Cote n° 93 · pages 37–235 · 32 commutative diagrams · Connexions de Gauss-Manin et équations de Picard-Fuchs. Opération de Cartier : copies de tapuscrit annoté (s.d.), tiré à part (1981), notes manuscrites (s.d.).
Inventory dating : [1972]-1981
Édition de démonstration

batch 2 · p. 37 — read it beside the facsimile1 / 32
LaTeX source
\begin{tikzcd}
{M^{(p)} = \underline{f}_{S}^{*}(M)} \arrow[r, "F_{M}", bend left=15] & M \arrow[l, "V_{M}", bend left=15]
\end{tikzcd}
batch 4 · p. 77 — read it beside the facsimile2 / 32
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\begin{tikzcd}[column sep=small]
\mathcal{E}_X \arrow[rr] \arrow[dr] & & \underline{\mathrm{Conn}}(\underline{\omega}_X) \\
& \underline{\omega}^*_{X/\mathbb{C}} \arrow[ur] &
\end{tikzcd}
batch 4 · p. 80 — read it beside the facsimile3 / 32
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
& \bar{X}_x & & \bar{X}_x \\
& U(x) \arrow[u, "\mathbb{C}\text{-an}"] \arrow[d, "\wr"] & B(x) \arrow[l, hook'] \arrow[r] \arrow[d, "\wr"] & V(x) \arrow[u] \arrow[d, "\wr"] \\
& D_x & \hat{T}_x \arrow[l, hook'] \arrow[r] \arrow[d] & D''_x \\
& & \Sigma_x &
\end{tikzcd}
batch 5 · p. 81 — read it beside the facsimile4 / 32
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathcal{E}_X \arrow[r] \arrow[d, "\exp"'] & \mathcal{E}_X/\mathbb{G}_a(\mathbb{C}) \overset{\sim}{\rightarrow} \underline{\omega}^{1*}_X \arrow[r, "\varphi\mapsto\varphi^n"] & \mathcal{E}_X/(\mu_n\cdot\mathbb{G}_a)(\mathbb{C}) \overset{\sim}{\rightarrow} \underline{\omega}^{n}_X \arrow[r, "{\psi\mapsto -\frac{1}{n}\frac{d\psi}{\psi}}"] & \mathcal{E}_X/(\mathbb{G}_m\cdot\mathbb{G}_a)(\mathbb{C}) \overset{\sim}{\rightarrow} \mathrm{Conn}(\underline{\omega}^1_X) \arrow[d, "D\mapsto c_D"] \\
\mathcal{E}^*_X \arrow[r] & \mathcal{E}^*_X/\mathbb{G}_m(\mathbb{C}) \overset{\sim}{\rightarrow} \underline{\omega}^{1*}_X \arrow[r, "\varphi\mapsto\varphi^2"] & \mathcal{E}^*_X/(\mathbb{Z}/2\mathbb{Z}\cdot\mathbb{G}_m)(\mathbb{C}) \simeq \underline{\omega}^{2*}_X & \mathcal{E}_X/\mathrm{GP}(1,\mathbb{C}) \overset{\sim}{\rightarrow} \mathrm{Connproj}(X)
\end{tikzcd}
batch 5 · p. 81 — read it beside the facsimile5 / 32
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\mathcal{E}_X \arrow[r, "d"] \arrow[d, "\exp"'] & {[\mathcal{E}_X/\mathbb{G}_a(\mathbb{C}) \overset{\sim}{\rightarrow} \underline{\omega}^{1*}_X]} \arrow[r, "\alpha_n"] & {[\mathcal{E}_X/(\mu_n\cdot\mathbb{G}_a)(\mathbb{C}) \overset{\sim}{\rightarrow} \underline{\omega}^{n*}_X]} \arrow[r, "\beta_n"] & {[\mathcal{E}_X/(\mathbb{G}_m\cdot\mathbb{G}_a)(\mathbb{C}) \overset{\sim}{\rightarrow} \mathrm{Conn}(\underline{\omega}^1_X)]} \arrow[d, "\gamma"] \\
