Proof. The stated map $i_*\colon H^1(Z,M)\to H^1(Z,M\otimes _{O_Z} \mathcal {E} nd(F))$ is induced by the natural morphism of $O_Z$ -modules

$$\begin{align*}i:M\to \mathcal{H} om_{O_Z}(F,M\otimes_{O_Z}F),\quad i(m)(f)=m\otimes f.\end{align*}$$

There is a canonical evaluation morphism

$$\begin{align*}\mathrm{ev}:\mathcal{H} om_{O_Z}(F,M\otimes_{O_Z}F)\otimes F\to M\otimes_{O_Z}F,\quad \phi\otimes f\mapsto \phi(f).\end{align*}$$

By construction of i, the left part of the diagram

is commutative. One has

$$\begin{align*}\mathrm{ev}\circ (i\otimes {\mathrm{Id}}_F)(m\otimes f)=\mathrm{ev}(i(m)\otimes f)=i(m)(f)=m\otimes f,\end{align*}$$

so $\mathrm {ev}\circ (i\otimes {\mathrm {Id}}_F)\colon M\otimes _{O_Z}F\to M\otimes _{O_Z}F$ is the identity. Therefore, the triangle on the right of the diagram is also commutative.

By definition, F admits an $\alpha $ -splitting if and only if the exact sequence (3) splits. The extension class of (3) is $\alpha \otimes F\in \mathrm {Ext}^1(F,M\otimes F)$ . By the five-term exact sequence associated with the spectral sequence,

$$\begin{align*}E_2^{i,j}=\mathrm{Ext}^i(O_Z,\mathcal{E} xt^j(F,M\otimes_{O_Z}F))\Rightarrow \mathrm{Ext}^{i+j}(F,M\otimes_{O_Z}F),\end{align*}$$

the composition of the two morphisms on the top of the diagram is injective. This injective map sends $i_*(\alpha )$ to $\alpha \otimes F$ , so $\alpha \otimes F=0$ is equivalent to $i_*(\alpha )=0$ .

For two $\alpha $ -splittings $\psi $ , $\psi ' $ on F, their difference $\psi '-\psi \colon F\to \mathcal {E}_{\alpha }\otimes _{O_Z}F$ is an $O_Z$ -linear morphism. Because of

$$\begin{align*}(\mu_{\alpha}\otimes{\mathrm{Id}}_F)\psi=(\mu_{\alpha}\otimes{\mathrm{Id}}_F)\psi'={\mathrm{Id}}_F,\end{align*}$$

$\psi '-\psi $ factors through $\ker (\mu _{\alpha }\otimes {\mathrm {Id}}_F)=M\otimes _{O_Z}F$ , hence $\psi '-\psi \in \mathrm {Hom}_{O_Z}(F,M\otimes _{O_Z}F)$ . For every $O_Z$ -linear morphism $\phi \colon F\to M\otimes _{O_Z}F$ , the sum $\psi +\phi\colon F\to \mathcal{E}_{\alpha}\otimes_{O_Z} F $ is an $\alpha $ -splitting on F. Therefore, the action of $\mathrm {Hom}_{O_Z}(F,M\otimes _{O_Z}F)$ on the set of $\alpha $ -splittings on F is simple transitive.

Proof. Locally, the vector bundle $\mathcal {E}_{\alpha }$ has a (holomorphic) frame $\{e_1,\dots ,e_r\}$ . For a local section ${f\in F}$ , write $\psi (f)=\sum _{i=1}^re_i\otimes f_i$ , where the $f_i$ are local sections of F. For every $1\le i\le r$ , write $\psi (f_i)=\sum _{j=1}^re_j\otimes f_j^{(i)}$ , where the $f_j^{(i)}$ are local sections of F. As $\psi :F\to \mathcal {E}_{\alpha }\otimes F$ is a section to $\mu _{\alpha }\otimes {\mathrm {Id}}_F:\mathcal {E}_{\alpha }\otimes F\to F$ , one has

(6) $$ \begin{align} f & =(\mu_{\alpha}\otimes{\mathrm{Id}}_F) \psi(f)=\sum_{i=1}^r\mu_{\alpha}(e_i)f_i; \end{align} $$

(7) $$ \begin{align} \hspace{7pt}f_i & =(\mu_{\alpha}\otimes{\mathrm{Id}}_F)\psi(f_i)=\sum_{j=1}^r\mu_{\alpha}(e_j) f_j^{(i)}. \end{align} $$

From (6), one has

(8) $$ \begin{align}\psi(f)=\sum_{i=1}^r\mu_{\alpha}(e_i)\psi(f_i).\end{align} $$

By construction, one has $\psi ^1\psi (f)=\sum _{i,j=1}^r(e_i\wedge e_j)\otimes f_j^{(i)}$ . Then

$$ \begin{align*} &(\omega_{\alpha}\otimes {\mathrm{Id}}_F)\psi^1\psi(f)=\sum_{i,j=1}^r\left(\mu_{\alpha}(e_i)e_j-\mu_{\alpha}(e_j)e_i\right)\otimes f_j^{(i)}\\ =&\sum_{i=1}^r\mu_{\alpha}(e_i)\sum_{j=1}^r(e_j\otimes f_j^{(i)})-\sum_{i=1}^re_i\otimes \left(\sum_{j=1}^r\mu_{\alpha}(e_j)f_j^{(i)}\right)\\ \stackrel{(a)}{=} &\sum_{i=1}^r\mu_{\alpha}(e_i)\psi(f_i)-\sum_{i=1}^re_i\otimes f_i\\ \stackrel{(b)}{=} &\psi(f)-\psi(f)=0, \end{align*} $$

where (a) and (b) use (7) and (8), respectively.

