omega)=\omega\,d\frac{\omega}{dt}$}}
\]

\[
\text{\struck{$D_s(\omega)-D_t(\omega)=\omega\,d\bigl(\frac{\omega}{ds}-\frac{\omega}{dt}\bigr)$}}
\]

\[
D_s\omega=ds\,d\frac{\omega}{ds}=\frac{ds}{dt}\,dt\,d\Bigl(\frac{\omega}{dt}\,\frac{dt}{ds}\Bigr)
=dt\Bigl[\frac{ds}{dt}\frac{dt}{ds}\,d\frac{\omega}{dt}+\frac{ds}{dt}\,\frac{\omega}{dt}\,d\frac{dt}{ds}\Bigr]
\]

\[
=\underbrace{dt\,d\frac{\omega}{dt}}_{D_t(\omega)}+\text{\struck{\ill{}}}\;\frac{ds}{dt}\,d\frac{dt}{ds}
\]

\[
D_s\omega=D_t\omega+\omega\,\frac{ds}{dt}\,\frac{d}{dt}\Bigl(\frac{dt}{ds}\Bigr)\qquad\text{donc}\qquad
s'\,d\frac{1}{s'}=-\frac{ds'}{s'}
\]

\[
\text{\struck{$c_{D_s\omega}=c_{D_t\omega}+2D$}}\qquad
D_s=D_t+\frac{ds\,d\frac{dt}{ds}}{dt}\quad\text{donc}
\]

\[
c_{D_s}=c_{D_t}+2\,dt\,d\Bigl(\frac{ds\,d\frac{dt}{ds}}{dt^2}\Bigr)+\Bigl(\frac{ds\,d\frac{dt}{ds}}{dt}\Bigr)^2
\]

\[
c_{D_0+\omega_1}=c_{D_0}+2D_0(\omega_1)+\omega_1^2
\]

\[
2D_0(\omega_1)+\omega_1^2\ \text{reste régulière}
\]

\[
2\frac{d\lambda}{dt}+\lambda^2\ \text{reste}
\]

\[
\lambda=\frac{2}{t}+\text{termes d'ordre}\geqslant1
\]

\[
2\lambda'+\lambda^2=0
\]

\[
\underline{\mathcal{O}}_{\mathbb{P}_X}/\mathcal{J}^4\overset{\varphi_3}{\simeq}P^3_{X/S}
\]

\[
\text{\struck{$(gf,y)\simeq(f,g^{-1}y)$}}\qquad (f,y)\simeq(\gamma\circ f,\gamma(y))\ \text{pour}\ \forall\,\gamma\in\mathrm{GP}(1,\mathbb{C}).
\]

\[
P|\mathcal{U}\simeq\mathbb{P}^1(\mathbb{C})\times\mathcal{U},\quad\text{la section s'identifie à}
\]

\[
P^{\mathrm{an}}=\underline{\mathrm{Isom}}(\mathbb{P}^1(\mathbb{C})_{X^{\mathrm{an}}},P)\ ]
\]

\[
\text{pr.}\;x\mapsto\widetilde{f}_{x_0}(x),\qquad\text{où}\ \widetilde{f}_{x_0}\ \text{est la section}
\]

\[
D:\underline{\omega}\to L
\]

\[
D=D_t+\varpi\quad(\varpi\in\Gamma(\underline{\omega})),\ \text{et l'équation sur }s\mapsto D_s=D_t+\varpi,\ \text{on a vu que}
\]

\[
D_s=D_t+s'\,d\frac{1}{s'}\qquad s'=\frac{ds}{dt},\qquad s'\,d\frac{1}{s'}=-\frac{ds'}{s'}
\]

\[
-\frac{ds'}{s'}=\varpi
\]

\[
\frac{s''}{s'}=-\lambda(t)\qquad\text{i.e.}\qquad(\log s')'=-\lambda(t)
\]

\[
\text{i.e.}\quad \log s'=-\Lambda(t)+\mathrm{c}^{\mathrm{te}}\qquad(\Lambda(t)\ \text{une primitive de}\ \lambda(t))
\]

\[
s'=\mathrm{c}^{\mathrm{te}}\exp\Lambda(t)\qquad s=\mathrm{c}^{\mathrm{te}}\int\exp-\Lambda(t)\,dt+\mathrm{c}^{\mathrm{te}}.
\]

\[
t\longmapsto(D_t,dt)\longmapsto D_t\qquad\text{les applications canoniques}
\]

\[
\text{correspondent :}\qquad \mathcal{T}\to\mathcal{T}/B_u\to\mathcal{T}/B
\]

\[
c_0=c_{D_t}+\varpi_2\qquad\varpi_2\in\Gamma(\underline{\omega}^2),\quad\varpi_2=\mu(t)\,dt^2
\]

\[
2D_t\omega+\omega^2=\varpi_2
\]

\[
(1)\qquad\boxed{2\lambda'(t)+\lambda(t)^2=\mu(t)}\qquad\Bigl(\text{où }\lambda'=\frac{d}{dt}\lambda\Bigr)
\]

\[
D\varphi=D_t\varphi+\omega\varphi=(\alpha'(t)+\alpha(t)\lambda(t))\,dt^2
\]

\[
(2)\qquad\boxed{\alpha'(t)+\alpha(t)\lambda(t)=0}\qquad\Bigl(\text{où }\alpha'=\frac{d\alpha}{dt}\Bigr)\qquad\text{i.e.}\quad\frac{\alpha'}{\alpha}=-\lambda
\]

\[
\text{i.e.}\quad\frac{\alpha'(t)}{\alpha(t)}=-\lambda(t),\quad\text{ou}\quad\log\alpha(t)=-\int\lambda(t)\,dt,
\]

\[
\text{ou}\quad\alpha(t)=c\exp-\int\lambda(t)\,dt\qquad c\ \mathrm{c}^{\mathrm{te}}\neq0
\]

\[
c_{D_t}\;\text{\struck{$-\frac{ds'}{s'}\,dt$}}=c_{D_t}+\varpi_2
\]

\[
2D_t\Bigl(-\frac{ds'}{s'}\Bigr)+\Bigl(-\frac{ds'}{s'}\Bigr)^2=\varpi_2\qquad\Bigl(s'=\frac{ds}{dt}\Bigr)
\]

\[
(3)\qquad\frac{s''}{s'}=-\lambda(t)\qquad s'=\frac{ds}{dt},\quad s''=\frac{ds'}{dt}
\]

\[
2\Bigl(-\frac{s''}{s'}\Bigr)'+\Bigl(-\frac{s''}{s'}\Bigr)^2=\mu(t)
\]

\[
\text{i.e.}\quad -2\,\frac{s'''s'-s''^2}{s'^2}+\frac{s''^2}{s'^2}=\mu
\]

\[
\frac{3s''^2-2s's'''}{s'^2}=-2\,\frac{s'''}{s'}+3\Bigl(\frac{s''}{s'}\Bigr)^2
\]

\[
(4)\qquad\boxed{G_t(s)\overset{\mathrm{dfn}}{=}-2\,\frac{s'''}{s'}+3\Bigl(\frac{s''}{s'}\Bigr)^2=\mu(t)}
\]

\[
\text{\struck{$\overline{s}=as$}}\qquad\overline{s}=\frac{as+b}{cs+d}\qquad a,b,c,d\ \text{constantes},
\]

\[
G_\tau(s)=G_t(s)\,\tau'^2
\]

\[
G_t(s)\overset{\mathrm{dfn}}{=}-2\,\frac{s'''}{s'}+3\Bigl(\frac{s''}{s'}\Bigr)^2=\mu(t)
\]

\[
t:\mathcal{X}\to\mathbb{C}
\]

\[
G(t,s)=\Bigl(-2\,\frac{s'''}{s'}+3\Bigl(\frac{s''}{s'}\Bigr)^2\Bigr)dt^2\in\Gamma(\mathcal{X},\underline{\omega}^{\otimes2})
\]

\[
(1)\qquad
\left\{
\begin{aligned}
&G(t,s+b)=G(t,s) &&\text{a)}\\
&G(t,as)=G(t,s) &&\text{b)}\\
&G\bigl(t,\tfrac1s\bigr)=G(t,s) &&\text{c)}
\end{aligned}
\right.
\]

\[
(2)\qquad G(t,s)+G(s,r)=G(t,r)
\]

\[
\mathcal{X}\to\mathbb{P}^1(\mathbb{C})
\]

\[
G\Bigl(t,\frac{as+b}{cs+d}\Bigr)=G(t,s)\qquad a,b,c,d\in\mathbb{C},\quad ad-bc\in\mathbb{C}^*
\]

\[
G(t,t)=0\qquad(t\ \text{uniformisante})
\]

\[
G(t,s)+G(s,t)=0,\quad\text{i.e.}\quad G(t,s)=-G(s,t)\qquad(s,t\ \text{uniformisantes})
\]

\[
(1\,\text{bis})\qquad
\left\{
\begin{aligned}
&G(at,s)=G(t,s)\\
&G(t+b,s)=G(t,s)\\
&G\bigl(\tfrac1t,s\bigr)=G(t,s)\qquad\bigl(t,\tfrac1t\ \text{unif}^{\text{tes}}\bigr)
\end{aligned}
\right.
\]

\[
s,t:\mathcal{X}\to\mathbb{P}^1(\mathbb{C})\qquad\text{morphismes étales}
\]

\[
(4)\qquad
\left\{
\begin{aligned}
&G\Bigl(\frac{at+b}{ct+d},s\Bigr)=G(t,s)\qquad a,b,c,d\in\mathbb{C}\ \text{avec}\ ad-bc\in\mathbb{C}^*\\
&\text{de plus,}\quad G\Bigl(t,\frac{as+b}{cs+d}\Bigr)=G(t,s)\qquad\text{id.}
\end{aligned}
\right.
\]

\[
\mathcal{X}\to\mathbb{P}^1(\mathbb{C}),
\]

\[
H=\mathrm{Aut}_{\mathbb{C}}\mathbb{P}^1(\mathbb{C})\simeq\mathrm{GP}(1,\mathbb{C})=\Bigl\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\Bigm|ad-bc\in\mathbb{C}^*\Bigr\}
\]

\[
\underline{G}:\mathcal{P}_X\times\mathcal{P}_X\to\underline{\omega}_X^{\otimes2}
\]

\[
G(g\cdot t,g'\cdot s)=G(t,s)\quad\text{si }g,g'\in H
\]

\[
s\longmapsto G(t,s)\qquad\mathcal{P}_X\to\underline{\omega}_X^{\otimes2}
\]

