\[ \underline{\mathrm{Pic}}_{X/S} = P \]

\[
(Rf_P)_{*}(\mathcal{L}_g) \in D_{\mathrm{parf}}(P)
\qquad [\, f_P : X \times_S P \to P \,].
\]

\[
{\det}^{*}(Rf_P)_{*}(\mathcal{L}_g) = M_g
\]

\[
\mathcal{L}_{g'} \simeq \mathcal{L}_g \otimes_{\mathcal{O}_P} g'^{*}_P(\mathcal{L}_g)^{-1}
\]

\[
(Rf_P)_{*}(\mathcal{L}_{g'}) \simeq (Rf_P)_{*}(\mathcal{L}_g)\, g'^{*}_P(\mathcal{L}_g)^{-1}
\]

\[
(*) \qquad M_{g'} \simeq M_g \, g'^{*}_P(\mathcal{L}_g)^{-\chi} = M_g \,(N_g N_{g'}^{-1})^{\chi}
\]

\[
\underline{\mathrm{NS}}_{P^{\delta}/S} \simeq \underline{\mathrm{NS}}_{P^{0}/S} .
\]

\[
\varphi(\delta) \in \underline{\mathrm{NS}}_{P^{0}/S}(S)
\]

\[
\boxed{\varphi : \underline{\mathrm{NS}}_{X/S} \longrightarrow \underline{\mathrm{NS}}_{B/S}}
\qquad (B = P^{0} = \underline{\mathrm{Pic}}^{0}_{X/S}).
\]

\[
\underline{\mathrm{Pic}}^{0}_{P^{\delta}/S} \simeq \underline{\mathrm{Pic}}^{0}_{B/S} ,
\qquad
\underline{\mathrm{NS}}_{P^{\delta}/S} \simeq \underline{\mathrm{NS}}_{B/S} \;].
\]

\[
A = \underline{\mathrm{Pic}}^{0}_{B/S} \quad (= \underline{\mathrm{Alb}}^{0}_{X/S})
\]

\[
\tilde{\lambda} : B \longrightarrow A
\]

\[
\underline{\mathrm{Pic}}^{\lambda}_{P^{\delta}/S} \simeq
\underline{\mathrm{Pic}}^{\lambda}_{B/S} \overset{A}{\times}
\underbrace{\bigl(P^{\delta} \overset{B}{\times} A\bigr)}
\]

\[ \lambda = \varphi(\delta) , \]

\[
\ell(g) \in \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S} \overset{A}{\times}
\bigl(P^{\delta} \overset{B}{\times} (A, \tilde{\lambda})\bigr)
\]

\[
X \xrightarrow{\ \ell_{\delta}\ } \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S}
\overset{A}{\times} \bigl(P^{\delta} \overset{B}{\times} (A, \widetilde{\varphi(\delta)})\bigr) ,
\]

\[
\underline{\mathrm{Alb}}^{1}_{X/S} \struck{(X/S)} \longrightarrow
\underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S} \overset{A}{\times}
\bigl(P^{\delta} \overset{B}{\times} (A, \widetilde{\varphi(\delta)})\bigr)
\]

\[
u_{\delta} : A \longrightarrow A
\]

\[
u_{\delta} = -\chi(\delta)\, \mathrm{id}_A ,
\]

\[
\underline{\mathrm{Alb}}^{-\chi(\delta)}_{X/S} \longrightarrow
\underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S} \overset{A}{\times}
\bigl(P^{\delta} \overset{B}{\times} (A, \widetilde{\varphi(\delta)})\bigr)
\]

\[
\boxed{L(\delta) \in \Gamma\Bigl(S, \underline{\mathrm{Alb}}^{\chi(\delta)}_{X/S}
\overset{A}{\times} \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S}
\overset{A}{\times} \bigl(P^{\delta} \overset{B}{\times} (A, \widetilde{\varphi(\delta)})\bigr)\Bigr)}
\]

\[
\underline{\mathrm{Alb}}^{\chi(\delta)}_{X/S} \overset{A}{\times}
\underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S} \overset{A}{\times}
\bigl(P^{\delta} \overset{B}{\times} (A, \widetilde{\varphi(\delta)})\bigr)
\]