Proof. Let $\mathrm {for}_{X\times Y}:\operatorname {\mathrm {Mod}}(O_{X\times Y})_{0-\mathrm {cxn}}\to \operatorname {\mathrm {Mod}}(O_{X\times Y})$ be the forgetful functor. All the functors (13), (14),

$$\begin{align*}p_X^*:\operatorname{\mathrm{Mod}}(O_X)\to \operatorname{\mathrm{Mod}}(O_{X\times Y}), \quad{\mathcal{P}}\otimes_{O_{X\times Y}}(-):\operatorname{\mathrm{Mod}}(O_{X\times Y})\to \operatorname{\mathrm{Mod}}(O_{X\times Y}),\end{align*}$$

as well as the forgetful functors involved in the proof are exact.

To prove the first isomorphism, it remains to show that there is an isomorphism of functors

$$\begin{align*}\mathrm{for}_YRp_{Y*}^{(\mathrm{cxn})}\cong Rp_{Y*}\mathrm{for}_{X\times Y}\colon D(\operatorname{\mathrm{Mod}}(O_{X\times Y})_{0-\mathrm{cxn}})\to D(O_Y).\end{align*}$$

Since $\mathrm {for}_Y:D(\operatorname {\mathrm {Mod}}(O_Y)_{\mathrm {cxn}}) \to D(O_Y)$ is exact, $\mathrm {for}_YRp_{Y*}^{(\mathrm {cxn})}:D(\operatorname {\mathrm {Mod}}(O_{X\times Y})_{0-\mathrm {cxn}})\to D(O_Y)$ is the right derived functor of

$$\begin{align*}\mathrm{for}_Y\circ p^{(\mathrm{cxn})}_{Y*}:\operatorname{\mathrm{Mod}}(O_{X\times Y})_{0-\mathrm{cxn}}\to \operatorname{\mathrm{Mod}}(O_Y).\end{align*}$$

From Remark 2.10, [45, Tags 0096 and 08BJ], $\mathrm {for}_{X\times Y}:\operatorname {\mathrm {Mod}}(O_{X\times Y})_{0-\mathrm {cxn}}\to \operatorname {\mathrm {Mod}}(O_{X\times Y})$ preserves K-injective complexes. Therefore, $Rp_{Y*}\mathrm {for}_{X\times Y}:D(\operatorname {\mathrm {Mod}}(O_{X\times Y})_{0-\mathrm {cxn}})\to D(O_Y)$ is the right derived functor of

$$\begin{align*}p_{Y*}\mathrm{for}_{X\times Y}:\operatorname{\mathrm{Mod}}(O_{X\times Y})_{0-\mathrm{cxn}}\to \operatorname{\mathrm{Mod}}(O_Y).\end{align*}$$

Since there is an isomorphism of functors

$$\begin{align*}\mathrm{for}_Y\circ p^{(\mathrm{cxn})}_{Y*}\cong p_{Y*}\circ\mathrm{for}_{X\times Y}\colon \operatorname{\mathrm{Mod}}(O_{X\times Y})_{0-\mathrm{cxn}}\to \operatorname{\mathrm{Mod}}(O_Y),\end{align*}$$

one obtains the first stated isomorphism of functors. Together with [Reference Liu25, Corollary 3.2.11], it shows that ${\mathfrak {F}}'$ preserves O-goodness. The other half about ${\mathfrak {F}}$ is similar.

Proof.

1. (i) The identification $Rp_{X*}{\mathcal {P}}\cong T^{-g}\mathbb {C}_{(0)}$ in $D^b(O_X)$ from [Reference Kempf21, Theorem 3.15] can be lifted to an isomorphism in $D^b(\operatorname {\mathrm {Mod}}(O_X)_{-1-\mathrm {sp}})$ . As stated in the last sentence of the proof of [Reference Viguier46, Proposition 2.1.21], a morphism of modules with splittings (or connections) is an isomorphism whenever the underlying morphism of O-modules is so. By [Reference Liu25, Theorem 1.3.2], applied to the Cartesian square

   
   the natural morphism $\epsilon _X^*Rp_{X*}{\mathcal {P}}\to Rp_{12*}(\epsilon _X\times 1_Y)^*{\mathcal {P}}$ in $D^b(\operatorname {\mathrm {Mod}}(O_{X\times X})_{(1,-1)-\mathrm {sp}})$ is mapped to an isomorphism in $D^b_{\mathrm {coh}}(O_{X\times X})$ , so itself is an isomorphism.

2. (ii) The proof is similar to that of Part (i).

3. (iii) The proof is similar to that of [Reference Liu25, Lemma 5.1.1].

Proof of Theorem 3.3

We shall use the projection formula and the flat base change theorem for modules equipped with connections or splittings. The comparison morphisms are constructed from the adjunction between the derived inverse image and the derived direct image in Remark 2.12. Similar to Proposition 3.2, one can prove that these comparison morphisms are compatible with the comparison morphisms of the underlying O-modules. Thus, ignoring the connections and splittings, one needs to prove that the comparison morphisms of O-modules are isomorphisms. (This type of arguments can also be found in the proof of [Reference Viguier46, Proposition 2.1.21 and Theorem 2.1.33].)

Let $p_i$ ( $i=1,2$ ) be the projections of $X\times X\to X$ . Let $p_{ij}$ be the projections of $X\times X\times Y$ . By the universal property of the Poincaré bundle, there is an isomorphism