\mathcal{E}^*_X \arrow[r] & {[\mathcal{E}^*_X/\mathbb{G}_m(\mathbb{C}) \simeq \underline{\omega}^{1*}_X]} \arrow[r, "\alpha'_2"] & {[\mathcal{E}^*_X/(\mathbb{Z}/2\mathbb{Z}\cdot\mathbb{G}_m)(\mathbb{C}) \simeq \underline{\omega}^{2*}_X]} & {[\mathcal{O}_X/\mathrm{GP}(1)(\mathbb{C}) \simeq \mathrm{Conn}(X)]}
\end{tikzcd}
batch 5 · p. 82 — read it beside the facsimile6 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small]
Y' \arrow[d] & \\
K(\pi_1(X),2) \arrow[r, "k_1'"] & K(\pi_2(X),4)
\end{tikzcd}
batch 5 · p. 82 — read it beside the facsimile7 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
X = \Omega Y \arrow[d] & \\
K(\pi_1(X),1) \arrow[r, "k_1"] & K(\pi_2(X),3)
\end{tikzcd}
batch 5 · p. 93 — read it beside the facsimile8 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & \underline{\omega} \arrow[r] \arrow[d, "\mathrm{id}"] & \mathcal{L} \arrow[r] \arrow[d] & \underline{O}_X \arrow[r] \arrow[d, "-2\,\mathrm{id}"] & 0\\
0 \arrow[r] & \underline{\omega} \arrow[r] & \mathrm{P}^1_{X/S}(\omega)\otimes\omega^{-1} \arrow[r] & \underline{O}_X \arrow[r] & 0
\end{tikzcd}
batch 5 · p. 93 — read it beside the facsimile9 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
0 \arrow[r] & \underline{\omega} \arrow[r] \arrow[d, "-\frac{1}{2}\mathrm{id}_{\underline{\omega}}"] & \mathcal{L} \arrow[r] \arrow[d] & \underline{O}_X \arrow[r] \arrow[d, "\mathrm{id}_{\underline{O}_X}"] & 0\\
0 \arrow[r] & \underline{\omega} \arrow[r] & \mathrm{P}^1_{X/S}(\omega)\otimes\omega^{-1} \arrow[r] & \underline{O}_X \arrow[r] & 0
\end{tikzcd}
batch 6 · p. 117 — read it beside the facsimile10 / 32
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\begin{tikzcd}
\mathcal{P}_{\mathcal{X}'} \arrow[r, "c^{\mathcal{X}'}"] \arrow[d, "\wr"'] & \mathrm{Conn}_{\mathcal{X}'} \arrow[d, "\wr"] \\
f^{-1}(\mathcal{P}_{\mathcal{X}}) \arrow[r, "c^{\mathcal{X}}"'] & f^{-1}(\mathrm{Conn}_{\mathcal{X}})
\end{tikzcd}
batch 6 · p. 119 — read it beside the facsimile11 / 32
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\begin{tikzcd}
U \arrow[r, "\text{ét}"] \arrow[d] & X \arrow[d, "f"] \\
V \arrow[r, "\text{ét}"'] & S
\end{tikzcd}
batch 7 · p. 126 — read it beside the facsimile12 / 32
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\begin{tikzcd}
\widetilde{X} \arrow[r] \arrow[dr] & X \times X \arrow[d, "\mathrm{pr}_2"] \\
 & X
\end{tikzcd}
batch 7 · p. 127 — read it beside the facsimile13 / 32
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\begin{tikzcd}
\widetilde{X} \subset \mathbb{P}_X \arrow[d] \\
X
\end{tikzcd}
batch 8 · p. 143 — read it beside the facsimile14 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
S \arrow[d] & S' \arrow[l] \arrow[d] \\
T & T' \arrow[l]
\end{tikzcd}