\[ P^{\delta} \simeq B \]

\[
{\det}^{*}(R\,\mathrm{pr}_{2*})\bigl(\mathcal{L}^{0} \otimes \mathrm{pr}_1^{*}(L)\bigr)
\]

\[
\varphi(\delta) = \text{termes de degré 1 dans }
\mathrm{pr}_{2*}\bigl(\exp(D + \mathrm{pr}_1^{*}\Delta)\, \mathrm{pr}_1^{*}(\mathrm{Todd}_{X/k})\bigr)
\]

\[
= \mathrm{pr}_{2*}\bigl(\exp D \cdot \mathrm{pr}_1^{*}(T_{X/k} \exp \Delta)\bigr)
= [\exp D]\,(T_{X/k} \exp \Delta)
\]

\[
\begin{cases}
D = \mathrm{cl}\,\mathcal{L} \in \mathrm{Gr}^{1}(X \times B) \\
\Delta = \mathrm{cl}\,\Delta \in \mathrm{Gr}^{1}(X) \\
\mathrm{Todd}_{X/k} \in \mathrm{Gr}^{*}(X)
\end{cases}
\]

\[
D \in H^{2}(X \times B) = H^{2}(X) + H^{1}(X) \otimes H^{1}(B) + H^{2}(B)
\]

\[
D \in H^{1}(X) \otimes H^{1}(B)
\]

\[
D = \sum e_i \otimes e'_i
\]

\[
\mathrm{Todd}_{X/k} \exp \Delta = \sum_{i=0}^{n} a^{i} \qquad a^{i} \in H^{2i}(X)
\]

\[
\exp D\, \mathrm{pr}_1^{*}(\mathrm{Todd}_{X/k} \exp \Delta) = \sum_{i,j} \mathrm{pr}_1^{*}(a^{i})\, D^{j} / j!
\]

\[
\varphi(\delta) = \tfrac{1}{2}\, \mathrm{pr}_{2*}\bigl(\mathrm{pr}_1^{*}(a_{n-1})\, D^{2}\bigr)
\]

\[
\varphi(\delta) = \tfrac{1}{2}\, \mathrm{pr}_{2*}(D^{2})
\]

\[
D = \sum (e_i \otimes f_i + f_i \otimes e_i)
\]

\[
D^{2} = \sum_{i,j} \underbrace{(e_i \otimes f_i + f_i \otimes e_i)(e_j \otimes f_j + f_j \otimes e_j)}
\]

\[
- [\, e_i f_j \otimes f_i e_j + f_i e_j \otimes e_i f_j \,]
\]

\[
= -\sum_i (\eta \otimes f_i e_i - \eta \otimes e_i f_i) = 2\eta \sum e_i f_i
\]

\[
\tfrac{1}{2} D^{2} = \eta \otimes \sum e_i f_i
\]

\[
\varphi(\delta) = \sum e_i \wedge f_i \in H^{2}(B)
\]

\[
\chi(\delta) = 1 - g + d
\]

\[
c : B \longrightarrow A
\]

\[
\underline{\mathrm{Pic}}^{-1}_{X/S} = P^{-1}
\quad \text{(\emph{attention au signe !})}
\]

\[
P^{\delta} \overset{B}{\times} (A, \underset{\substack{\parallel \\ c}}{\widetilde{\varphi(\delta)}})
\;\text{devient}\; P^{\delta} = P^{d} .
\]

\[
\Phi : \mathrm{chow}(A)_{\mathbb{Q}} \xrightarrow{\ \sim\ } \mathrm{chow}(B)_{\mathbb{Q}}
\]

\[
\varphi(\delta) = \sum \frac{1}{i!}\, \underbrace{\Phi(\Delta)}_{\text{classe d'une courbe}}{}^{*i}
= \frac{1}{(n-1)!}\, \Phi(\Delta)^{*(n-1)} .
\]

\[
\varphi : \underline{\mathrm{NS}}_{A/S} \longrightarrow \underline{\mathrm{NS}}_{B/S}
\]

\[ \tilde{\delta} : A \longrightarrow B \]

\[ \widetilde{\varphi(\delta)} : B \longrightarrow A \]