\[
H\backslash\mathcal{P}_X\hookrightarrow\underline{\omega}_X^{\otimes2}
\]

\[
H\backslash\mathcal{P}_{\mathcal{X}}\xrightarrow{\ \sim\ }\underline{\omega}_X^{\otimes2}\qquad s\longmapsto G(t,s)=\Bigl(-2\,\frac{s'''}{s'}+3\Bigl(\frac{s''}{s'}\Bigr)^2\Bigr)dt^2\ \bigr)
\]

\[
t\longmapsto c_t:\ \mathcal{P}_X\xrightarrow{\ c^X\ }\mathrm{Conn}_{\mathcal{X}}
\]

\[
\boxed{G(t,s)=c_s-c_t}
\]

\[
c^X:H\backslash\mathcal{P}_{\mathcal{X}}\xrightarrow{\ \sim\ }\mathrm{Conn}_{\mathcal{X}}
\]

\[
f:\mathcal{X}'\to\mathcal{X}
\]

\[
\mathcal{P}_{\mathcal{X}'}\simeq f^{-1}(\mathcal{P}_{\mathcal{X}})\qquad\mathrm{Conn}_{\mathcal{X}'}\simeq f^{-1}(\mathrm{Conn}_{\mathcal{X}})
\]

\[
c_t=t^*(\mathcal{C})=\text{image inverse par }t\text{ de la conn.\ projective can.\ }\mathcal{C}
\]

\[
t,s:X\to\mathbb{E}^1_S
\]

\[
t,s:X\to\mathbb{P}^1_S
\]

\[
\text{(\uncertain{groupe} }H_S=\underline{\mathrm{Aut}}_S(\mathbb{P}^1_S))
\]

\[
G(gt,g's)=G(t,s)\qquad(g,g'\in\Gamma(S,H))
\]

\[
G:\mathcal{P}_{X/S}\times\mathcal{P}_{X/S}\to\underline{\omega}^{\otimes2}_{X/S}
\]

\[
G(g\cdot s,g'\cdot t)=G(s,t)\quad\text{si}\quad
\left\{
\begin{aligned}
&s,t\in\Gamma(\mathcal{U},\mathcal{P}_X)\\
&g,g'\in\Gamma(V,H_V)
\end{aligned}
\right.
\]

\[
G(s,t)+G(t,r)=G(s,r).
\]

\[
c^X:\mathcal{P}_{X/S}\to\mathrm{Conn}_{X/S}
\]

\[
G(t,s)=c^X(s)-c^X(t)
\]

\[
c^X(t)=t^*(\mathcal{C})
\]

\[
G(t,s)=c(s)-c(t)=s^*(c_{\mathcal{X}_0})-t^*(c_{\mathcal{X}_0})
\]

\[
g\in\mathrm{Aut}(\mathcal{X}_0)\quad\text{tel que}\quad s=g\circ t
\]

\[
f^{-1}(H_S)\backslash\mathcal{P}_X\to\underline{\omega}^{\otimes2}_{X/S}\qquad s\longmapsto G(t,s)
\]

\[
{}_H\backslash\mathcal{P}_X\xrightarrow{\ c^X\ }\mathrm{Conn}_{X/S}
\]

\[
X\xrightarrow{\ F_X\ }X^{(p/S)}\qquad(\text{Frobenius})
\]

\[
F_X^{-1}(H_{X^{(p/S)}})\backslash\mathcal{P}_X\hookrightarrow\mathrm{Conn}_{X/S}
\]

\[
D^{(n)}_tf=\frac{1}{n!}\,\frac{d^nf}{dt^n}
\]

\[
G(t,s)=12\Bigl(-\frac{D^{(3)}_ts}{D^{(1)}_ts}+\Bigl(\frac{D^{(2)}_ts}{D^{(1)}_ts}\Bigr)^2\Bigr)=12\,G_0(t,s)
\]

\[
G_0(t,s)=-\frac{D^{(3)}_ts}{D^{(1)}_ts}+\Bigl(\frac{D^{(2)}_ts}{D^{(1)}_ts}\Bigr)^2
\]

\[
\underline{\omega}^{\otimes2}_{X/S}\xrightarrow{\ 12\,\mathrm{id}\ }\underline{\omega}^{\otimes2}_{X/S},
\]

\[
\mathcal{P}_{X/S}\xrightarrow{\ c^X_0\ }\mathrm{Conn}_{0,X/S}
\]

\[
c^X_0(s)-c^X_0(t)=G_0(t,s)
\]

\[
H^{1}\mathcal{P}_{X/S} \longleftrightarrow \mathrm{Conn}^{0}_{X/S}
\]

\[
\mathrm{Conn}^{0}_{X/S} \longrightarrow \mathrm{Conn}_{X/S}
\]

\[
\underline{\omega}_{X/S}^{\otimes 2} \xrightarrow{\ 12\,\mathrm{id}\ } \underline{\omega}_{X/S}^{\otimes 2}
\]

\[
c_{0}^{X}(\ill{}) = \tfrac{1}{12}\, c^{X}(\ill{}) \quad \ldots\,)
\]

\[
H_{\mathcal{X}}(\mathcal{P}/S) \backslash \mathcal{P}_{X/S} \longleftrightarrow \mathrm{Conn}^{0}_{X/S}
\]

\[
s = a_0 + a_1 t + \cdots + a_{p-1} t^{p-1}, \qquad a_i \in A \ (0 \leqslant i \leqslant p-1),
\]

\[
D_t s = (ds)/(dt), \qquad D^{(i)}_t s = \tfrac{1}{i!}(D_t)^{i}s \quad\text{si } 1 \leqslant i \leqslant 3,
\]

\[
c_0(s) = -\frac{D^{(3)}_t s}{D^{(1)}_t s} + \Bigl(\frac{D^{(2)}_t s}{D^{(1)}_t s}\Bigr)^{2} \quad \text{si } D^{(1)}_t s \in B^{*}.
\]

\[
(D^{(2)}s)^{2} = (D^{(1)}s)(D^{(3)}s).
\]

\[
s = \frac{at + b}{ct + d} \quad\text{i.e.}\quad s(ct + d) = at + b
\]

\[
s' = \frac{s}{1 + a_2 s} = \sum_{n \geqslant 0} (-1)^{n} a_2^{n} s^{n+1}
= s - a_2 s^{2} + \cdots \equiv t \bmod t^{3}
\]

\[
s = t + \text{termes d'ordre} \geqslant 3 .
\]

\[
c_0(s) = 0 \Longrightarrow s = t .
\]

\[
s = t + a t^{d} + \cdots \quad (\text{termes d'ordre} \geqslant d+1),
\]

\begin{align*}
D^{(3)}s &\equiv a\binom{d}{3} t^{d-3} \bmod (t^{d-2})\\
D^{(2)}s &\equiv a\binom{d}{2} t^{d-2} \bmod (t^{d-1}) \text{ et a fortiori mod } t^{d-2}\\
D^{(1)}s &\equiv 1 \bmod (t^{d-1}) \text{ et a fortiori mod } t^{d-2}\\
D^{(3)}s\, D^{(1)}s &\equiv a\binom{d}{3} t^{d-3} \bmod t^{d-2}\\
(D^{(2)}s)^{2} &\equiv 0 \bmod t^{d-2}
\end{align*}

\[
a\binom{d}{3} t^{d-3} \equiv 0 \bmod t^{d-2}
\]

\[
\frac{d(d-1)(d-2)}{6}\, a = 0 \text{ dans } A .
\]

\[
a\binom{d}{3} = 0 \quad\text{i.e.}\quad a\, c_{3,d-3} = 0,
\]

\[
\text{pour}\quad
\begin{aligned}
d-3 &= d_0 + d_1 p\\
3 &= d'_0 + d'_1 p
\end{aligned}
\qquad \Bigl|\ 0 \leqslant d_0, d_1, d'_0, d'_1 \leqslant p-1
\]

\[
c_{3,3} = 1 \Longrightarrow a = 0 \qquad \text{OK}
\]

\[
d - 3 = d - p = d_0 + (d_1 - 1)p, \quad\text{et}
\]

\[
c_{d,d-3} \equiv c_{d_0,d_0}\, c_{d_1,d_1-1} \quad \text{dans } \mathbb{F}_p,
\]

\[
\in \mathbb{F}_p^{*} \text{ ssi }
\begin{aligned}
2d_0 &\leqslant p-1 = 2\\
2d_1 - 1 &\leqslant p-1 = 2
\end{aligned}
\]

\[
\operatorname{Lie}\operatorname{Aut}_{\mathbb{C}} P = H^{0}(P, \mathcal{V}_{P/\mathbb{C}}) \simeq H^{0}(P, \Omega^{-1}_{P/\mathbb{C}})
\]

\[
X \xhookrightarrow{\ i_X\ } \mathbb{P}(X)(\mathbb{C})
\]

\[
\widetilde{X}(x) \longrightarrow X = X(x)
\]

\[
V = \mathbb{V}(\underbrace{\Omega^{1}_{X/\mathbb{C}}}_{\omega_{X/\mathbb{C}}}) = \mathbb{W}(\underbrace{t_{X/\mathbb{C}}}_{\text{fibré tangent}})
\]

\[
\widehat{\mathbb{P}}_X \xrightarrow[\ \sim\ ]{\ \hat{p}_X\ } \widehat{X}_X
\]

\[
\text{\struck{$(\mathbb{P}_X^{\mathrm{an}} \supset \sigma'(X))$}}\quad \widetilde{X} \overset{\mathrm{df}}{=} \mathbb{P}_X^{\mathrm{an}} - \sigma'^{\mathrm{an}}(X^{\mathrm{an}}) \xrightarrow{\ p\ } (X_X)^{\mathrm{an}},
\]

\[
0 \longrightarrow \underline{\omega}_{X/S} \longrightarrow E \longrightarrow \underline{\mathcal{O}}_X \longrightarrow 0
\]

\[
E \simeq \mathcal{J}/\mathcal{J}^{3} \otimes \omega^{-1}
\]

\[
\text{\struck{$\mathcal{P}$}} \quad \mathcal{C}_{X/S} \quad \text{torseur sous } \underline{\omega}^{\otimes 2} = \underline{\Omega}^{1\,\otimes 2}_{X/S}
\]

\[
\mathcal{V} \simeq (\operatorname{Sym}^{2} E) \otimes \underline{\omega}^{-1}
\]

\[
c : \mathcal{V} \longrightarrow \mathcal{V} \otimes \underline{\omega} \simeq \operatorname{Sym}^{2} E
\]