(22) $$ \begin{align} p_{13}^*{\mathcal{P}}^{-1}\otimes p_{23}^*{\mathcal{P}}\cong (\epsilon_X\times {\mathrm{Id}}_Y)^*{\mathcal{P}} \end{align} $$

in $\operatorname {\mathrm {Mod}}(X\times X\times Y)_{(1,-1)-\mathrm {cxn}}$ . Now we adapt the proof of [Reference Rothstein37, Theorem 3.2] to the analytic case and add more details. There are isomorphisms

$$ \begin{align*} {\mathfrak{F}}{{\mathfrak{F}}}'& =Rp_{X*}({\mathcal{P}}^{-1}\otimes p_Y^*Rp_{Y*}({\mathcal{P}}\otimes p_X^*-))\\ &\stackrel{(\textrm{a})}{\xrightarrow{\sim}} Rp_{X*}({\mathcal{P}}^{-1}\otimes Rp_{13*}p_{23}^*({\mathcal{P}}\otimes p_X^*-))\\ &\stackrel{(\textrm{b})}{\xrightarrow{\sim}} Rp_{X*}Rp_{13*}(p_{13}^*{\mathcal{P}}^{-1}\otimes p_{23}^*({\mathcal{P}}\otimes p_X^*-))\\ &\cong Rp_{1*}Rp_{12*}((p_{13}^*{\mathcal{P}}^{-1}\otimes p_{23}^*{\mathcal{P}})\otimes p_{23}^*p_X^*-)\\ &\stackrel{(\textrm{c})}{\cong} Rp_{1*}Rp_{12*}((\epsilon_X\times {\mathrm{Id}}_Y)^*{\mathcal{P}}\otimes p_{12}^*p_2^*-)\\ &\stackrel{(\textrm{d})}{\xleftarrow{\sim}} Rp_{1*}(Rp_{12*}(\epsilon_X\times {\mathrm{Id}}_Y)^*{\mathcal{P}}\otimes^L p_2^*- )\\ &\stackrel{(\textrm{e})}{\cong} Rp_{1*}(T^{-g}O_{\Delta_X}\otimes p_2^*-)\stackrel{(f)}{\cong} T^{-g} \end{align*} $$

of functors $D_{O-{\mathrm {good}}}({\operatorname {\mathrm {Mod}}}(O_{X})_{\mathrm {sp}})\to D_{O-{\mathrm {good}}}({\operatorname {\mathrm {Mod}}}(O_{X})_{\mathrm {sp}})$ . Here, (a) uses the goodness of ${\mathcal {P}}\otimes p_X^*(-)$ over $O_{X\times Y}$ and applies the flat base change theorem [Reference Liu25, Theorem 1.3.2] to the Cartesian square

The isomorphisms (b) and (d) rely on the projection formula [Reference Spaltenstein44, Proposition 6.18]. Finally, (c), (e), and (f) are from (22), Lemma 3.4(i), and Lemma 3.4(iii), respectively.

Thus, the second stated isomorphism is proved. Similarly, using Lemma 3.4(ii), one can prove the first stated isomorphism.

Proof. Suppose to the contrary that $\mathrm {Supp}(F)$ is infinite. By [Reference Grauert and Remmert7, p. 76], $\mathrm {Supp}(F)$ is an analytic set in X. Then $\dim \mathrm {Supp}(F)\ge 1$ . Let C be an irreducible component of $\mathrm {Supp}(F)$ of maximal dimension. Write $i:C\to X$ for the inclusion.

By [Reference Liu25, Lemma 6.3.5], there is a connected compact Kähler manifold Z and a morphism $h:Z\to X$ such that $h(Z)=C$ and $F":=F'/T(F')$ is a vector bundle on Z of positive rank r, where $F'=h^*F$ . As X is a complex torus, its Albanese is X itself. The morphism of Albanese tori $\mathrm {alb}(h):\operatorname {\mathrm {Alb}}(Z)\to X$ contains a translate of C in its image. Since $\dim C>0$ , $\mathrm {alb}(h):\operatorname {\mathrm {Alb}}(Z)\to X$ is nonzero. Therefore, the dual morphism of complex tori $h^*:\mathrm {Pic}^0(X)\to \mathrm {Pic}^0(Z)$ is also nonzero. However, we claim that its tangent map at origin $h^*:\mathfrak {g}\to H^1(Z,O_Z)$ is zero.

Let $\mathcal {E}'=h^*\mathcal {E}$ . By [45, Tag 05NJ], because $O_X$ is flat over itself, pulling back (9) to Z and tensoring with $F"$ , one has a short exact sequence of $O_Z$ -modules

(23) $$ \begin{align}0\to \mathfrak{g}^*\otimes_{\mathbb{C}} F"\to \mathcal{E}'\otimes_{O_Z} F"\to F"\to0.\end{align} $$

Since $\mathcal {E}'$ is a vector bundle on Z, one has

$$\begin{align*}\frac{\mathcal{E}'\otimes F'}{T(\mathcal{E}'\otimes F')}=\mathcal{E}'\otimes F".\end{align*}$$

Then a $1$ -splitting $\psi \colon F\to \mathcal {E}\otimes _{O_X} F$ induces a splitting $\psi '\colon F"\to \mathcal {E}'\otimes _{O_Z} F"$ of (23).

Let $\beta :O_Z\to \mathcal {E} nd(F")$ be the natural morphism. By Lemma 2.3, the composition

$$\begin{align*}\mathrm{End}(\mathfrak{g})\overset{{\mathrm{Id}}_{\mathfrak{g}^*}\otimes h^*}{\to}\mathfrak{g}^*\otimes_{\mathbb{C}}H^1(Z,O_Z)\overset{{\mathrm{Id}}_{\mathfrak{g}^*}\otimes H^1(Z,\beta)}{\to} \mathfrak{g}^*\otimes_{\mathbb{C}}H^1(Z,\mathcal{E} nd(F")) \end{align*}$$

sends $1\in \mathrm {End}(\mathfrak {g})$ to $0$ . Therefore, the map $H^1(Z,\beta )\circ h^*:\mathfrak {g}\to H^1(Z,\mathcal {E} nd(F"))$ is zero. The morphism $\tau :\mathcal {E} nd(F")\to O_Z$ which takes trace satisfies $\tau \beta =r\cdot {\mathrm {Id}}_{O_Z}$ . Then $h^*=\frac {1}{r}\tau _*H^1(Z,\beta )h^*=0$ as maps $\mathfrak {g}\to H^1(Z,O_Z)$ . The claim follows. This gives a contradiction.