\[
\boxed{\widetilde{\varphi(\delta)}\, \tilde{\delta} = -\chi(\delta)\, \mathrm{id}_A ,
\qquad \tilde{\delta}\, \widetilde{\varphi(\delta)} = -\chi(\delta)\, \mathrm{id}_B}
\]

\[
\chi(\delta') = \chi(\delta)^{n-1} \qquad \delta' = -\varphi(\delta)
\]

\[
\tilde{\delta}' = \tilde{\delta}^{-1} \chi(\delta) ,
\qquad
\tilde{\delta}''^{-1} = \tilde{\delta}'^{-1} \chi(\delta')
= \tilde{\delta}\, \chi(\delta)^{-1} \chi(\delta') = \tilde{\delta}\, \chi(\delta)^{n-2}
\]

\[
\varphi(\varphi(\delta)) = \delta\, \chi(\delta)^{n-2}
\]

\[
\underline{\mathrm{Pic}}^{-\delta'}_{B/S} \overset{A}{\times}
\bigl(\underline{\mathrm{Pic}}^{\delta}_{A/S} \overset{B}{\times} (A, \tilde{\delta}')\bigr)
\]

\[
(*) \qquad
\boxed{\underline{\mathrm{Pic}}^{\delta'}_{B/S} \simeq
\underline{\mathrm{Pic}}^{\delta}_{A/S} \overset{B}{\times} (A, \tilde{\delta}')}
\]

\[
\underline{\mathrm{Pic}}^{\delta}_{A/S} \simeq \underline{\mathrm{Pic}}^{-\delta}_{A/S}
\]

\[
\underline{\mathrm{Alb}}^{\chi(\delta)}_{X/S} \times
\underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S} \overset{A}{\times}
\bigl(P^{\delta} \overset{B}{\times} (A, \widetilde{\varphi(\delta)})\bigr)
= \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S} \overset{A}{\times}
\bigl(P^{\delta} \overset{B}{\times} (A, \widetilde{\varphi(\delta)})\bigr)
\]

\[
\underline{\mathrm{Pic}}^{\lambda^{n-1}\delta'}_{B/S} \simeq
\bigl(\underline{\mathrm{Pic}}^{\delta'}_{B/S}\bigr)^{(\lambda^{n-1})}
\]

\[
\underline{\mathrm{Pic}}^{-\lambda\delta}_{A/S} \overset{B}{\times} (A, \lambda^{n-1}\tilde{\delta}')
\simeq \bigl(\underline{\mathrm{Pic}}^{-\delta}_{A/S} \overset{B}{\times} (A, \tilde{\delta}')\bigr)^{(\lambda^{n})} .
\]

\[
\bigl(\underline{\mathrm{Pic}}^{\delta'}_{B/S}\bigr)^{\otimes(\lambda^{n} - \lambda^{n-1})}
\simeq \text{torseur trivial (canonique)}
\]

\[
\bigl(\underline{\mathrm{Pic}}^{\delta'}_{B/S}\bigr)^{\otimes 2}
\simeq \text{torseur trivial} \quad \text{(canonique)}
\]

\[
\bigl(\underline{\mathrm{Pic}}^{\delta}_{A/S}\bigr)^{\otimes(2\chi(\delta)^{n-2})}
\simeq \text{torseur trivial (canon !)}
\]

\[
\bigl(\underline{\mathrm{Pic}}^{\delta}_{A/S}\bigr)^{\otimes(\chi(\delta)^{n-2})}
\simeq \text{torseur trivial (canonique)}
\]

\[
\underline{\mathrm{Pic}}^{\delta'}_{B/S} \simeq
\underline{\mathrm{Pic}}^{\delta}_{A/S} \overset{B}{\times} (A, \tilde{\delta}')
\]

\[
\underline{\mathrm{Pic}}^{\delta''}_{A/S} \simeq
\underline{\mathrm{Pic}}^{\delta'}_{B/S} \overset{A}{\times} (B, \tilde{\delta}'')
\simeq \underline{\mathrm{Pic}}^{\delta}_{A/S} \overset{B}{\times} (B, \tilde{\delta}''\tilde{\delta}')
\]