\[
c|\underline{\omega} = d^{1}_{\underline{\omega}, X/S} : \underline{\omega} \longrightarrow \mathcal{P}^{1}_{X/S}(\underline{\omega}) .
\]

\[
H^{0}(\hat{X}, \underline{\omega}^{\otimes 2}_{\hat{X}}(D))
\]

\[
\text{\struck{$0 \to \underline{\lambda} \to \underline{\lambda}(D) \to$}} \quad \underline{\omega}(D)/\underline{\omega}^{2} \longrightarrow 0
\]

\[
0 \longrightarrow \underline{\lambda} \longrightarrow \underline{\lambda}(D) \longrightarrow \underline{\lambda}(D)/\underline{\lambda} \longrightarrow 0
\]

\[
0 \longrightarrow \underline{\mathcal{O}}_{\hat{X}} \longrightarrow \underline{\mathcal{O}}_{\hat{X}}(D) \longrightarrow \underline{\mathcal{O}}_{\hat{X}}(D)/\underline{\mathcal{O}}_{\hat{X}} \longrightarrow 0
\]

\[
\underline{\lambda}(D)/\underline{\lambda} \simeq (\underline{\lambda}\,\underline{\omega}^{-1})|D
\]

\[
0 \to H^{0}(\hat{X}, \underline{\lambda}) \to H^{0}(\hat{X}, \underline{\lambda}(D)) \to H^{0}(D, \underline{\lambda}\,\underline{\omega}^{-1}|D) \to H^{1}(\hat{X}, \underline{\lambda})
\]

\[
0 \to H^{0}(\hat{X}, \underline{\omega}^{2}) \to H^{0}(X, \underline{\omega}^{2})^{\mathrm{reg}} \to H^{0}(D, \underline{\omega}|D) \to 0
\]

\[
H^{1}(\hat{X}, \underline{\lambda}) \longrightarrow H^{1}(\hat{X}, \underline{\lambda}(D)) \quad \text{est inj.\ ?}
\]

\[
H^{0}(\hat{X}, \underline{\omega}^{-1}) \longleftarrow H^{0}(\hat{X}, \underline{\omega}^{-1}(-D))
\]

\[
0 \to H^{0}(\hat{X}, \underline{\omega}^{2}) \to H^{0}(X, \underline{\omega}^{2})^{\mathrm{rég}} \to H^{0}(D, \underline{\omega}|D) \to \check{Q}(X) \to 0
\]

\[
\text{Alors } H^{0}(\hat{X}, \underline{t}_{\hat{X}/\mathbb{C}}) = 0
\]

\[
\begin{array}{ll}
\nu = 1 & \dim H^{0}(\hat{X}, \underline{t}_{\hat{X}/\mathbb{C}}(-D)) = 2\\
\nu = 2 & \phantom{\dim H^{0}(\hat{X}, \underline{t}(-D))} = 1\\
\nu \geqslant 3 & \phantom{\dim H^{0}(\hat{X}, \underline{t}(-D))} = 0
\end{array}
\]

\[
H^{0}(X, \underline{\omega}^{2})^{\mathrm{rég}} \simeq \underline{\Omega}^{1}_{M_{0,\nu}}(s) \quad \ldots
\]

\[
\underset{\omega_0}{\underline{\omega}_0} \subset \underset{\underline{\mathcal{O}}_S}{\mathcal{L}_0} \subset \underset{\omega_0^{-1}\ \text{sur } X}{\mathcal{V}_0}, \qquad (P_0, \sigma_0) \quad P_0 = \mathbb{P}(\mathcal{L}_0)
\]

\[
\mathcal{L}_0 = (\underline{\mathcal{O}}_H/\mathcal{J}^{3})^{+}, \quad \mathcal{J} \text{ id.\ d'augm.\ de } H
\]

\[
\mathcal{V}_0 \simeq \underline{\omega}_0 \oplus \underline{\mathcal{O}}_S \oplus \underline{\omega}_0^{-1}
\]

\[
\omega_2 + \omega_1 + 1 \in \Gamma(\underline{\omega}_0^{2} + \underline{\omega}_0 + \underline{\mathcal{O}}_S).
\]

\[
0 \longrightarrow \underline{\omega}_X \longrightarrow F \longrightarrow \underline{\mathcal{O}}_X \longrightarrow 0
\]

\[
(1)\qquad 0 \longrightarrow E \otimes \Omega^1_{S/T} \longrightarrow P^1_{S/T}(E) \longrightarrow E \longrightarrow 0
\]

\[
(2)\qquad d^1_E : E \longrightarrow P^1_{S/T}(E)
\]

\[
(3)\qquad \xi = d^1_E x + \omega \qquad
\begin{cases} \omega \in \Gamma(\ ,E \otimes \Omega^1_{S/T}) \\ x \in \Gamma(\ ,E) \end{cases}
\]

\[
f\xi = f\omega + f\,d^{(1)}_E x = f\omega + d^{(1)}_E(fx) + [f - d^{(1)}(f)]\,d^1_E(x)
\]

\[
(4)\qquad f(d^1_E x + \omega) = d^1_E fx + (f\omega - x \otimes df) .
\]

\[
(5)\qquad d^0_E : E \longrightarrow E \otimes \Omega^1_{S/T}
\]

\[
(6)\qquad \delta_E x = d^1_E(x) - d^0_E(x) ,
\]

\[
d^1_E(fx) + (-f d^0_E x - x \otimes df) = d^1_E(fx) - d^0_E(fx) ,
\]

\[
(7)\qquad d^0_E(fx) = f d^0_E(x) + x \otimes df .
\]

\[
(7')\qquad d^0_E(xf) = d^0_E(x) f + x\,df .
\]

\[
(8)\qquad d^\bullet_E : E \otimes \Omega^\bullet_{S/T} \longrightarrow E \otimes \Omega^\bullet_{S/T}
\]

\[
(9)\qquad d^\bullet_E(\varphi\omega) = d_E(\varphi)\,\omega + (-1)^{\deg\varphi}\,\varphi\, d(\omega) ,
\]

\[
(10)\qquad d^\bullet_E \circ d^\bullet_E = 0 ,
\]

\[
(11)\qquad H^*_{DR}(S,E) = \mathbb{H}^*(S, E \otimes \Omega^\bullet_{S/T}) .
\]

\[
(12)\qquad H^*_{DR}(S,E) \Longleftarrow E_1^{pq} = H^q(S, E \otimes \Omega^p_{S/T})
\]

\[
(13)\qquad H^*(\Gamma(S, E \otimes \Omega^\bullet_{S/T})) \Longrightarrow H^*_{DR}(S,E)
\]

\[
(14)\qquad H^i_{DR}(S,E) \longrightarrow H^i(S,E) ,
\]

\[
(15)\qquad E \otimes \Omega^\bullet_{S/T} \longrightarrow E .
\]

\[
(16)\qquad d^0_E(x) = 0 \qquad\text{i.e.}\qquad \delta_E x = d^1_E x
\]

\[
\mathbb{H}^1(S,K^\bullet) \longrightarrow H^1(S,K^0)
\]

\[
\bar d^0 : P^0 \longrightarrow K^1
\]

\[
\bar d^0(xf) = \bar d^0(x)\, d^0(f) \qquad x \in P^0(\cdot),\ f \in K^0(\cdot) .
\]

\[
(17)\qquad d_P : P \longrightarrow (E \otimes \Omega^1_{S/T}) ,
\]

\[
(18)\qquad
\begin{cases}
d_P(xf) = d_P(x)\, d^0_E(f) & x \in P(\cdot),\ f \in \mathcal{O}_S(\cdot) . \\
d_P(P) \subset Z^1(E \otimes \Omega_{S/T}) & \text{i.e. } d^1_E d_P = 0 .
\end{cases}
\]

\[
(19)\qquad H^*_{\mathrm{strat}}(S,E) \overset{\mathrm{déf}}{=} H^*((S/T)_{\mathrm{strat}}, E_{\mathrm{strat}}) .
\]

\[
(20)\qquad \delta^\infty_E : E \longrightarrow P^\infty_{S/T}(E)
\]

\[
P^\infty_{S/T}(E)^+ \longrightarrow P^1_{S/T}(E)^+ \simeq E \otimes \Omega^1_{S/T} ,
\]

\[
(23)\qquad d^0_E : E \longrightarrow E \otimes \Omega^1_{S/T} , \qquad d^0_E = d^+_E \bmod \ldots
\]

\[
(24)\qquad H^*_{\mathrm{strat}}(S,E) \longrightarrow H^*_{DR}(S,E) ,
\]

\[
(25)\qquad 0 \longrightarrow E \xrightarrow{\ \delta = \delta^\infty_E\ } P^\infty_{S/T}(E) \xrightarrow{\ p\ } P^\infty_{S/T}(E)/\delta(E) \longrightarrow 0 ,
\]

\[
(26)\qquad P \xrightarrow{\ \lambda\ } P^\infty_{S/T}(E)/\delta E
\]

\[
\text{\struck{$\lambda(x + \xi) = \lambda(x) + d^\infty_{S/T}(\xi)$}}
\]

\[
(27)\qquad \lambda(x + \xi) = \lambda(x) + p(d^\infty_{S/T}(\xi)) \qquad x \in P(\cdot),\ \xi \in E(\cdot) .
\]

\[
(28)\qquad \lambda(P) \subset \bigl(P^\infty_{S/T}(E)/E\bigr)^{\natural} ,
\]

\[
(26')\qquad P \xrightarrow{\ \lambda\ } \bigl(P^\infty_{S/T}(E)/\delta E\bigr)^{\natural} .
\]

\[
P^\infty_{S/T}(E)^+ \hookrightarrow P^\infty_{S/T}(E)
\]

\[
(29)\qquad P^\infty_{S/T}(E)^+ \simeq P^\infty_{S/T}(E)/\delta E ,
\]

\[
(29')\qquad P^\infty_{S/T}(E) \xrightarrow{\ p\ } P^\infty_{S/T}(E)/\delta E \xrightarrow{\ \sim\ } P^\infty_{S/T}(E)^+
\]

\[
\text{\struck{$d^+_E$}}\qquad \varphi \longmapsto \varphi - \delta_E(\varphi) .
\]

\[
(30)\qquad \lambda^+ : P \longrightarrow P^\infty_{S/T}(E)^+ ,
\]

\[
(31)\qquad \lambda^+(x + \xi) = \lambda^+(x) + \underbrace{\bigl[d^\infty_{S/T}(\xi) - \delta^\infty_E(\xi)\bigr]}_{d^+_{S/T}(\xi)} .
\]

