A Factorization Theorem for Curves with Vanishing Self-Intersection

Abstract

Let C be a connected divisor in a compact Kähler manifold such that the self-intersection of C, computed with respect to a Kähler metric, vanishes. Assume that the normal closure of the image of $\pi _{1}(C)$ in $\pi _{1}(Y)$ has infinite index. Then there exists a holomorphic map f from Y to a curve B such that C is a fiber. The conclusion holds if one assumes that the image of $\pi _{1}(C)$ is amenable but $\pi _{1}(Y)$ is non-amenable.

Appendices

Appendix 1

In this section, we prove a version of the Galois $\partial {\overline{\partial }}$-lemma for uniform Sobolev spaces, in particular currents with compact support. This will complete the exposition given in [21] where the domains of the operators were left unprecise.

1.1 Appendix 1.1: Preliminaries

We follow the notations of [21, Sect. 3.2]. Complements and references should be find in [43, 46].

Let $X\rightarrow Y$ be a Galois covering of a compact Kähler manifold Y, let $\Gamma$ be the covering transformations group, let $\Lambda ^{p}:=\Lambda ^{p}T^{*}X$ be the vector bundle of p-forms. The Laplace operator $\Delta$ is essentially self-adjoint on $L^{2}( X,\oplus _{p}\Lambda ^{p})$ and ${\mathrm {Dom}}(\Delta _{\max })={\mathrm {Dom}}(\Delta _{\min })=S^{2}_{2}( X)$ is the uniform sobolev space of order two on X. Let $\partial ^{*}_{f}$ (resp. ${\overline{\partial }}^{*}_{f}$) be the formal adjoint of $\partial$ (resp. ${\overline{\partial }}$) and let S be one of the operator $\partial +\partial ^{*}_{f}$ or ${\overline{\partial }}+{\overline{\partial }}^{*}_{f}$. Then $S: L^{2}( X,\oplus _{p}\Lambda ^{p})\rightarrow L^{2}( X,\oplus _{p}\Lambda ^{p})$ is essentially self-adjoint and $S^{2}=\Delta$ for the metric is Kähler and complete. In particular ${\mathrm {Dom}}(\Delta )\subset {\mathrm {Dom}}(S)$.

Let $A=(1+S^{2})^{\frac{1}{2}}:L^{2}( X,\oplus _{p}\Lambda ^{p})\rightarrow L^{2}( X,\oplus _{p}\Lambda ^{p})$ be defined through the functional calculus. Then $A^{m}:S^{k}(X,\oplus _{p}\Lambda ^{p})\rightarrow S^{k-m}(X,\oplus _{p}\Lambda ^{p})$ defines an isomorphism (see e.g., [43, Th. 5.5]).

For $s\in {\mathbb {R}}$, let $(H^{s},||.||_{s})=({\mathrm {Dom}}(A^{s}), ||A^{s}.||_{L^{2}})$ be the completion of the space of the smooth sections with compact support with respect to the Sobolev norm $||\alpha ||_{s}=||A^{s}\alpha ||_{L^{2}}$. Let $s\in {\mathbb {R}}$, the following isomorphism holds $(S^{j}( X,\oplus _{p}\Lambda ^{p}),\, ||.||_{S^{j}( X,\oplus _{p}\Lambda ^{p})}) \overset{\sim }{\rightarrow } (H^{s},||.||_{s})$.

Then $A^{k}: H^{s+k}\rightarrow H^{s}$ is bounded with adjoint $A^{-k}$ and $\cap _{s\in {\mathbb {R}}}H^{s}$ is dense in any $H^{p}$. Let B be one of the four operators $\partial$, ${\overline{\partial }}$, ${\overline{\partial }}^{*}_{f}$, $\partial ^{*}_{f}$. Then B commutes to $A^{s}$ on smooth sections with compact support, for $A^{s}$ is a weak limit of a sequence of polynomial in $\Delta$. Hence $\forall s\in {\mathbb {R}}$, B extends uniquely as $B_{s}:H_{s}\rightarrow H_{s-1}$ such that $A^{k}B_{s}=B_{s-k}A^{k}$ on $H^{s}$. Moreover $(\partial _{s})^{*}=A^{-1} (\partial ^{*}_{f})_{s}A^{-1}=A^{-2} (\partial ^{*}_{f})_{s-1}$ and $({\overline{\partial }}_{s})^{*}=A^{-1} ({\overline{\partial }}^{*}_{f})_{s}A^{-1}=A^{-2}({\overline{\partial }}^{*}_{f})_{s-1}$. This last point uses that $H^{1}\subset {\mathrm {Dom}}(S)$ and that the formal adjoint is equal to the Hilbertian adjoint.