Proof of Proposition 3.5

By Proposition 3.2, $\mathrm {for}_X:D(\operatorname {\mathrm {Mod}}(O_X)_{\mathrm {sp}})\to D(O_X)$ sends ${\mathfrak {F}}(E,\nabla )$ to an object isomorphic to $\mathcal {F}(E)$ . From [Reference Liu25, Corollary 3.2.11], $\mathcal {F}(E)$ lies in $D^b_{\mathrm {coh}}(O_X)$ . Then by Lemma 3.6, for every integer i, the support of $H^i\mathcal {F}(E)$ is finite. As there exist only finitely many $i\in \mathbb {Z}$ with $H^i\mathcal {F}(E)\neq 0$ , there is an open subset $U\subset X$ with $P_y|_U\cong O_U$ which contains the supports of all $H^i\mathcal {F}(E)$ . By Lemma 7.17 below, there is an isomorphism ${\mathfrak {F}}(T_y^*(E,\nabla ))\cong {\mathfrak {F}}(E,\nabla )\otimes _{O_X} P_y$ in $D(\operatorname {\mathrm {Mod}}(O_X)_{\mathrm {sp}})$ . By the choice of U, ${\mathfrak {F}}(E,\nabla )\otimes _{O_X} P_y$ is isomorphic to ${\mathfrak {F}}(E,\nabla )$ . Then by Theorem 3.3, $T_y^*(E,\nabla )$ is isomorphic to $(E,\nabla )$ in $\operatorname {\mathrm {Mod}}(O_Y)_{\mathrm {cxn}}$ .

Proof. By (24), the graded ring associated with the degree filtration of $\mathcal {A}_X$ is

(27) $$ \begin{align}G\mathcal{A}_X:=\oplus_{m\ge0}\mathcal{A}_X(m)/\mathcal{A}_X(m-1)=\mathrm{Sym}(\mathfrak{g})\otimes_{\mathbb{C}}O_X=O_X[x_1,\dots,x_g].\end{align} $$

Here for each chosen pair $(U,\rho )$ as in Section 4.1, $x_i|_U\in \Gamma (U,\mathcal {A}_X(1)/\mathcal {A}_X(0))\subset \Gamma (U,G\mathcal {A}_X)$ is the image of $x_i^{\rho }\in \Gamma (U,\mathcal {A}_X(1))$ . From [Reference Björk5, Theorem 1.26, p. 460], $\mathcal {A}_X$ is stalkwise Zariskian. The other part follows from [Reference Björk5, Proposition 1.27, p. 460, Theorem 2.7, p. 465]. (See also the proof of [Reference Björk6, Theorem 1.2.5].)

Fact 4.6. The resulting functor $\operatorname {\mathrm {Mod}}(\mathcal {A}_X)\to \operatorname {\mathrm {Mod}}(O_X)_{\mathrm {sp}}$ induces an equivalence of abelian categories from $\operatorname {\mathrm {Mod}}(\mathcal {A}_X)$ to the full subcategory of $\operatorname {\mathrm {Mod}}(O_X)_{\mathrm {sp}}$ of objects $(F,\psi )$ with $[\psi ,\psi ]=0$ .

Proof. The proof is similar to that of Proposition 3.2, as $\mathcal {A}_X$ (resp. $D_Y$ ) is flat over $O_X$ (resp. $O_Y$ ).

Proof. With Proposition 5.3, the proof is similar to that of Theorem 3.3.

Proof. The result follows from Proposition 5.3, Theorem 5.4, and Fact 1.1(i) as in the proof of [Reference Rothstein36, Theorem 6.1] (cf. [Reference Laumon23, Proposition 3.1.2 and Corollary 3.2.4]).

Proof. By [Reference Hotta and Tanisaki13, Example 1.6.4], one has $D_Y\otimes _{O_Y} \mathbb {C}_{(y)}=i_{y+}\mathbb {C}$ . The result follows from Theorem 5.4, Proposition 5.5, Fact 1.3, and [Reference Liu25, Lemma 2.0.9].

Proof. As $\pi \colon X^{\natural }\to X$ is a submersion and $\mathrm {Supp}(F)$ is finite, there exists an open neighborhood $U\subset X$ of $\mathrm {Supp}(F)$ and a morphism of complex manifolds $s:U\to X^{\natural }$ that is a local section to $\pi :X^{\natural }\to X$ . Let $\iota :U\hookrightarrow X$ be the inclusion. Applying $\pi _*$ to the morphism of sheaves of rings $O_{X^{\natural }}\to s_*O_U$ , one gets a morphism $\pi _*O_{X^{\natural }}\to \iota _*O_U$ . As $\mathcal {A}_X$ is an $O_X$ -subalgebra of $\pi _*O_{X^{\natural }}$ , this endows $ \iota _*O_U$ with an $\mathcal {A}_X$ -module structure.Footnote ¹ Since ${\mathrm {Id}}_F\otimes \iota ^{\#}:F\to F\otimes _{O_X}\iota _*O_U$ is an isomorphism of $O_X$ -modules, F also obtains an $\mathcal {A}_X$ -module structure.

Proof of Theorem 5.7

Let E be a homogeneous vector bundle on Y. Set $\hat {E}:=H^g\mathcal {F}(E)\in \operatorname {\mathrm {Mod}}(O_X)$ . According to [Reference Liu25, Proposition 6.3.3] and Fact 1.1, one has $E=H^0\mathcal {F}'(\hat {E})$ and $\mathrm {Supp}(\hat {E})$ is a finite subset of X. By Lemma 5.8, $\hat {E}$ has an $\mathcal {A}_X$ -module structure. By Proposition 5.3, the $O_Y$ -module underlying $H^0\tilde {\mathcal {F}}^{\natural }(\hat {E})\in \operatorname {\mathrm {Mod}}(D_Y)$ is isomorphic to E, so E carries an integrable connection.

Proof. It follow from Proposition 3.5 and Theorem 5.7.

Proof. The implications (ii) $\Rightarrow $ (iii), (v) $\Rightarrow $ (vi), and (vi) $\Rightarrow $ (vii) are direct. See [Reference Björk6, Sec. 1.4.10] for (i) $\Rightarrow $ (ii).