\[
(32)\qquad d^0_P : P \longrightarrow E \otimes \Omega^1_{S/T} , \qquad d^0_P(x) = \lambda^+(x) \bmod \ldots
\]

\[
d^0_P(x + \xi) = d^0_P(x) + d^0_E(\xi) ,
\]

\[
P^\infty_{S/T}(E)^{+\natural} \xrightarrow{\ \sim\ } Z^1(E \otimes \Omega^\bullet_{S/T}) ,
\]

\[
(33)\qquad 0 \longrightarrow E' \xrightarrow{\ u\ } E \xrightarrow{\ v\ } E'' \longrightarrow 0 .
\]

\[
(34)\qquad P = P' \times^{E'} E .
\]

\[
(35)\qquad \lambda' : P' \longrightarrow \bigl(P^\infty_{S/T}(E)/\delta E\bigr)^{\natural}
\]

\[
(36)\qquad \lambda'(x' + \xi') = \lambda'(x') + p\,d^\infty_E(u\xi') \qquad x' \in P'(\cdot),\ \xi' \in E'(\cdot) .
\]

\[
(35')\qquad \lambda'^{+} : P' \longrightarrow \bigl(P^\infty_{S/T}(E)^+\bigr)^{\natural} , \qquad \text{(cf. transport de structure)}
\]

\[
(36')\qquad \lambda'^{+}(x' + \xi') = \lambda'^{+}(x') + \underbrace{d^+_E(u\xi')}_{= d^\infty_E(u\xi') - \delta^\infty_E(u\xi')} .
\]

\[
(37)\qquad \cdots \longrightarrow P^\infty_{S/T}(E') \xrightarrow{\ u\ } P^\infty_{S/T}(E) \xrightarrow{\ v\ } P^\infty_{S/T}(E'') \longrightarrow 0 ,
\]

\[
(38)\qquad \varphi = \varphi(\lambda') \in \Gamma\bigl(S, P^\infty_{S/T}(E'')/v\delta(E)\bigr)
\]

\[
\simeq \Gamma\bigl(S, P^\infty_{S/T}(E'')^+/v\delta u(E')\bigr) .
\]

\[
(38')\qquad \varphi \in \Gamma\Bigl(S, \bigl(P^\infty_{S/T}(E'')/v\delta(E)\bigr)^{\natural}\Bigr) \simeq \Gamma\bigl(S, P^\infty_{S/T}(E'')^{+\natural}/\delta u(E')\bigr) .
\]

\[
(39)\qquad D : E'' \longrightarrow F
\]

\[
(40)\qquad u_D : P^\infty_{S/T}(E'') \longrightarrow F .
\]

\[
(41)\qquad \bar u_D : P^\infty_{S/T}(E'')/v\delta(E) \longrightarrow F .
\]

\[
(42)\qquad \text{\struck{$\ldots$}}\ \mu_D(P,\sigma) = \bar u_D\bigl(\varphi(P,\sigma)\bigr) \in \Gamma(S,F) ,
\]

\[
(43)\qquad P' \xrightarrow{\ d_{P'}\ } E \otimes \Omega^1_{S/T}
\]

\[
(44)\qquad
\begin{cases}
d_{P'}(x' + \xi') = d_{P'}(x') + d^0_E\bigl(u(\xi')\bigr) \\
d_{P'}(P') \subset Z^1(E \otimes \Omega^\bullet_{S/T}) , \ \text{i.e. } d^1_E \circ d_{P'} = 0 .
\end{cases}
\]

\[
(45)\qquad E \otimes \Omega^1_{S/T} \xrightarrow{\ \Delta\ } F
\]

\[
(46)\qquad \nu_\Delta(P', d_{P'}) \in \Gamma(S,F) .
\]

\[
F = P^\infty_{S/T}(E \otimes \Omega^1_{S/T})/I\bigl(P^\infty_{S/T}(E')\bigr) ,
\]

\[
(47)\qquad D_0 : E \xrightarrow{\ d^0_E\ } E \otimes \Omega^1_{S/T} \xrightarrow{\ \Delta\ } F ,
\]

\[
(48)\qquad \nu_\Delta = \mu_D ,
\]

\[
(1) \qquad \text{\struck{$0 \to$}}\; 0 \to f^{*}(\Omega^{1}_{S/T}) \to \Omega^{1}_{X/T} \to \Omega^{1}_{X/S} \to 0
\]

\[
(2) \qquad \mathrm{Gr}^{p}_{X/S}(\Omega^{\bullet}_{X/T}) \simeq f^{*}(\Omega^{p}_{S/T}) \otimes \Omega^{\bullet}_{X/S}[-p] .
\]

\[
(2') \qquad \mathrm{Gr}^{p}_{X/S}(\Omega^{\bullet}_{X/T}) \simeq \mathbf{L}f^{*}(\Omega^{p}_{S/T}) \overset{\mathbf{L}}{\otimes} \Omega^{\bullet}_{X/S}[-p] ,
\]

\[
\mathrm{Gr}^{p}_{X/S}\, \mathbf{R}f_{*}(\Omega^{\bullet}_{X/T}) \simeq \mathbf{R}f_{*}(\mathrm{Gr}^{p}_{X/S}(\Omega^{\bullet}_{X/T})) .
\]

\[
(3) \qquad \mathrm{Gr}^{p}_{X/S}\, \mathbf{R}f_{*}(\Omega^{\bullet}_{X/T}) \simeq \Omega^{p}_{S/T} \overset{\mathbf{L}}{\otimes} \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S})[-p] .
\]

\[
(4) \qquad d^{(p)}_{1} : \mathrm{Gr}^{p}_{X/S}\, \mathbf{R}f_{*}(\Omega^{\bullet}_{X/T}) \to \mathrm{Gr}^{p+1}_{X/S}\, \mathbf{R}f_{*}(\Omega^{\bullet}_{X/T})[1] ,
\]

\[
(5) \qquad d^{(p)}_{1} : \Omega^{p}_{S/T} \overset{\mathbf{L}}{\otimes} \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S})[-p] \to \Omega^{p+1}_{S/T} \overset{\mathbf{L}}{\otimes} \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S})[-p] ,
\]

\[
(6) \qquad \text{\struck{$\ill{}$}}\; d^{(p)}_{1}[p] : \Omega^{p}_{S/T} \overset{\mathbf{L}}{\otimes} \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S}) \to \Omega^{p+1}_{S/T} \overset{\mathbf{L}}{\otimes} \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S}) .
\]

\[
(7) \qquad d^{(p+1)}_{1}\, d^{(p)}_{1} = 0 .
\]

\[
(6_{0}) \qquad d^{(0)}_{1} : \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S}) \to \Omega^{1}_{S/T} \otimes \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S}) ,
\]

\[
(8) \qquad d^{p,q}_{1}[p] : \mathcal{H}^{q}(\Omega^{p}_{S/T} \otimes \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S})) \to \mathcal{H}^{q}(\Omega^{p+1}_{S/T} \otimes \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S})) ,
\]

\[
(8') \qquad d^{p,q}_{1}[p] : \Omega^{p}_{S/T} \otimes \mathbf{R}^{q}f_{*}(\Omega^{\bullet}_{X/S}) \to \Omega^{p+1}_{S/T} \otimes \mathbf{R}^{q}f_{*}(\Omega^{\bullet}_{X/S}) ,
\]

\[
(9) \qquad K^{\bullet} = \mathbf{R}f_{*}(\Omega^{\bullet}_{X/T}) ,
\]

\[
(10) \qquad \mathbf{R}\Gamma_{S}(K^{\bullet}) \simeq \mathbf{R}\Gamma_{X}(\Omega^{\bullet}_{X/T}) \simeq \mathbf{R}H_{DR}(X/T) .
\]

\[
(11) \qquad \mathbf{R}\Gamma_{S}(K^{\bullet}) \Longleftarrow E^{p,q}_{2} = \mathbb{H}^{p}(S, \alpha \mapsto \mathcal{H}^{\alpha+q}(\mathrm{Gr}^{\alpha}(K^{\bullet}))) ,
\]

\[
(12) \qquad H^{*}_{DR}(X/T) \Longleftarrow E^{p,q}_{2} = \mathbb{H}^{p}(S, \alpha \mapsto \mathcal{H}^{q}(\Omega^{\alpha}_{S/T} \overset{\mathbf{L}}{\otimes} \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S}))) ,
\]

\[
(12') \qquad H^{*}_{DR}(X/T) \Longleftarrow E^{p,q}_{2} = \mathbb{H}^{p}(S, \alpha \mapsto \Omega^{\alpha}_{S/T} \otimes \mathbf{R}^{q}f_{*}(\Omega^{\bullet}_{X/S}))
\]

\[
= H^{p}_{DR}(S/T, \mathbf{R}^{q}f_{*}(\Omega^{\bullet}_{X/S})) ,
\]

\[
(13) \qquad \xi \in H^{n}_{DR}(X/T)
\]

\[
(14) \qquad H^{n}_{DR}(X/T) \to H^{0}(S, \mathbf{R}^{n}f_{*}(\Omega^{\bullet}_{X/S}))
\]

\[
(15) \qquad \eta \in \mathbb{H}^{1}(S, \alpha \mapsto \mathcal{H}^{n-1}(\Omega^{\alpha}_{S/T} \overset{\mathbf{L}}{\otimes} \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S}))) .
\]

\[
(16) \qquad \eta \in H^{1}_{DR}(S, \mathbf{R}^{n-1}f_{*}(\Omega^{\bullet}_{X/S})) \qquad (\Omega^{1}_{S/T} \text{ plat}) .
\]

\[
(17) \qquad P \xrightarrow{d_{p}} \Omega^{1}_{S/T} \otimes \mathbf{R}^{n-1}f_{*}(\Omega^{\bullet}_{X/S})
\]

\[
(18) \qquad \begin{cases} d_{p}(x+\xi) = d_{p}(x) + d^{0}(\xi) \\ \operatorname{Im} d_{p} \subset Z^{1}(\Omega^{\bullet}_{S/T} \otimes \mathbf{R}^{n-1}f_{*}(\Omega^{\bullet}_{X/S})) . \end{cases}
\]

\[
(19) \qquad P \xrightarrow{\delta^{\infty}_{p}} P^{\infty}_{S/T}(\mathbf{R}^{n-1}f_{*}(\Omega^{\bullet}_{X/S}))^{+}
\]

\[
(20) \qquad \delta^{\infty}_{p}(x+\xi) = \delta^{\infty}_{p}(x) + d^{+}_{S/T}(\xi)
\]