batch 9 · p. 161 — read it beside the facsimile15 / 32
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\begin{tikzcd}
X \arrow[d, "f"] \\
S \arrow[d, "g"] \\
T
\end{tikzcd}
batch 9 · p. 169 — read it beside the facsimile16 / 32
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\begin{tikzcd}[column sep=small]
B \arrow[r] \arrow[rr, bend left=30, "F_{A}"] & \widetilde{B} = B \otimes_{A} \widetilde{A} \arrow[r] \arrow[d] & B \\
A \arrow[u] \arrow[r] & \widetilde{A} = (A, F_{A}) \arrow[ur, no head] &
\end{tikzcd}
batch 9 · p. 170 — read it beside the facsimile17 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
B \arrow[r] & \widetilde{B} \arrow[r] & B \\
B_{i} \arrow[u] \arrow[r, no head] & \widetilde{B}_{i} \arrow[u, no head] \arrow[r, no head] & B_{i} \arrow[u] \\
\Lambda \arrow[u] \arrow[r, no head] & \widetilde{\Lambda} \arrow[u, no head] \arrow[r, no head] & \Lambda \arrow[u] \\
A \arrow[u, no head] \arrow[r, no head] & \widetilde{A} \arrow[u, no head] &
\end{tikzcd}
batch 9 · p. 172 — read it beside the facsimile18 / 32
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\begin{tikzcd}
X \arrow[d] & X^{(p/S)} \arrow[l] \arrow[d, "{f = f_{X/S}}"] & X \arrow[l] \arrow[ll, bend right=30, "F_{X}"'] \arrow[dl, no head] \\
S & S \arrow[l, "F_{S}"] &
\end{tikzcd}
batch 9 · p. 173 — read it beside the facsimile19 / 32
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\begin{tikzcd}
X \arrow[d] & X' \arrow[l] \arrow[d] \\
S & S' \arrow[l]
\end{tikzcd}
batch 9 · p. 173 — read it beside the facsimile20 / 32
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\begin{tikzcd}[column sep=small, row sep=small]
 & X \arrow[dl] \arrow[ddl] & & X' \arrow[ll] \arrow[dl] \arrow[ddl] \\
X^{(p/S)} \arrow[d] & & X'^{(p/S')} \arrow[ll] \arrow[d] & \\
S & & S' \arrow[ll] &
\end{tikzcd}
batch 9 · p. 179 — read it beside the facsimile21 / 32
LaTeX source
\begin{tikzcd}
X \arrow[d, "f"'] \arrow[r, "F_{X}"] & X \arrow[d] \\
S \arrow[r, "F_{S}"] & S
\end{tikzcd}
batch 10 · p. 185 — read it beside the facsimile22 / 32
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
 & & 0 \arrow[d] & & \\
 & & R^1f_*(\mathcal{O}_{X/S}) & & \\
0 \arrow[r] & R^1f_*(\mathcal{O}_X)^{(p)} \arrow[r, "\alpha"] \arrow[ur, "F\ (\mathrm{Frob})"] & \mathcal{H}^1_{DR} \arrow[r, "\beta"] \arrow[u] & f_*(\Omega^1_{X/S})^{(p)} \arrow[r] & 0\\
 & & f_*(\Omega^1_{X/S}) \arrow[u] \arrow[ur, "C\ (\text{op. de Cartier})"'] & & \\
 & & 0 \arrow[u] & &
\end{tikzcd}
batch 10 · p. 186 — read it beside the facsimile23 / 32
LaTeX source
\begin{tikzcd}
X \arrow[r, "f^{(i)}"] & X^{[i]}\\
T_0 \arrow[u, "\varphi_0"] \arrow[r, "i"'] & T \arrow[u, "u"']
\end{tikzcd}
batch 10 · p. 186 — read it beside the facsimile24 / 32