1. (iii)⃒(iv) For every relatively compact open subset U of Z, consider the filtration $\{M_k\}$ in the definition. By induction on k, one proves that each $M_k$ is $D_U$ -coherent.

2. (iv)⃒(v) By [Reference Björk6, Corollary 1.4.6], as every quotient $M_v/M_{v-1}$ admits a good filtration, it is countably quasi-good. By induction on v and using [Reference Kashiwara and Schapira18, Lemma 2.1.1], one proves that every $M_v$ is countably quasi-good. Therefore, for every integer v, there is an increasing sequence $\{M_v^k\}_{k\ge 1}$ of coherent $O_U$ -submodules of $M_v$ with $M_v=\cup _{k\ge 1}M_v^k$ . For every integer $k\ge 1$ , let $M^k:=\sum _{\substack{i\le k,\\ v\le k}}M_v^i$ . By [45, Tag 01BY], $M^k$ is a coherent $O_U$ -submodule of $M_k$ . Then

   $$\begin{align*}M=\cup_{v\in\mathbb{Z}}M_v=\cup_{v\in \mathbb{Z}}\cup_{i\ge 1}M_v^i=\cup_{k\ge 1}M^k,\end{align*}$$
   so M is countably quasi-good.

3. (vii)⃒(i) Let U be a relatively compact open subset of Z. Because M is a finite-type $D_Z$ -module, for every $x\in \bar {U}$ , there is a relatively compact open neighborhood $U(x)\subset Z$ of x, an integer $n(x)\ge 1,$ and sections

   $$\begin{align*}\{s^x_i\}_{1\le i\le n(x)}\subset \Gamma(U(x),M)\end{align*}$$
   generating the $D_{U(x)}$ -module $M|_{U(x)}$ . By compactness of $\bar {U}$ , the open cover $\{U(x)\}_{x\in \bar {U}}$ of $\bar {U}$ has a finite subcover $\{U(x_j)\}_{1\le j\le r}$ . Then $V=\cup _{j=1}^rU(x_j)$ is a relatively compact open subset of Z containing U. By Condition (vii), one may write $M|_V=\sum _{\alpha \in I}G_{\alpha }$ , where I is an index set, and each $G_{\alpha }$ is a coherent $O_V$ -submodule of $M|_V$ .

   For every $x\in \bar {U}$ , there is an open neighborhood $V(x)\subset U(x)$ of x, such that for each $1\le i\le n(x)$ , $s^x_i|_{V(x)}$ is in $\Gamma (V(x),G_{\alpha (x,i)})$ for some index $\alpha (x,i)\in I$ . By compactness of $\bar {U}$ again, the open cover $\{V(x)\}_{x\in \bar {U}}$ has a finite subcover $\{V(x^{\prime }_k)\}_{1\le k\le m}$ . Then

   $$\begin{align*}F:=\sum_{\substack{1\le k\le m,\\1\le i\le n(x^{\prime}_k)}}G_{\alpha(x^{\prime}_k,i)}\end{align*}$$
   is a finite-type $O_V$ -submodule of $M|_V$ . By Lemma 6.8 below, it is coherent over $O_V$ . Moreover, one has $D_U\cdot F|_U=M|_U$ .

Proof. The proof is similar to that of Proposition 6.3.

Proof. It follows from [Reference Liu25, Proposition 3.1.5(2)], [Reference Björk6, Theorem 1.2.5], and Lemma 4.5.

Proof. The first half is a combination of [Reference Kashiwara15, Proposition 4.23], [45, Tag 01BY], and [45, Tag 0754]. The second half follows from [Reference Yekutieli49, Proposition 7.4.5].

Proof. Take $M\in D_{O-\mathrm {good}}(D_W)$ . By [Reference Sabbah38, Remark 3.3.4(4)], $f_+:D(D_W)\to D(D_Z)$ has finite cohomological dimension. So by [Reference Hartshorne11, Ch. I, Proposition 7.3(iii)], to prove $f_+M\in D_{O-\mathrm {good}}(D_Z)$ , one may assume $M\in \operatorname {\mathrm {Mod}}(D_W)$ . Define a morphism $i:W\to W\times Z,\quad w\mapsto (w,f(w))$ , which is a closed embedding. Let $q:W\times Z\to Z$ be the projection. By [Reference Sabbah38, Theorem 3.3.6(1)], one has $f_+=q_+i_+$ . By [Reference Björk6, Proposition 2.4.8], one has $f_+M=Rq_*\operatorname {\mathrm {DR}}_{W\times Z/Z}(i_+M)[\dim Z]$ . By [Reference Liu25, Theorem 3.2.1], as each term of the relative de Rham complex $\operatorname {\mathrm {DR}}_{W\times Z/Z}(i_+M)$ is $O_{W\times Z}$ -good and supported on W, one has $Rq_*\operatorname {\mathrm {DR}}_{W\times Z/Z}(i_+M)\in D_{\mathrm {good}}(O_Z)$ .

Proof. By [Reference Mebkhout30, Proposition 2.1.9] and its analog for $\mathcal {A}_X$ , for every point $x\in Z$ , there exists an open neighborhood U of x in Z and a good filtration on $M|_U$ . By [Reference Björk6, Corollary 1.4.6], $M|_U$ is the sum of an increasing sequence of coherent $O_U$ -submodules. Hence, $M|_U$ is good over $O_U$ . By [Reference Liu25, Lemma A.4.1(1)], the $O_U$ -module $M|_U$ is pseudo-coherent. As pseudo-coherence is a local property, M is pseudo-coherent over $O_Z$ .