\[
\begin{cases}
\delta'' = \delta\, \chi(\delta)^{n-2} \\
\tilde{\delta}''\tilde{\delta}' = \chi(\delta')\, \mathrm{id}_B = \chi(\delta)^{n-1}\, \mathrm{id}_B
\end{cases}
\]

\[
\underline{\mathrm{Pic}}^{\delta\chi(\delta)^{n-2}}_{B/S} \simeq
\underline{\mathrm{Pic}}^{\delta\chi(\delta)^{n-1}}_{B/S}
\]

\[
\bigl(\underline{\mathrm{Pic}}^{\delta}_{B/S}\bigr)^{\otimes[\chi(\delta)^{n-1} - \chi(\delta)^{n-2}]}
\simeq \text{torseur trivial (isom.\ canonique)}
\]

\[
\bigl(\underline{\mathrm{Pic}}^{\delta}_{B/S}\bigr)^{\otimes \chi(\delta)^{n-2}(\lambda^{n}\chi(\delta) - \lambda^{n-1})}
\simeq \text{torseur trivial}
\]

\[
\bigl(\underline{\mathrm{Pic}}^{\delta}_{B/S}\bigr)^{\otimes 2\chi(\delta)^{n-2}}
\simeq \text{torseur trivial}
\]

\[
\bigl(\underline{\mathrm{Pic}}^{\delta}_{B/S}\bigr)^{\otimes \chi(\delta)^{n-2}}
\simeq \text{torseur trivial si $\chi(\delta)$ pair}
\]

\[
\bigl(\underline{\mathrm{Pic}}^{\delta}_{B/S}\bigr)^{(\chi(\delta)-1)\chi(\delta)^{n-2}}
\simeq \text{torseur trivial}.
\]

\[
{\det}^{*} R\,\mathrm{pr}_{2*}\bigl(\mathrm{pr}_1^{*}(\mathcal{F}^{\bullet}) \otimes \mathcal{L}_g\bigr) = M_g
\]

\[
M_{g'} \simeq M_g\, g'^{*}_{P}(\mathcal{L}_g)^{-\chi_{\mathcal{F}}}
= M_g\, (N_g N_{g'}^{-1})^{\chi_{\mathcal{F}}}
\]

\[
\varphi_{\mathcal{F}^{\bullet}} : \underline{\mathrm{NS}}_{X/S} \longrightarrow \underline{\mathrm{NS}}_{B/S}
\]

\[
X \xrightarrow{\ \ell^{\mathcal{F}^{\bullet}}_{\delta}\ }
\underline{\mathrm{Pic}}^{\varphi_{\mathcal{F}}(\delta)}_{B/S} \overset{A}{\times}
\bigl(P^{\delta} \overset{B}{\times} (A, \widetilde{\varphi_{\mathcal{F}}(\delta)})\bigr)
\]

\[
\underline{\mathrm{Alb}}^{-\chi_{\mathcal{F}}(\delta)}_{X/S} \longrightarrow
\underline{\mathrm{Pic}}^{\varphi_{\mathcal{F}}(\delta)}_{B/S} \times
\bigl(\underline{\mathrm{Pic}}^{\delta}_{X/S} \overset{B}{\times} (A, \widetilde{\varphi_{\mathcal{F}}(\delta)})\bigr)
\]

\[
\boxed{L_{\mathcal{F}}(\delta) \in \Gamma\Bigl(S, \underline{\mathrm{Alb}}^{\chi_{\mathcal{F}}(\delta)}_{X/S}
\overset{A}{\times} \underline{\mathrm{Pic}}^{\varphi_{\mathcal{F}}(\delta)}_{B/S}
\times \bigl(\underline{\mathrm{Pic}}^{\delta}_{X/S} \overset{B}{\times} (A, \widetilde{\varphi_{\mathcal{F}}(\delta)})\bigr)\Bigr)}
\]

\[
\chi(L_{\delta} \otimes \mathcal{F}^{\bullet}) \quad\text{et}\quad
{\det}^{*} R\,\mathrm{pr}_{2*}\bigl(\mathcal{L} \otimes \mathrm{pr}_1^{*}(L_{\delta} \otimes \mathcal{F})\bigr)
\]