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\begin{tikzcd}
X \arrow[r, "\delta"] & Z\\
T_0 \arrow[u, "\varphi_0"] \arrow[r, "i"'] & T \arrow[u, "u"']
\end{tikzcd}
batch 10 · p. 200 — read it beside the facsimile25 / 32
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\begin{tikzcd}
X \arrow[r, "F^{(\nu)}_{X/S}"] & X^{(p^\nu/S)} \arrow[r, "\text{can}"] \arrow[d, dashed] & X \arrow[d, dashed]\\
 & S \arrow[r, "F_S^{\nu}"'] & S
\end{tikzcd}
batch 11 · p. 207 — read it beside the facsimile26 / 32
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
(3.1) \quad 0 \arrow[r] & P^{cip}_{X/S} \arrow[r] & P^{ci}_{X/S} \arrow[r, "K"] & f^{\mathrm{fl}}_{*}(\Omega^{1\otimes p}_{X/S})
\end{tikzcd}
batch 11 · p. 214 — read it beside the facsimile27 / 32
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
0 \arrow[r] & {}'E^{20}_{2} \arrow[r] \arrow[d, "\simeq"] & \mathbb{H}^{2}(X, {}^{1}\Omega^{*}_{X/S}) \arrow[r] \arrow[d] & {}'E^{11}_{2} \arrow[r] \arrow[d] & {}'E^{30}_{2} \arrow[d, "\simeq"] \\
{} & H^{2}(H^{0}(X, \Omega^{*})) \arrow[r] & H^{2}_{\mathrm{DR}}(X) \arrow[r] & H^{1}(H^{1}(X, \Omega^{*})) \arrow[r] & E^{30}_{2}
\end{tikzcd}
batch 11 · p. 216 — read it beside the facsimile28 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
X \arrow[r, "F_{X/S}"] \arrow[d] & X^{p/S} \arrow[r] \arrow[dl] & X \arrow[dl] \\
S \arrow[r] & S &
\end{tikzcd}
batch 11 · p. 217 — read it beside the facsimile29 / 32
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\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
S & U_{0} \arrow[l, "\psi_{0}"'] \arrow[d, "j"'] & T_{0} \arrow[l, dashed, "u_{0}"'] \arrow[d, "i"] \arrow[ll, bend right=30, "\varphi_{0}"'] \\
{} & U & T \arrow[l, dashed, "u"']
\end{tikzcd}
batch 11 · p. 220 — read it beside the facsimile30 / 32
LaTeX source
\begin{tikzcd}[column sep=small, row sep=small, nodes={font=\scriptsize}]
\underline{\Gamma}(G) \arrow[r, "g^{-1}dg"] \arrow[rr, bend right=30, "\tilde{\tilde{D}}"'] & Z^{1}C_{S/R}(\underline{g}) \arrow[r, "\tilde{D}"] & \mathcal{M}
\end{tikzcd}
batch 12 · p. 233 — read it beside the facsimile31 / 32
LaTeX source
\begin{tikzcd}
  X_0 \arrow[d, "f_0"] & X' \arrow[d, "f'"] & X'' \arrow[dl, "{f''}"] \\
  T_0 \arrow[r, hook] & T &
\end{tikzcd}
batch 12 · p. 235 — read it beside the facsimile32 / 32
LaTeX source
\begin{tikzcd}[column sep=small]
  \cdots \arrow[r] & 0 \arrow[r] & \mathscr{G} \arrow[r, "i"] \arrow[d, "\varphi^{-1}"]
    & \overline{\mathscr{G}} \arrow[r] \arrow[d, "\varphi^0"] & 0 \arrow[r] & \cdots \\
  & \mathscr{H}^{-2} \arrow[r] & \mathscr{H}^{-1} \arrow[r, "\delta"]
    & \mathscr{H}^0 \arrow[r] & \mathscr{H}^1 \arrow[r] & \cdots
\end{tikzcd}