Proof. By [45, Tag 01BY (1)], N is coherent over $\mathcal {R}$ . For every relatively compact open subset U of X and every $x\in \bar {U}$ , there is an open neighborhood $U(x)\subset X$ of x, an integer $n(x)>0,$ and sections $\{s_i(x)\}_{i=1}^{n(x)}\subset \Gamma (U(x),N)$ generating the $\mathcal {R}|_{U(x)}$ -module $N|_{U(x)}$ . The open cover $\{U(x)\}_{x\in \bar {U}}$ of $\bar {U}$ has a finite subcover $\{U(x_j)\}_{j=1}^m$ . Let $N_0$ be the $O_U$ -submodule of $N|_U$ generated by the finitely many local sections

$$\begin{align*}\{s_i(x_j)\}_{1\le j\le m, 1\le i\le n(x_j)}.\end{align*}$$

Then $N_0$ is a finite-type $O_U$ -module. By Lemma 6.8, because $M|_U$ is good over $\mathcal {R}|_U$ , the $O_U$ -module $N_0$ is coherent. By construction, one has $\mathcal {R}|_U\cdot N_0=N|_U$ . Therefore, the $\mathcal {R}$ -module N is good by Propositions 6.3 (in the case $\mathcal {R}=D_Z$ ) and 6.4 (in the case $\mathcal {R}=\mathcal {A}_X$ ).

Proof.

1. (i) For every coherent $O_Y$ -module F, we prove $\tilde {\mathcal {F}}(D_Y\otimes ^L_{O_Y} F)\in D^b_{\mathrm {good}}(\mathcal {A}_X)$ .

By Proposition 5.5, one has $\tilde {\mathcal {F}}(D_Y\otimes ^L_{O_Y} F)\cong\mathcal {A}_X\otimes ^L_{O_X}\mathcal {F}(F)$ . By Fact 1.3, one has $\mathcal {F}(F)\in D_{\mathrm {coh}}^b(O_X)$ . From Lemma 6.5, one gets $\mathcal {A}_X\otimes ^L_{O_X}\mathcal {F}(F)\in D^b_{\mathrm {good}}(\mathcal {A}_X)$ .

1. (ii) We prove that for every good $D_Y$ -module M and every integer i, the $\mathcal {A}_X$ -module $H^i\tilde {\mathcal {F}}(M)$ is good.

We use the descending induction on $i\in \mathbb {Z}$ . By Proposition 5.3, the $O_X$ -module underlying $H^i\tilde {\mathcal {F}}(M)$ is isomorphic to $H^i\mathcal {F}(M)$ . By Lemma 6.11, if $i>2g$ , then $H^i\mathcal {F}(M)=0$ and hence $H^i\tilde {\mathcal {F}}(M)=0$ .

Assume the statement for $i+1$ . By Proposition 6.3, there is a coherent $O_Y$ -submodule $F\subset M$ with $D_Y\cdot F=M$ . Let $M'$ be the kernel of the natural epimorphism $D_Y\otimes _{O_Y}F\to M$ , so that

(29) $$ \begin{align}0\to M'\to D_Y\otimes_{O_Y}F\to M\to0 \end{align} $$

is a short exact sequence in $\operatorname {\mathrm {Mod}}(D_Y)$ . By Lemma 6.5, the $D_Y$ -module $D_Y\otimes _{O_Y}F$ is good. By Lemma 6.6, so is $M'$ . From (29), one has an exact sequence

(30) $$ \begin{align}\begin{aligned} &H^i\tilde{\mathcal{F}}(M')\to H^i\tilde{\mathcal{F}}(D_Y\otimes_{O_Y}F)\to H^i\tilde{\mathcal{F}}(M)\\ \to &H^{i+1}\tilde{\mathcal{F}}(M')\to H^{i+1}\tilde{\mathcal{F}}(D_Y\otimes_{O_Y}F)\end{aligned} \end{align} $$

in $\operatorname {\mathrm {Mod}}(\mathcal {A}_X)$ . By (i), the $\mathcal {A}_X$ -module $H^j\tilde {\mathcal {F}}(D_Y\otimes _{O_Y}F)$ is good for $j\in \{i,i+1\}$ . By the inductive hypothesis, so is $H^{i+1}\tilde {\mathcal {F}}(M')$ .

Let $G:=\ker \left ( H^{i+1}\tilde {\mathcal {F}}(M')\to H^{i+1}\tilde {\mathcal {F}}(D_Y\otimes _{O_Y}F)\right ) $ . By Lemma 6.6, the $\mathcal {A}_X$ -module G is good (hence of finite type). The sequence (30) yields an exact sequence

$$\begin{align*}H^i\tilde{\mathcal{F}}(D_Y\otimes_{O_Y}F)\to H^i\tilde{\mathcal{F}}(M)\to G\to0,\end{align*}$$

so $H^i\tilde {\mathcal {F}}(M)$ is a finite-type $\mathcal {A}_X$ -module for every good $D_Y$ -module M. In particular, as $M'$ is good over $D_Y$ , $H^i\tilde {\mathcal {F}}(M')$ is a finite-type $\mathcal {A}_X$ -module.

Let $N:=\mathrm {im}\left (H^i\tilde {\mathcal {F}}(M')\to H^i\tilde {\mathcal {F}}(D_Y\otimes _{O_Y}F)\right )$ . It is a finite-type $\mathcal {A}_X$ -submodule of the good $\mathcal {A}_X$ -module $H^i\tilde {\mathcal {F}}(D_Y\otimes _{O_Y}F)$ . By Lemma 6.9, the $\mathcal {A}_X$ -module N is good. The sequence (30) yields an exact sequence

$$\begin{align*}0\to N\to H^i\tilde{\mathcal{F}}(D_Y\otimes_{O_Y}F)\to H^i\tilde{\mathcal{F}}(M)\to H^{i+1}\tilde{\mathcal{F}}(M')\to H^{i+1}\tilde{\mathcal{F}}(D_Y\otimes_{O_Y}F).\end{align*}$$

By Lemma 6.6, the $\mathcal {A}_X$ -module $H^i\tilde {\mathcal {F}}(M)$ is good. The induction is completed.