\[
\mathrm{Gr}^{p}_{X/S}(\Omega^{\bullet}_{X/T} \otimes E) \simeq \mathrm{Gr}^{p}_{X/S}(\Omega^{\bullet}_{X/T}) \otimes E
\]

\[
(21) \qquad \mathrm{Gr}^{p}_{X/S}(\mathbf{R}f_{*}(\Omega^{\bullet}_{X/T} \otimes E)) \simeq \Omega^{p}_{S/T} \overset{\mathbf{L}}{\otimes} \mathbf{R}f_{*}(\Omega^{\bullet}_{X/S} \otimes E)[-p] ,
\]

\[
(22) \qquad H^{*}_{DR}(X/T, E) \Longleftarrow E^{p,q}_{2} = H^{p}_{DR}(S/T, \mathcal{H}^{q}_{DR}(X/S, E)) .
\]

\[
\begin{cases} \Omega^{1}_{B/A} \xrightarrow{\sim} \Omega^{1}_{B/\Lambda} & \text{d'où } \Omega^{*}_{B/A} \xrightarrow{\sim} \Omega^{*}_{B/\Lambda} \\ \text{donc } H^{*}_{DR}(B/A) \simeq H^{*}_{DR}(B/\Lambda) & \text{(comme $\Lambda$-algèbres)} \end{cases}
\]

\[
B \simeq \Lambda[X]/(X^{p} - a) .
\]

\[
\Omega^{1}_{B/A} \simeq \Omega^{1}_{B/\Lambda} \quad \text{admet base } (dx_{i})_{i \in I}
\]

\[
\Omega^{*}_{B/\Lambda} \simeq \bigotimes_{I,\Lambda} \Omega^{*}_{B_{i}/\Lambda} \qquad \text{(tous de $\Lambda$-alg.\ différentielles)}
\]

\[
B_{i} \simeq \Lambda[x_{i}] \simeq \Lambda[X_{i}]/(X_{i}^{p} - a_{i}) \qquad (a_{i} = x_{i}^{p} \in \Lambda) .
\]

\[
H^{*}_{DR}(B/A) \simeq H^{*}_{DR}(B/\Lambda) \simeq \bigotimes_{I,\Lambda} H^{*}_{DR}(B_{i}/\Lambda)
\]

\[
\psi : \Omega^{*}_{B/A} \to Z^{*}_{B/A} \qquad (\text{cycles de } \Omega^{*}_{B/A})
\]

\[
\partial(xy) = \psi_{0}(x)\,\partial y + \psi_{0}(y)\,\partial x
\]

\[
(xy)^{p-1}d(xy) = x^{p}y^{p-1}dy + y^{p}x^{p-1}dx
\]

\[
\widetilde{\psi} : \Omega^{*}_{B/A} \otimes_{B} \widetilde{B} \simeq \Omega^{*}_{\widetilde{B}/\widetilde{A}} \to Z^{*} \ldots
\]

\[
Z^{*}_{B/A} \to H^{*}_{DR}(B/A) \ldots
\]

\[
\boxed{\varphi : \Omega^{*}_{\widetilde{B}/\widetilde{A}} \to H^{*}_{DR}(B/A)}
\]

\[
\Omega^{*}_{B/A} \simeq \bigotimes_{i \in I} \Omega^{*}_{B_{i}/\Lambda} \otimes_{B_{i}} B
\]

\[
\Omega^{*}_{\widetilde{B}/\widetilde{A}} \simeq \bigotimes_{i \in I} (\Omega^{*}_{\widetilde{B}_{i}/\widetilde{\Lambda}} \otimes_{\widetilde{B}_{i}} \widetilde{B})
\]

\[
H^{*}_{DR}(B/A) \simeq \bigotimes_{i \in I} H^{*}_{DR}(B_{i}/\Lambda) .
\]

\[
\varphi_{i} : \Omega^{*}_{\widetilde{B}_{i}/\widetilde{\Lambda}} \otimes_{\widetilde{B}_{i}} \widetilde{B}_{i} \to H^{*}_{DR}(B_{i}/\Lambda)
\]

\[
\text{base } \widetilde{1},\ d\widetilde{x}_{i} \qquad\qquad \text{base } \dot{1},\ \overline{x_{i}^{p-1}dx_{i}}
\]

\[
Z^{*}(B/A) \simeq B^{*}(B/A) \oplus [\Omega^{*}_{\widetilde{B}/\widetilde{A}} \otimes_{\widetilde{B}} \Lambda]
\]

\[
\boxed{\widetilde{C} : H^{*}_{DR}(B/A) \xrightarrow{\sim} \Omega^{*}_{\widetilde{B}/\widetilde{A}} \otimes_{\widetilde{B}} \Lambda \ (= \Omega^{*}_{B/A} \otimes_{B} \Lambda)}
\]

\[
\widetilde{C} : Z^{*}(B/A) \to \Omega^{*}_{\widetilde{B}/\widetilde{A}}
\]

\[
\widetilde{C} : Z^{*}(B/A) \to \Omega^{*}_{\widetilde{B}/\widetilde{A}}
\]

\[
\begin{cases} \widetilde{C}(\omega + \omega') = C(\omega) + C(\omega') \\ \widetilde{C}(\omega\omega') = C(\omega)\,C(\omega') \\ \widetilde{C}(\lambda\omega) = \lambda\,C(\omega) \quad \text{si } \lambda \in \Lambda \\ \widetilde{C}(1) = 1 \end{cases}
\]

\[
\widetilde{C}\left(\frac{dx}{x}\right) = x^{-p}\,\widetilde{C}(x^{p-1}dx) = \text{\struck{$\ill{}$}}\; F_{A}(x^{-1})\,C(x^{p-1}dx)
\]

\[
= (\widetilde{x}^{-1}d\widetilde{x}) \otimes_{\widetilde{B}} \Lambda
\]

\[
\Omega^{*}_{B} \xrightarrow[\sim]{\varphi} H^{*}_{DR}(B) .
\]

\[
\widetilde{C} : Z^{*}(B) \to \Omega^{*}_{B}
\]

\[
\begin{cases} C(x^{p-1}dx) = dx \\ C(dx/x) = dx/x \end{cases}
\]

\[
\psi : \Omega^{*}_{X^{(p/S)}/S} \to f_{*}(Z^{*}(\Omega^{*}_{X/S}))
\]

\[
\psi : \Omega^{*}_{X^{(p/S)}/S} \to f_{*}(\mathcal{H}^{*}(\Omega^{*}_{X/S})) \simeq \mathcal{H}^{*}(f_{*}(\Omega^{*}_{X/S})) ,
\]

\[
\widetilde{\psi} : \Omega^{*}_{X^{(p/S)}/S} \otimes_{\underline{O}_{X^{(p/S)}}} \underline{\Lambda} \to \text{\struck{$f_{*}$}}\; \mathcal{H}^{*}(f_{*}(\Omega^{*}_{X/S}))
\]

\[
\widetilde{C} : \mathcal{H}^{*}(f_{*}(\Omega^{*}_{X/S})) \to \Omega^{*}_{X^{(p/S)}/S} \otimes_{\underline{O}_{X^{(p/S)}}} \underline{\Lambda}
\]

\[
\widetilde{C} : Z^{*}(f_{*}(\Omega^{*}_{X/S})) \to \Omega^{*}_{X^{(p/S)}/S} \otimes_{\underline{O}_{X^{(p/S)}}} \underline{\Lambda}
\]

\[
\Theta_{D_{i}}^{p} = 0 \qquad (i \in I)
\]

\[
\begin{cases} [D_{i}, D_{j}] = 0 \\ [x_{i}, x_{j}] = 0 \\ [D_{i}, x_{j}] = \delta_{ij} \\ x_{i}^{p} = a_{i} \\ D_{i}^{p} = 0 \end{cases}
\]

\[
\Delta_{i} = \Theta_{D_{i}} = \Theta^{0}_{D_{i}} + \omega(D_{i}).\mathrm{id}
\]

\[
(\Delta^{0}_{i} + c_{i})^{p} = 0 .
\]

\[
(\Delta^{0}_{i} + c_{i})^{p} = \underbrace{\Delta^{0\,p}_{i}}_{=\,0 \text{ par hyp.}} + c_{i}^{p} + D_{i}^{p-1}c_{i}
\]

\[
(*) \qquad c_{i}^{p} + D_{i}^{p-1}c_{i} = 0 \qquad i \in I
\]

\[
(**) \qquad D_{i}c_{j} - D_{j}c_{i} = 0 \qquad i, j \in I
\]

\[
(***) \qquad \widetilde{C}\omega = \text{\struck{$\ill{}$}}\; \sum_{i} -D_{i}^{p-1}c_{i}\,(d\widetilde{x}_{i} \otimes 1_{\Lambda})
\]

\[
\widetilde{\omega} = \sum c_{i}^{p}\,(d\widetilde{x}_{i} \otimes 1_{\Lambda})
\]

\[
-\widetilde{C}\omega + \widetilde{\omega} = \sum (c_{i}^{p} + D_{i}^{p-1}c_{i})(d\widetilde{x}_{i} \otimes 1_{\Lambda})
\]

\[
\underline{O}_{X}^{*} \xrightarrow{f \mapsto df/f} \underline{\Omega}^{1}_{X/S}
\]

\[
\widetilde{C} : F_{X/S\,*}(Z\Omega^{*}_{X/S}) \to \Omega^{*}_{X^{(p/S)}}
\]

\[
\widetilde{C} : F_{X/S\,*}(\underbrace{\underline{\Omega}^{n}_{X/S}}_{\substack{\omega_{X/S} \\ \text{module dualisant relatif}}}) \to \underbrace{\underline{\Omega}^{n}_{X^{(p/S)}}}_{\omega_{X^{(p/S)}}}
\]

\[
\boxed{\operatorname{res}\left(\frac{\widetilde{C}\omega}{\widetilde{f}_{1} \ldots \widetilde{f}_{n}}\right) = \operatorname{res}\left(\frac{\omega}{f_{1}^{p} \ldots f_{n}^{p}}\right)}
\]

\[
\operatorname{res}\frac{\widetilde{C}\omega}{\widetilde{f}} = \operatorname{res}\frac{\omega}{f^{p}} \quad \text{i.e.} \quad \boxed{\operatorname{res}\widetilde{C}\overline{\omega} = \operatorname{res}\overline{\omega}}
\]

\[
\text{\struck{$\left(\operatorname{res}\dfrac{C\omega}{f}\right)^{p} = \operatorname{res}\dfrac{\omega}{f^{p}}$}} \qquad \boxed{(\operatorname{res} C\overline{\omega})^{p} = \operatorname{res}\overline{\omega}}
\]