\[
\chi_{\mathcal{F}}(\delta) = \chi(L_{\delta} \otimes \mathcal{F}^{\bullet})
= \pi\bigl(\mathrm{ch}(L_{\delta} \otimes \mathcal{F})\, \mathrm{Todd}(X)\bigr)
\]

\[
\varphi_{\mathcal{F}}(\delta) = \text{terme de degré 1 dans }
(\exp D) . [\mathrm{ch}_X(L_{\delta} \otimes \mathcal{F})\, \mathrm{Todd}(X)]
\]

\[
\mathrm{ch}_X(L_{\delta} \otimes \mathcal{F}) =
(1 + \underset{\substack{\mid \\ c(L_{\delta})}}{\delta})
(\underset{\substack{\parallel \\ \mathrm{rang}\,\mathcal{F}^{\bullet}}}{\rho} + \underset{\substack{\parallel \\ \det'(\mathcal{F})}}{\gamma})
= 1 + (\delta + \gamma)
\]

\[
\mathrm{Todd}(X) = 1 + \underset{\text{degré } 1-g}{\tfrac{1}{2}K}
\]

\[
\mathrm{ch}_X(L_{\delta} \otimes \mathcal{F}) \otimes \mathrm{Todd}(X) = 1 + \tau ,
\qquad \text{avec } \deg \tau = 1 - g + d + c
\]

\[
\boxed{\chi_{\mathcal{F}}(\delta) = 1 - g + d + c}
\]

\[
\boxed{\varphi_{\mathcal{F}}(\delta) = \varphi(\delta) = \tfrac{1}{2}\, \mathrm{pr}_{2*}(D^{2})
= \text{polarisation canonique de } B}
\]

\[
\underline{\mathrm{Alb}}^{\chi_{\mathcal{F}}(\delta) = \chi(\delta) + c}_{X/S}
\overset{A}{\times} \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S}
\overset{A}{\times} \bigl(\underline{\mathrm{Pic}}^{\delta}_{X/S} \overset{B}{\times} (B, \widetilde{\varphi(\delta)})\bigr)
\]

\[
\chi_{\mathcal{F}}(\delta) = \pi\bigl(\mathrm{ch}(L_{\delta}\mathcal{F})\bigr) = \deg a_n
\]

\[
\mathrm{ch}(L_{\delta}\mathcal{F}) = \sum_{1}^{n} a_i \qquad a_i \in \mathrm{Chow}(A)
\]

\[
\varphi_{\mathcal{F}}(\delta) = (\exp D) . (a_{n-1}) = \mathrm{pr}_{2*}\bigl(\tfrac{1}{2} D^{2} a_{n-1}\bigr) .
\]

\[
\boxed{\ell' : X \times \underline{\mathrm{Pic}}_{X/S} \longrightarrow \underline{\mathrm{Pic}}_{B/S}}
\quad \text{(additif en $\underline{\mathrm{Pic}}_{X/S}$)} .
\]

\[
{\det}^{*}\bigl(\mathrm{pr}_{2*}(\mathrm{pr}_1^{*}(L) \otimes W)\bigr)
\]

\[
L(\lambda, g') = L(\lambda, g)\, g'^{*}(L(\lambda, g))^{-1} =
\]

\[
W(g') = W(g)\, g'^{*}_{B}(W(g))^{-1}
\]

\[
\mathrm{pr}_1^{*}(L(\lambda, g'))\, W(g') = \mathrm{pr}_1^{*}(L(\lambda, g))\, W(g)\, \mathcal{D}
\qquad
\mathcal{D} = g'^{*}_{B}(L(\lambda, g))^{-1} \otimes_{\mathcal{O}_X} g'^{*}_{B}(W(g))^{-1}
\]

\[
\ell_{g'}(\lambda) = \ell_g(\lambda) + \bigl\lbrace \text{classe dans } \underline{\mathrm{Pic}}_{B/S}
\text{ de } g'^{*}_{B}(W(g))^{-\chi(\lambda)} = (N_{g'} N_g^{-1})^{-\chi(\lambda)}
\]