From (ii), Lemma 6.6, and [Reference Hartshorne11, Ch. I, Proposition 7.3(i)], $\mathcal {F}\colon D(D_Y)\to D(\mathcal {A}_X)$ restricts to a functor $ D^b_{\mathrm {good}}(D_Y)\to D^b_{\mathrm {good}}(\mathcal {A}_X)$ . Similarly, using Proposition 6.4, one can prove that $\tilde {\mathcal {F}}^{\natural }$ restricts to a functor $D^b_{\mathrm {good}}(\mathcal {A}_X)\to D^b_{\mathrm {good}}(D_Y)$ . By Theorem 5.4, the restrictions are equivalences.

Proof. By left exactness of the functor $p_{Y*}:\operatorname {\mathrm {Mod}}(O_{X\times Y})\to \operatorname {\mathrm {Mod}}(O_Y)$ , one has $H^i\mathcal {F}'(F)=0$ for every integer $i<0$ . For every $y\in Y$ , let M be the restriction of ${\mathcal {P}}\otimes _{O_{X\times Y}} p_X^*F$ to $X\times y$ as a sheaf. By the proper base change theorem (see, e.g., [Reference Milne31, Theorem 17.2]), for every integer j, one has $H^j\mathcal {F}'(F)_y=H^j(X\times y,M)$ . By [Reference Kashiwara and Schapira16, Proposition 3.2.2(iv)], if $j>2g$ , then $H^j(X\times y,M)=0$ . Therefore, $H^j\mathcal {F}'(F)=0$ . The other part is similar.

Proof. Using Theorem 6.10 and Malgrange’s theorem [Reference Malgrange26, p. 35], one can prove the result by following Schnell’s strategy of [Reference Schnell41, Proposition 18.2] in the algebraic case.

Proof. Let F be a coherent $O_Z$ -module and $N:=D_Z\otimes _{O_Z}F$ . By [Reference Björk6, Equation (ii), p. 122], there is $G\in D_{\mathrm {coh}}^b(O_Z)$ with $\Delta ^{D_Z}N=D_Z\otimes _{O_Z}G$ . By Lemma 6.5, $\Delta ^{D_Z}N\in D^b_{\mathrm {good}}(D_Z)$ .

Take $M\in D^b_{\mathrm {good}}(D_Z)$ . By [Reference Hartshorne11, Ch. I, Proposition 7.3(i)], to prove $\Delta ^{D_Z}M\in D^b_{\mathrm {good}}(D_Z)$ , one may assume $M\in \mathrm {Good}(D_Z)$ . By Proposition 6.3 and the exactness of Spencer’s complex [Reference Björk6, Theorem 1.5.8], for every relatively compact open subset $U\subset Z$ , there is a finite length exact sequence in $\operatorname {\mathrm {Mod}}(D_U)$ :

$$\begin{align*}0\to D_U\otimes_{O_U}F^{-n}\to\dots\to D_U\otimes_{O_U}F^0\to M|_U\to0,\end{align*}$$

where each $F^i$ is a coherent $O_U$ -module. For every i, one has $\Delta ^{D_U}(D_U\otimes _{O_U}F^i)\in D^b_{\mathrm {good}}(D_U)$ . By Lemma 6.6, one has $(\Delta ^{D_Z}M)|_U=\Delta ^{D_U}(M|_U)\in D^b_{\mathrm {good}}(D_U)$ . Hence, $\Delta ^{D_Z}M$ lies in $D^b_{\mathrm {good}}(D_Z)$ .

Proof. For algebraic varieties, analogous results are stated as [Reference Hotta and Tanisaki13, Corollary 2.6.8(iii) and Proposition 3.2.1]. From [Reference Hotta and Tanisaki13, p. 101], all the arguments in [Reference Hotta and Tanisaki13, Section 2.6] are valid for analytic D-modules.

Proof. By [Reference Hartshorne11, Ch. I, Proposition 7.1(ii)], one may assume $M\in \mathrm {Coh}(O_Z)$ . By [45, Tag 08DL] and [Reference Griffiths and Harris8, p. 696], one may shrink Z such that M admits a globally free resolution $F^{\bullet }\to M$ , where the complex $F^{\bullet }$ is

$$\begin{align*}0\to O_Z^{k_n}\to\dots \to O_Z^{k_1}\to O_Z^{k_0}\to0\end{align*}$$

with $O_Z^{k_i}$ placed in degree $-i$ . The morphism (31) is identified with

$$\begin{align*}K\otimes^L_{O_Z}\mathcal{H} om_{O_Z}(F^{\bullet},L)\to \mathcal{H} om_{O_Z}(F^{\bullet},K\otimes^L_{O_Z}L),\end{align*}$$

which is an isomorphism.

Proof. By [Reference Rothstein36, Equation (6.2)], there exist isomorphisms

$$\begin{align*}\Delta^{\mathcal{A}_X}(\mathcal{A}_X\otimes^L_{O_X}-):= T^gR{\mathcal{H}om}_{\mathcal{A}_X}(\mathcal{A}_X\otimes^L_{O_X}-,\mathcal{A}_X)\cong T^gR{\mathcal{H}om}_{O_X}(-,\mathcal{A}_X)\end{align*}$$

of functors $ D_{\mathrm {coh}}^b(O_X)^{\mathrm {op}}\to D_{\mathrm {coh}}^b(\mathcal {A}_X)$ . By Lemma 7.4, this functor is isomorphic to $T^gR{\mathcal {H}om}_{O_X}(-,O_X)\otimes _{O_X}^L\mathcal {A}_X\cong \mathcal {A}_X\otimes ^L_{O_X}(\Delta ^{O_X}-)$ .

Proof of Proposition 7.3

The result can be deduced from Lemma 7.5, Proposition 5.5, and [Reference Liu25, Proposition 6.1.5], in the same way that [Reference Rothstein36, Proposition 6.3] is proved.

Proof. The result follows from Proposition 7.3 and Lemma 7.2(ii), in the same way how Theorem 6.5 follows from Propositions 6.3 and 6.4 in [Reference Rothstein36].