\[
F_{i} : F_{*}(\Omega^{i}_{X/S}) \to \Omega^{i}_{X^{(p/S)}/S} ,
\]

\[
F^{*} : \Omega^{i}_{X^{(p/S)}/S} \to F_{*}\Omega^{i}_{X/S}
\]

\[
C = F_{X/S\,*} : F_{X/S\,*}(\omega_{X/S}) \to \omega_{X^{(p/S)}/S} \simeq \widetilde{\omega}_{X/S}
\]

\[
\omega_{X/S} \to \omega^{(p)}_{X/S}
\]

\[
\text{\struck{$[H^{i}(X, E) \to]\ H^{i}(X^{(p/S)}, \widetilde{E}) \to H^{i}(X, E^{(p)})$}}
\]

\[
(\varphi) \qquad \mathbf{R}f_{*}(\widetilde{E}) \xrightarrow{\ \varphi\ } \mathbf{R}f_{*}(E^{(p)})
\]

\[
\phantom{(\varphi) \qquad} \simeq \mathbf{R}f_{*}(E)^{(p)}
\]

\[
\mathbf{R}f_{*}(\check{E}^{(p)} \otimes \Omega^{n}_{X/S}[n]) \to \mathbf{R}f_{*}(\check{\widetilde{E}} \otimes \Omega^{n}_{\widetilde{X}/S}[n])
\]

\[
(**) \qquad F_{X/S\,*}(\check{E}^{(p)} \otimes \Omega^{n}_{X/S}) = \check{\widetilde{E}} \otimes F_{X/S\,*}(\Omega^{n}_{X/S})
\]

\[
\phantom{(**) \qquad} \to \check{\widetilde{E}} \otimes \Omega^{n}_{X^{(p/S)}/S}
\]

\[
R^{n}f_{*}(\widetilde{E}) \longrightarrow R^{n}f_{*}(E^{(p)})
\]

\[
\phantom{R^{n}f_{*}(\widetilde{E})} \simeq R^{n}f_{*}(E)^{(p)}
\]

\[
f_{*}(\check{E}^{(p)} \otimes \Omega^{n}_{X/S}) \to g_{*}(\check{\widetilde{E}} \otimes \Omega^{n}_{X^{(p/S)}/S})
\]

\[
\phantom{f_{*}(\check{E}^{(p)} \otimes \Omega^{n}_{X/S}) \to{}} \simeq g_{*}(\check{E} \otimes \Omega^{n}_{X/S})^{(p)}
\]

\[
R^{i}f_{*}(\underline{O}_{X})^{(p)} \to R^{i}f_{*}(\underline{O}_{X})
\]

\[
\widetilde{C} : f_{*}(\Omega^{n}_{X/S}) \to g_{*}(\Omega^{n}_{X^{(p/S)}/S}) \underset{\text{s.\ réserve}}{\simeq} f_{*}(\Omega^{n}_{X/S})^{\otimes p}
\]

\[
R^{n-i}f_{*}(\Omega^{n}_{X/S}) \to R^{n-i}f_{*}(\Omega^{n}_{X/S})^{(p)}
\]

\[
R^{1}f_{*}(\underline{O}_{X^{(p/S)}}) \longrightarrow R^{1}f_{*}(\underline{O}_{X})
\]

\[
\underline{\mathrm{Lie}}\,\underline{\mathrm{Pic}}_{X^{(p/S)}/S} \longrightarrow \underline{\mathrm{Lie}}\,\underline{\mathrm{Pic}}_{X/S}
\]

\[
\Vert
\]

\[
(\underline{\mathrm{Lie}}\,\underline{\mathrm{Pic}}_{X/S})^{(p/S)}
\]

\[
\underline{\mathrm{Pic}}_{X^{(p/S)}/S} \simeq \underline{\mathrm{Pic}}^{(p/S)}_{X/S} \xrightarrow{\ V\ } \underline{\mathrm{Pic}}_{X/S}
\]

\[
  Q = f_{X/S*}(\mathcal{O}_X)/\operatorname{Im}\mathcal{O}_{X^{p/S}}
    = f_{X/S*}(\mathcal{O}_X)/\Lambda
    \simeq \mathcal{O}_X/f^{-1}\mathcal{O}_S[\mathcal{O}_X^{p}]
\]

\[
  (*)\qquad Q\times Q \longrightarrow
  \Omega^1_{X^{p/S}/S}\otimes_{\mathcal{O}_{X^{p/S}}}\Lambda
\]

\[
  \varphi : f_{X/S*}(\mathcal{O}_X)\times f_{X/S*}(\mathcal{O}_X)
  \longrightarrow \Omega^1_{X^{p/S}/S}\otimes_{\mathcal{O}_{X^{p/S}}}\Lambda
\]

\[
  \varphi(f,g) = \widetilde{C}(f\,dg)
\]

\[
  \widetilde{C}(x^i\,dx^j) = \widetilde{C}(j\,x^{i+j-1}\,dx) =
  \begin{cases} 0 & \text{si } i+j\neq p-1\\ j & \text{si } i+j = p-1\end{cases}
\]

\[
  \begin{pmatrix} & & p-1\\ & \cdots & \\ 1 & & \end{pmatrix}
  \qquad\text{(antidiagonale $1, 2, \dots, p-1$, zéros ailleurs),}
\]

\[
  Q \xrightarrow{\ \sim\ } \check{Q}\otimes\bigl(\Omega^1_{X^{p/S}/S}\otimes\Lambda\bigr)
\]

\[
  \det Q \simeq \det\check{Q}\otimes
  \bigl(\Omega^{1\ \otimes(p-1)}_{X^{p/S}/S}\otimes\Lambda\bigr)
\]

\[
  (**)\qquad (\det Q)^{\otimes 2} \simeq
  \Omega^{1\ \otimes(p-1)}_{X^{p/S}/S}\otimes\Lambda
\]

\[
  \text{\struck{$\det Q^{\otimes}\simeq$}}\ \Omega^1_{X^{p/S}/S}\otimes\Lambda
  \simeq (\det Q)^{\otimes 2}
\]

\[
  \psi : \det(Q) \simeq
  \Omega^{1\ \otimes(\frac{p-1}{2})}_{X^{p/S}/S}\otimes\Lambda
  \qquad\Bigl[\simeq \Omega^{1\ \otimes(\frac{p-1}{2})}_{X/S}
  \otimes_{\mathcal{O}_X}\Lambda\Bigr].
\]

\[
  (-1)^{(p-1)-1}\ 1.2\dots(p-1) = \text{\struck{$-(1.2\dots(p-1))$}}\,.
\]

\[
  (***)\qquad \boxed{\ \Psi(\dot{x}\wedge\dot{x}^2\wedge\dots\wedge\dot{x}^{p-1})
  = \widetilde{dx}^{\,\otimes\frac{p-1}{2}}\otimes\Lambda\ }
\]

\[
  f_*(\Omega^1_{X/S}) = \underline{\omega} \xrightarrow{\ \mathrm{dfn}\ }
  g_*(\Omega^1_{X^{p/S}/S}) \simeq f_*(\Omega^1_{X/S})^{(p)} =
  \underline{\omega}^{(p)}.
\]

\[
  \widetilde{C} : \underline{\omega}\longrightarrow\underline{\omega}^{(p)}
\]

\[
  \widetilde{C}^{(p^i)} : \underline{\omega}^{(p^i)} \longrightarrow
  (\underline{\omega}^{(p)})^{(p^i)} = \underline{\omega}^{(p^{i+1})},
\]

\[
  F_*^{(\nu)} = \underbrace{\widetilde{C}^{(p^{\nu-1})}\,
  \widetilde{C}^{(p^{\nu-2})}\cdots\widetilde{C}^{p}\,\widetilde{C}}_{\nu\ \text{facteurs}}
  : \underline{\omega}\longrightarrow\underline{\omega}^{(p^\nu)}
\]

\[
  F_*^{(\nu)} : \underline{\omega}\longrightarrow\underline{\omega}
\]

\[
  \widetilde{C}\omega = \omega^{(p)}
\]

\[
  \text{\struck{$\widetilde{C}^{(p^\nu)}$}}\ F^{(\nu)}\omega = \omega^{(p^\nu)}\,,
\]

\[
  \boxed{\ F_*^{(\nu)}\omega = \omega\ }
\]

\[
  f_{X/S} : X\longrightarrow X^{p/S}
\]

\[
  f_{X/S}^{p^i/S} : X^{p^i/S}\longrightarrow X^{p^{i+1}/S}
\]

\[
  F_{X/S}^{(\nu)} = f_{X/S}^{(p^{\nu-1})}\,f_{X/S}^{(p^{\nu-2})}\cdots f_{X/S}^{p}\,f_{X/S}
  : X\longrightarrow X^{(p^\nu/S)}
\]

\[
  F_*^{(\nu)} = \operatorname{Tr}_{F^{(\nu)}_{X/S}} = \bigl(F^{\nu}_{X/S}\bigr)_*
\]

\[
  F^{\nu}_{X/S} : X\longrightarrow X
\]

\[
0 \to \underline{\omega} \to \mathcal{H}^{1}_{\mathrm{DR}}(X/S) \to R^{1}f_{*}(\underline{\mathcal{O}}_{X/S})
\]

\[
\mathcal{H}^{1}_{\mathrm{DR}}(X/S) \to \mathcal{H}^{1}_{\mathrm{DR}}(X^{(p^{\nu}/S)}) \simeq \mathcal{H}^{1}_{\mathrm{DR}}(X/S)^{(p^{\nu})},
\]

\[
(1.1) \qquad 0 \to f^{\mathrm{fl}}_{*}(\underline{\Omega}^{1}_{X/S})/\mathrm{Im}\, f^{\mathrm{fl}}_{*}(\underline{\mathcal{O}}^{*}_{X/S}) \to P^{c}_{X/S} \to \underline{\mathrm{Pic}}_{X/S} \to R^{1}f^{\mathrm{fl}}_{*}(\underline{\Omega}^{1}_{X/S})
\]

\[
(1.2) \qquad 0 \to f^{\mathrm{fl}}_{*}(\underline{\Omega}^{1}_{X/S}) \to u^{*}(P^{c}) \to B \to 1
\]

\[
(1.3) \qquad 0 \to f^{\mathrm{fl}}_{*}(\underline{\Omega}^{1}_{X/S}) \to (P^{c}_{X/S})^{0} \to B \to 0, \qquad B = \underline{\mathrm{Pic}}^{0}_{X/S}
\]