\[
(*) \qquad = \ell_g(\lambda) + (\psi_X(g') - \psi_X(g))(-\chi(\lambda))
\]

\[
\varphi : \underline{\mathrm{NS}}_{X/S} \longrightarrow \underline{\mathrm{NS}}_{B/S}
\]

\[
\boxed{\ell_{\delta} : X \times \underline{\mathrm{Pic}}^{\delta}_{X/S} \longrightarrow \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S}}
\]

\[
(**) \qquad \ell_g(\lambda + \beta) = \ell_g(\lambda) + \widetilde{\varphi(\delta)}(\beta)
\qquad \bigl(\lambda \in \underline{\mathrm{Pic}}^{\delta}_{X/S}(S)\bigr)
\]

\[
{\det} \mathfrak{F}_{\mathcal{L}}(L \otimes L_{\beta}) \simeq {\det} \mathfrak{F}_{\mathcal{L}}(L) \otimes
\underbrace{L'_{-\varphi(\delta)(\beta)}}_{\text{faisceau sur $B$ associé à $-\varphi(\delta)(\beta)$}} .
\]

\[
\mathfrak{F}_{\mathcal{L}}(L \otimes L_{\beta}) = \mathfrak{F}_{\mathcal{L}}(L) * \varepsilon_{-\beta}
\]

\[
{\det} \mathfrak{F}_{\mathcal{L}}(L \otimes L_{\beta}) \simeq {\det} \mathfrak{F}_{\mathcal{L}}(L) * \varepsilon_{-\beta}
\]

\[
X \longrightarrow \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S} \overset{A}{\times}
\underline{\mathrm{Pic}}^{\delta}_{X/S} \overset{B}{\times} (A, \widetilde{\varphi(\delta)})
\]

\[
B \xrightarrow{\ i\ } \underline{\mathrm{Pic}}_{X/S} ,
\]

\[
X \xrightarrow{\ \uncertain{\tau}\ } Q
\]

\[
X \times \underline{\mathrm{Pic}}_{X/S} \longrightarrow \underline{\mathrm{Pic}}_{B/S}
\]

\[
X \times \underline{\mathrm{Pic}}^{\delta}_{X/S} \longrightarrow \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S}
\]

\[
X \longrightarrow \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S} \overset{A}{\times}
\bigl(P \overset{B}{\times} (A, \widetilde{\varphi(\delta)})\bigr)
\]

\[
Q^{-\chi(\delta)} \longrightarrow \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S} \overset{A}{\times}
\bigl(P \overset{B}{\times} (A, \widetilde{\varphi(\delta)})\bigr)
\]

\[
Q^{\chi(\delta)} \overset{A}{\times} \underline{\mathrm{Pic}}^{\varphi(\delta)}_{B/S}
\overset{A}{\times} \bigl(P \overset{B}{\times} (A, \widetilde{\varphi(\delta)})\bigr)
\]

\[
  \psi : X \longrightarrow \underline{\mathrm{Alb}}^{1}_{X/S} ,
\]

\[
  \delta : X \times_S X \longrightarrow A \qquad \delta(x,y) = \psi(y) - \psi(x)
\]

\[
  \delta(x,x) = 0 .
\]

\[
  \delta(x,y) = N_y N_x^{-1} \quad \text{dans } \underline{\mathrm{Pic}}^{0}_{B/S}(S)
\]

\[
  X \xrightarrow{\ \psi\ } \underline{\mathrm{Pic}}^{1}_{X/S} ,
\]

\[
  c : B \xrightarrow{\ \sim\ } A = \underline{\mathrm{Pic}}^{0}_{B/S} ,
\]

\[
  c(\psi(y) - \psi(x)) = c(\mathrm{cl}(y(S)) - \mathrm{cl}(x(S))) = N_y N_x^{-1}
\]

\[
  u(t) = \text{classe de } t.\Theta - \Theta .
\]

\[
  u((y) - (x)) = \text{classe de } \ldots
\]

\[
  P \xrightarrow{\ \sigma_P\ } P^{-1} ,
\]

\[
  K : P \xleftarrow{\ \sim\ } P^{-1}
\]

\[
  K \in \Gamma(S, P^{\overset{A}{\otimes} 2}) .
\]