Proof. One can prove the first isomorphism using [Reference Liu25, Proposition 3.1.2(2), Theorem 4.2.7, and Equation (19)]. The third isomorphism can be proved using [Reference Björk6, Theorems 2.8.1, 2.8.7, and 2.11.8], (26), and Theorem 6.10. By Theorem 5.4 (resp. Theorem 6.10), the second (resp. last) isomorphism follows from the first (resp. third).

Proof. Using [Reference Liu25, Lemma 6.1.3], Lemma 6.7, and [Reference Sabbah38, Theorem 3.3.6(1)], one can argue as in [Reference Hotta and Tanisaki13, Proposition 1.5.30] to prove the result.

Proof. There are isomorphisms of functors $\operatorname {\mathrm {Mod}}(\mathcal {A}_X)\to \operatorname {\mathrm {Mod}}(O_{X^{\prime \prime }})$ :

$$ \begin{align*} \mathrm{for}_{X^{\prime\prime}}\tilde{u}^*&:=(u^{-1}-)\otimes_{u^{-1}\mathcal{A}_X}\mathcal{A}_{X^{\prime\prime}}\\ &\stackrel{(\textrm{a})}{=} (u^{-1}-)\otimes_{u^{-1}\mathcal{A}_X}(u^{-1}\mathcal{A}_X\otimes_{u^{-1}O_X}u^{\prime*}\mathcal{A}_{X^{\prime}})\\ &\cong(u^{-1}-)\otimes_{u^{-1}O_X}u^{\prime*}\mathcal{A}_{X^{\prime}}\\ &\cong(u^{-1}-\otimes_{u^{-1}O_X}O_{X^{\prime\prime}}) \otimes_{O_{X^{\prime\prime}}}u^{\prime*}\mathcal{A}_{X^{\prime}}\\ &\cong(u^*\mathrm{for}_X-)\otimes_{O_{X^{\prime\prime}}}u^{\prime*}\mathcal{A}_{X^{\prime}},\end{align*} $$

where (a) uses (34). Similarly, there is an isomorphism of functors $\mathrm {for}_{X^{\prime \prime }}\tilde {u}^{\prime *}\cong u^*\mathcal {A}_X\otimes _{O_{X^{\prime \prime }}}u^{\prime *}\mathrm {for}_{X^{\prime }}(-) :\operatorname {\mathrm {Mod}}(\mathcal {A}_{X^{\prime }})\to \operatorname {\mathrm {Mod}}(O_{X^{\prime \prime }})$ . One has isomorphisms of bifunctors

$$ \begin{align*} &\mathrm{for}_{X^{\prime\prime}}(-\boxtimes_{\mathcal{A}}+):=(\tilde{u}^*-)\otimes_{\mathcal{A}_{X^{\prime\prime}}}(\tilde{u}^{\prime*}+)\\ \cong&\ (u^*\mathrm{for}_X-\otimes_{O_{X^{\prime\prime}}}u^{\prime*}\mathcal{A}_{X^{\prime}})\otimes_{u^*\mathcal{A}_X\otimes_{O_{X^{\prime\prime}}}u^{\prime*}\mathcal{A}_{X^{\prime}}}(u^*\mathcal{A}_X\otimes_{O_{X^{\prime\prime}}}u^{\prime*}\mathrm{for}_{X^{\prime}}+)\\ \cong&\ (u^*\mathrm{for}_X-)\otimes_{O_{X^{\prime\prime}}}(u^{\prime*}\mathrm{for}_{X^{\prime}}+)\\ =:&\ (\mathrm{for}_X-)\boxtimes_O(\mathrm{for}_{X^{\prime}}+).\\[-42pt] \end{align*} $$

Proof. It follows from [Reference Liu25, Proposition 6.1.2], Lemma 7.11, and Proposition 5.3.

Proof. Let $i:\{0\}\hookrightarrow Y$ be the inclusion of the origin. Then $D_Y\otimes _{O_Y}\mathbb {C}_{(0)}=i_+\mathbb {C}$ . There are isomorphisms

$$ \begin{align*} (i_+\mathbb{C})*_D(-)&:=\mu_+((i_+\mathbb{C})\boxtimes_O-)\\ &\stackrel{(\textrm{a})}{\cong} \mu_+(i\times {\mathrm{Id}}_Y)_+(\mathbb{C}\boxtimes_O-)\\ &\stackrel{(\textrm{b})}{\cong} {\mathrm{Id}}_{Y+}={\mathrm{Id}}_{D(D_Y)} \end{align*} $$

of functors $D(D_Y)\to D(D_Y)$ , where (a) and (b) use Lemma 7.10(i) and [Reference Sabbah38, Theorem 3.3.6(1)], respectively. Therefore, $D_Y\otimes _{O_Y}\mathbb {C}_{(0)}$ is the unit. The other axioms can be verified as in [Reference Weissauer47, pp. 10–11].

Proof. It follows from (35) and Theorems 5.4 and 6.10.

Proof. By [45, Tag 08DJ], for every object $N\in D^b_{\mathrm {good}}(\mathcal {A}_X)$ , the functor $(-)\otimes ^L_{\mathcal {A}_X}N:D^b_{\mathrm {good}}(\mathcal {A}_X)\to D^b_{\mathrm {good}}(\mathcal {A}_X)$ admits a right adjoint $R{\mathcal {H}om}_{\mathcal {A}_X}(N,-)$ . By [Reference Huybrechts14, p. 84], the right adjoint is naturally isomorphic to $T^{-g}\Delta ^{\mathcal {A}_X}(N)\otimes ^L_{\mathcal {A}_X}(-)$ . The result follows from Propositions 7.3 and 7.15.

Proof. Arguing as in [Reference Mukai33, Equation (3.1)], one can deduce the isomorphisms from the projection formula.

Proof. By Example 5.2 and Theorem 5.4, one has $\tilde {\mathcal {F}}(L,\nabla )=T^{-g}\pi _*\mathbb {C}_{(z)}$ . Then (36) follows from the last isomorphism in Proposition 7.15. By Theorem 5.4, (36) implies (37).