\[
(2.1) \qquad 0 \to P^{ci}_{X/S} \to P^{c}_{X/S} \to f^{\mathrm{fl}}_{*}(Z^{2}_{X/S}) \subset f^{\mathrm{fl}}_{*}(\Omega^{2}_{X/S})
\]

\[
\begin{aligned}
(2.2) \qquad 0 &\to f^{\mathrm{fl}}_{*}(\underline{Z}^{1}_{X/S})/\mathrm{Im}\, f^{\mathrm{fl}}_{*}(\underline{\mathcal{O}}^{*}_{X/S}) \to P^{ci}_{X/S} \to \underline{\mathrm{Pic}}_{X/S} \\
&\to \mathbb{R}^{2}f^{\mathrm{fl}}_{*}(\mathcal{O} \to \underline{\Omega}^{1}_{X/S} \to \underline{\Omega}^{2}_{X/S} \to \cdots)
\end{aligned}
\]

\[
B^{*} \simeq A, \quad \underline{\mathrm{Lie}}(B^{*})^{\vee} = \underline{\mathrm{Lie}}(A)^{\vee} \simeq g^{0}_{*}(\Omega^{1}_{A/S}) \simeq g^{1}_{*}(\Omega^{1}_{A^{1}/S}) \simeq f_{*}(\Omega^{1}_{A^{1}/S}) \ldots
\]

\[
(2.3) \qquad u^{*}(P^{c}_{X/S}) \to P^{ci}_{X/S},
\]

\[
(2.4) \qquad f^{\mathrm{fl}}_{*}(\Omega^{1}_{X/S}) = f^{\mathrm{fl}}_{*}(\underline{Z}^{1}_{X/S})
\]

\[
(2.5) \qquad f^{*}V = f_{*}(\underline{\Omega}^{2}_{X/S})
\]

\[
\mathrm{Hom}(u^{*}(P^{c}), V) \to \mathrm{Hom}(f^{\mathrm{fl}}_{*}(\Omega^{1}_{X/S})/-, V)
\]

\[
(2.6) \qquad (P^{c0}) \subset P^{ci} \quad \text{i.e.} \quad P^{c0} = P^{ci0}.
\]

\[
P^{cip}_{X/S} \subset P^{ci}_{X/S}
\]

\[
K \in \Gamma(X, \underline{\mathrm{End}}(E) \otimes (\Omega^{1}_{X/S})^{\otimes p}) \ ?
\]

\[
\begin{aligned}
(3.2) \qquad 0 &\to \underbrace{[f^{\mathrm{fl}}_{*}(\underline{\mathcal{O}}^{*}_{X/S}) \to f^{\mathrm{fl}}_{*}(\underline{Z}^{1}_{X/S}) \to f^{\mathrm{fl}}_{*}(\underline{\Omega}^{1\otimes p}_{X/S})]}_{\underline{\mathrm{Cart}}_{X/S} \text{ « faisceau de Cartier »}} \to P^{cip}_{X/S} \\
&\to \underline{\mathrm{Pic}}^{ci}_{X/S} \to f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})/\mathrm{Im}\, f^{\mathrm{fl}}_{*}(\underline{Z}^{1}_{X/S}),
\end{aligned}
\]

\[
K_{0} : f^{\mathrm{fl}}_{*}(\underline{Z}^{1}_{X/S}) \to f^{\mathrm{fl}}_{*}(\underline{\Omega}^{1\otimes p}_{X/S})
\]

\[
(3.3) \qquad K_{0}(\omega) = \omega^{\otimes p} - F^{*}_{X/S}(\tilde{C}\omega) \quad \text{pour } d\omega = 0,
\]

\[
S' \mapsto \mathrm{Pic}(X_{S'}^{p/S'\,!}).
\]

\[
(3.4) \qquad P^{cip}_{X/S} \simeq \underline{\mathrm{Pic}}_{X^{(p/S)}/S} \simeq \underline{\mathrm{Pic}}_{X/S}^{(p/S)}
\]

\[
(3.5) \qquad V : \underline{\mathrm{Pic}}^{p/S}_{X/S} \to \underline{\mathrm{Pic}}_{X/S}
\]

\[
(3.6) \qquad \mathrm{Ker}(V_{\underline{\mathrm{Pic}}_{X/S}}) \simeq
\left\{
\begin{array}{l}
\text{faisceau des sections de } f^{\mathrm{fl}}_{*}(\underline{\Omega}^{1}_{X/S}) \text{ telles} \\
\text{que } d\omega = 0,\ C\omega = \tilde{\omega}, \text{ modulo les sous-} \\
\text{faisceau des } df/f,\ f \text{ dans } f^{\mathrm{fl}}_{*}(\underline{\mathcal{O}}^{*}_{X/S})
\end{array}
\right.
\]

\[
(\underline{\mathrm{Pic}}_{X/S})^{(p/S)\,0} = (\underline{\mathrm{Pic}}^{0}_{X/S})^{p/S} = B^{(p/S)},
\]

\[
(3.7) \qquad P^{cip\,0}_{X/S} \xleftarrow{\sim} P^{cip\,0}_{A^{1}/S} \simeq P^{cip\,0}_{A^{0}/S} \simeq B^{(p/S)}
\]

\[
P^{cin} \subset P^{ci},
\]

\[
\overline{f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})} \subset f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S}),
\]

\[
(4.1) \qquad 0 \to P^{cip} \to P^{ci}_{X/S} \to f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})/\overline{f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})}
\]

\[
(4.2) \qquad 0 \to P^{cip}_{X/S} \to P^{cin}_{X/S} \to \overline{f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})},
\]

\[
(4.3) \qquad 0 \to P^{cip\,0}_{X/S} \to P^{cin\,0}_{X/S} \to \overline{f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})}
\]

\[
(4.4) \qquad 0 \to \mathcal{V} \to P^{cin}_{X/S} \to \underline{\mathrm{Pic}}_{X/S}
\]

\[
f^{\mathrm{fl}}_{*}(\underline{Z}^{1}_{X/S})/\mathrm{Im}\, f^{\mathrm{fl}}_{*}(\mathcal{O}^{*}_{X}) \to f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})/\overline{f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})},
\]

\[
(4.5) \qquad 0 \to \underline{\mathrm{Cart}}_{X/S} \to \mathcal{V} \to \overline{f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})}
\]

\[
(4.6) \qquad 0 \to \mathcal{V} \to [P^{cin}_{X/S}]^{0} \to \underline{\mathrm{Pic}}^{0}_{X/S} \to 0
\]

\[
\begin{aligned}
0 &\to H^{1}(H^{0}(X, \Omega^{*}_{X/S})) \to H^{1}_{\mathrm{DR}}(X/S) \to \mathrm{Ker}[H^{1}(X, \mathcal{O}_{X}) \xrightarrow{d} H^{1}(X, \Omega^{1}_{X})] \\
&\to H^{2}(H^{0}(X, \Omega^{*})) \to H^{2}_{\mathrm{DR}}(X) \to H^{1}(H^{1}(X, \Omega^{*})) \to E^{30}_{2}
\end{aligned}
\]

\[
E^{10}_{2} = \frac{Z^{1}(X, \Omega^{*}_{X/S})}{\mathrm{Im}\, C^{0}(X, \Omega^{*}_{X/S})}, \quad
E^{20}_{2} = \frac{Z^{2}(X, \Omega^{*}_{X/S})}{\mathrm{Im}\, C^{1}(X, \Omega^{*}_{X/S})},
\]

\[
E^{11}_{2} = \frac{\mathrm{Ker}[H^{1}(X, \Omega^{1}) \to H^{1}(X, \Omega^{2})]}{\mathrm{Im}[H^{1}(X, \Omega^{0}) \to H^{1}(X, \Omega^{1})]},
\]

\[
\begin{aligned}
\text{\struck{$\xi \mapsto$}}\ v'(\xi) &= v(\xi) + \underline{\omega(\xi)^{p} + \theta_{\xi}^{p-1}\omega(\xi) - \omega(\xi^{p})} \\
&= v(\xi) + \underbrace{\omega(\xi)^{p} - \langle C\omega, \tilde{\xi} \rangle 1_{\underline{L}}}
\end{aligned}
\]

\[
S \xleftarrow{\varphi_{0}} U_{0} \xrightarrow{f} U \qquad (S \text{ lisse})
\]

\[
v'(\xi) = v(\xi) + \langle \omega^{\otimes p} - F^{*}_{X/S}(C\omega), \xi^{\otimes p} \rangle, \quad \text{i.e.\ \uncertain{intrinsèquement}}
\]

\[
v' = v + \underbrace{(\omega^{\otimes p} - F^{*}_{X/S}C\omega)}_{F^{*}_{X/S}(\tilde{\omega} - C\omega)}
\]

\[
\Gamma(X, \Omega^{1\,\otimes p}_{X/S})/\mathrm{Im}[Z^{1}(X, \Omega^{*}_{X/S}) \to \Gamma(X, \Omega^{1\,\otimes p}_{X/S})]
\]

\[
Z^{1}(X, \Omega^{*}_{X/S}) \to \Gamma(X, \Omega^{1\,\otimes p}_{X/S}), \qquad \omega \mapsto F^{*}_{X/S}(\tilde{\omega} - C\omega) = \omega^{\otimes p} - F^{*}_{X/S}(C\omega)
\]

\[
\underline{\mathrm{Pic}}^{\mathrm{conn\,int}}_{X/S} \to \underline{\mathrm{Pic}}_{X/S}
\]

\[
(f_{*}\underline{Z}^{1}_{X/S}) \xrightarrow{\gamma} f_{*}(\Omega^{1\,\otimes p}_{X/S}), \qquad f_{*}\underline{Z}^{1}_{X/S} = \mathrm{Ker}[f_{*}(\Omega^{1}_{X/S}) \xrightarrow{d} f_{*}(\Omega^{2}_{X/S})]
\]

\[
\left( S \xleftarrow{\psi_{0}} U_{0} \xrightarrow{j^{(n)}} U^{(n)} \right),
\]

\[
H^{\cdot}(\mathbb{R}^{\cdot} f_{*}(\Omega^{\cdot}_{X/S})) \to \text{\struck{$\mathbb{R}$}}\ \mathcal{H}^{0}(\Omega^{\cdot}_{X/S}) \ldots
\]

\[
\underline{\Gamma}(G)/\underline{E} \longrightarrow Z^{1}C^{\cdot}_{S/R}(\underline{g}^{*})/d^{0}(\underline{E})
\]

\[
D : \underline{g} \to \mathcal{M}
\]

\[
\underline{g} \xrightarrow{c = d^{0}} Z^{1}C_{S/R}(\underline{g}) \xrightarrow{\tilde{D}} \mathcal{M}, \qquad \tilde{D} \circ d^{0} = D
\]

\[
  I(s) = \int_{C_s} \omega_s = \int_{a(s)}^{b(s)} \omega_s
\]

\[
  I'(s) = I(s) + \int_{Z_s} \omega_s . \tag{1}
\]

\[
  \sum D_{i,s} \int_{a(s)}^{b(s)} \omega_{i,s} . \tag{2}
\]

\[
  \sum D_{i,s} \int_{Z_s} \omega_{i,s}
  \quad \text{\struck{$= \int_{Z_s} \sum D_{i,s}\, \omega_{i,s}$}} \tag{3}
\]

\[
  \mu(a,b) = \sum_i D_{i,s} \int_{a(s)}^{b(s)} \omega_{i,s} \tag{4}
\]

\[
  \mu(a,b) + \mu(b,c) = \mu(a,c) . \tag{5}
\]

\[
  \mathscr{F} = f_*(Z\Omega^1_{X/S}) / d f_*(\mathcal{O}_X) , \tag{6}
\]

\[
  \sum D_i \omega'_i = 0 . \tag{7}
\]

\[
  \sum \overline{D}_i\, \omega_i = \varpi \tag{8}
\]

\[
  \sum D_i\, \omega'_i = \varpi' \quad \text{dans } \mathscr{F} \tag{9}
\]

\[
  \varpi \;(= \textstyle\sum \overline{D}_i\, \omega_i) = d\varphi ,
  \qquad \varphi \in \Gamma(X, \mathcal{O}_X) . \tag{10}
\]

\[
  \sum_i D_{i,s} \int_{Z_s} \omega_{i,s} = \int_{Z_s} \varpi \tag{11}
\]

\[
  D_s \int_{Z_s} \omega_s = \int_{Z_s} \overline{D}_s\, \omega_s \tag{12}
\]

\[
  D_s \int_{a(s)}^{b(s)} \omega_s = \int_{a(s)}^{b(s)} (\overline{D}\omega)_s ,
  \tag{13}
\]

\[
  D_s \int_{C_s} \omega_s = \int_{C_s} (\overline{D}\omega)_s
  \tag{13 bis}
\]

\[
  \int_{a(s)}^{b(s)} \overline{D}\, d\varphi
  = \int_{a(s)}^{b(s)} d\, \overline{D}\varphi
  = (\overline{D}\varphi)(b(s)) - (\overline{D}\varphi)(a(s)) ,
\]

\[
  \mu(a,b) = \int_{a(s)}^{b(s)} \Bigl(\sum \overline{D}_i\, \omega_i\Bigr)_s
  = \varphi(b(s)) - \varphi(a(s)) , \tag{14}
\]

\[
  \sum D_{i,s} \int_{C_s} \omega_{i,s} = 0
\]

\[
  \sum_i D_{i,s} \int_{a(s)}^{b(s)} \omega_{i,s} = 0
\]

\[
  \sum D_{i,s} \int_{Z_s} \omega_i
  = \int_{Z_s} \Bigl(\sum \overline{D}_i\, \omega'_i\Bigr)_s
\]

\[
  \boxed{\; D_s \int_{Z_s} \omega'_s = \int_{Z_s} (\overline{D}\omega')_s \;}
  \tag{15}
\]

\[
  \mu(a,b) = \varphi(b(s)) - \varphi(a(s))
\]

\[
  a_0(s_0) = a(s_0) , \qquad b_0(s_0) = b(s_0) .
\]

\[
  \mu(a,b) = \mu(a_0, b_0) + \mu(b_0, b) - \mu(a_0, a)
\]

\begin{align*}
  \mu(a,b)(s_0) &= \bigl[\varphi(b(s_0)) - \varphi(a(s_0))\bigr] \\
  &\quad + \Bigl(\sum_i D_{i,s} \int_{b_0(s)}^{b(s)} \omega_{i,s}
  - \sum_i D_{i,s} \int_{a_0(s)}^{a(s)} \omega_{i,s}\Bigr)_{s = s_0}
\end{align*}

\[
  \text{\struck{$d_a \in \Gamma(S, a^*(\mathscr{V}_{X/S}) \otimes \Omega^1_S)$}}
\]

\[
  \text{\struck{$a^*(\omega) \in \Gamma(S, a^*(\Omega^1_{X/S}))$}}
\]

\[
  \text{\struck{$a^*(\omega) \in \Gamma(S, \Omega^1_S)$}}
\]

\[
  \mathrm{Diff}^i(P,F) \xrightarrow{u \mapsto D \circ u}
  \mathrm{Diff}^{i+1}(P,G) \xrightarrow{v \mapsto D' \circ v}
  \mathrm{Diff}^{i+2}(P,H)
\]

\[
  \mathrm{Diff}^i(H,P) \xrightarrow{u \mapsto u \circ D'}
  \mathrm{Diff}^{i+1}(G,P) \xrightarrow{v \mapsto v \circ D}
  \mathrm{Diff}^{i+2}(F,P)
\]

\[
  E \otimes \Omega^*_{S/T} = \Omega^*_{S/T}(E)
\]

\begin{align*}
  0 \leftarrow \mathrm{Hom}(E,P) &\leftarrow \mathrm{Diff}^i(E,P)
  \xleftarrow{u \mapsto u \circ d^0_E} \\
  &\qquad \mathrm{Diff}^{i-1}(E \otimes \Omega^1_{S/T}, P)
  \xleftarrow{u \mapsto u \circ d^1_E} \cdots
\end{align*}

\[
  \mathrm{Diff}^{(i)}_+(E,P) \simeq
  \mathrm{Diff}^{i-1}(E \otimes \Omega^1_{S/T}, P) /
  \mathrm{Im}\, \mathrm{Diff}^{i-2}(E \otimes \Omega^2_{S/T}, P)
\]

\[
  \mathrm{Diff}^{(i)}_+(E,P) \hookrightarrow
  \mathscr{H}\!om\,\mathrm{add}(\mathrm{Ker}\, d^1_E, P)
\]

\[
  P^i(E) / \mathcal{O}_S \cdot \delta_i(E) ,
\]

\[
  \delta_i : E \to P^i(E)
\]

\[
  P^i(M) \to P
\]

\[
  \rho_i : E \to P^i(M)
\]

\[
  E \xrightarrow{\delta_i} P^i(E) \to P^i(M) .
\]

\[
  \mathscr{P}_i = P^i(M) / \mathcal{O}_S \cdot \rho_i(E) ,
\]

\[
  \rho_\infty : E \to P^\infty(M)
\]

\[
  X' \times_T T_0 \simeq X_0 , \qquad X'' \times_T T_0 \simeq X_0 ,
\]

\[
  \mathscr{G} \simeq \underline{\mathrm{Hom}}(\Omega^1_{X_0/T_0}, f_0^*(I))
  = \mathscr{V}_{X_0/T_0} \otimes f_0^*(\underline{I})
\]

\[
  \mathscr{P} \times \Omega^\bullet_{X'/T} \to \Omega^\bullet_{X''/T}
  \qquad \text{i.e.}
\]

\begin{align*}
  \mathscr{P} &\xrightarrow{u} \underline{\mathrm{Isom}}_{\text{de complexes}}
    (\Omega^\bullet_{X'/T}, \Omega^\bullet_{X''/T})
    \subset Z^0\, \underline{\mathrm{Hom}}^\bullet
    (\Omega^\bullet_{X'/T}, \Omega^\bullet_{X''/T}) \\
  \mathscr{G} &\xrightarrow{v} \underline{\mathrm{Aut}}_{\text{de complexes}}
    (\Omega^\bullet_{X''/T})
    \subset Z^0\, \underline{\mathrm{Hom}}^\bullet
    (\Omega^\bullet_{X''/T}, \Omega^\bullet_{X''/T}) \\
  \mathscr{G} &\xrightarrow{k} \underline{\mathrm{Hom}}^{(-1)}
    (\Omega^\bullet_{X''/T}, \Omega^\bullet_{X''/T})
\end{align*}

\[
  \begin{cases}
    u(pg) = u(p)\, v(g) \\
    v(g) - \mathrm{id} = d\, k(g) + k(g)\, d
  \end{cases}
  \qquad \text{si } p \in \Gamma(U, \mathscr{P}),\ g \in \Gamma(U, \mathscr{G})
\]

\begin{align*}
  \mathscr{P} &\xrightarrow{u} \underline{\mathrm{Hom}}^0(K', K'') \\
  \mathscr{G} &\xrightarrow{v} \underline{\mathrm{Hom}}^\bullet(K'', K'') \\
  \mathscr{G} &\xrightarrow{k} \underline{\mathrm{Hom}}^{-1}(K', K'') ,
\end{align*}

\begin{gather*}
  u(pg) = u(p)\, v(g) \\
  v(g + g') = v(g) + v(g') \\
  v(g) = d\, k(g) + k(g)\, d
\end{gather*}

\[
  K' \to K''
\]

\[
  \mathbb{R}\Gamma_X(K') \to \mathbb{R}\Gamma_X(K'')
\]

\[
  0 \to \mathscr{G} \xrightarrow{i} \overline{\mathscr{G}}
  \xrightarrow{j} \mathbb{Z} \to 0
\]

\[
  j^{-1}(1) \simeq \mathscr{P}
\]

\[
  \mathscr{H} = \underline{\mathrm{Hom}}^\bullet(K', K'')
\]

\[
  \cdots \to 0 \to \mathscr{G} \to \overline{\mathscr{G}} \to 0 \to 0
\]

\[
  \varphi : \mathscr{L}^\bullet \to \mathscr{H}^\bullet
\]

\[
  \varphi^{-1} = k , \qquad \varphi^0 | \mathscr{P} = u
\]

\[
  \underline{\mathcal{O}}_X \xrightarrow{\ \sim\ } \text{\struck{$\mathscr{L}$}}
  \ \mathscr{L} \to \mathscr{H} \to \mathbb{R}\,\underline{\mathrm{Hom}}(K', K'')
\]

\[
  \text{\struck{\ill{}}}\ \mathbb{R}\Gamma_X(\underline{\mathcal{O}}_X)
  \to \text{\struck{\ill{}}}\ \mathbb{R}\,\underline{\mathrm{Hom}}(K', K'') ,
\]

\[
  \boxed{\; \mathbb{R}\Gamma_X(\underline{\mathcal{O}}_X) \to
  \mathbb{R}\,\underline{\mathrm{Hom}}(K', K'') \;}
\]