These commutation relations granted, we will drop the indice s referring to the extension of these operators to the uniform Sobolev spaces $H^{s}$.

Recall ([17, p. 9], [35, 46]) that a unbounded $\Gamma$-operator $f: A_{1}\rightarrow A_{2}$ is $\Gamma$-Fredholm if for some $\lambda _{0}>0$ the spectral projection $E_{\lambda _{0}}$ of the operator $ff^{*}+f^{*}f:A_{1}\oplus A_{2}\rightarrow A_{1}\oplus A_{2}$ satisfies $\dim _{\Gamma }E(\lambda _{0})<+\infty$.

Theorem 6.1

(Atiyah [5], see also [17], [35, Lemma 2.66], [46])

• (i) The operator $\Delta$ is $\Gamma$-Fredholm: for any $\lambda >0$, let $E_{\lambda }$ be the spectral projection of $\Delta$ on the interval $[0,\lambda ]$. Then

  $$\begin{aligned} \dim _{\Gamma }E_{\lambda }L^{2}(X,\oplus _{p}\Lambda ^{p})<+\infty . \end{aligned}$$

• (ii) The operators ${\overline{\partial }}+{\overline{\partial }}^{*}: H^{s}\rightarrow H^{s-1}$ and $\partial +\partial ^{*}: H^{s}\rightarrow H^{s-1}$ are $\Gamma$-Fredholm.

The second assertion is reduced to the first for $S^{*}S=\Delta (1+\Delta )^{-1}$ is $\Gamma$-Fredholm by i).

1.2 Appendix 1.2: A Galois $\partial {\overline{\partial }}$-Lemma for Currents

Theorem 6.2

(A Galois $\partial {\overline{\partial }}$-lemma) Let $X\rightarrow X/\Gamma =Y$ be a Galois covering of a compact Kähler manifold Y with Galois group $\Gamma$. Let $s\in {\mathbb {R}}$ and $\alpha$ be a (p, q)-form in $H^{s}$ which is closed and orthogonal to the square integrable harmonic forms. Then there exists $r\in \text {M}(\Gamma )$ almost invertible (i.e., injective with dense range) and a $(p-1,q-1)$-form $u\in H^{s+2}$ such that

$$\begin{aligned} r.\alpha= & {} \partial {\overline{\partial }}u. \end{aligned}$$

Proof

The form $\alpha$ is both ${\overline{\partial }}$ and $\partial$-closed for it is d-closed of pure type. Generalized Hodge–Kodaira decomposition (see [46, Lemma 2.6]) implies that $H^{s}={\mathcal {H}}\oplus \overline{{\mathrm {Ran}}(\partial +\partial ^{*})\circ A^{-1}}$ with ${\mathcal {H}}$ the harmonic space. By hypothesis $\alpha$ belongs to $\overline{{\mathrm {Ran}}(\partial +\partial ^{*})\circ A^{-1}}$. But $A^{-1}$ is an isomorphism, hence $(\partial +\partial ^{*})\circ A^{-1}$ is $\Gamma$-Fredholm. From [21, Lemma 2.15], there exists $r_{1}\in \text {M}(\Gamma )$ almost invertible and $s_{1}\in H^{s}\ominus {\mathcal {H}}$ such that

$$\begin{aligned} r_{1}.\alpha = (\partial +\partial ^{*})\circ A^{-1} s_{1}=\partial \circ A^{-1} s_{1} \end{aligned}$$

for $\alpha \in \mathrm {Ker}(\partial )=({\mathrm {Ran}}(\partial ^{*}))^{\perp }$. Generalized Hodge–Kodaira decomposition implies that $s_{1}\in \overline{{\mathrm {Ran}}({\overline{\partial }}+{\overline{\partial }}^{*})\circ A^{-1}}$. As $({\overline{\partial }}+{\overline{\partial }}^{*})\circ A^{-1}$ is $\Gamma$-Fredholm, there exists $r_{2}\in \text {M}(\Gamma )$ almost invertible and $s_{2}\in H^{s}\ominus {\mathcal {H}}$ such that $r_{2}.s_{1}=({\overline{\partial }}+{\overline{\partial }}^{*})\circ A^{-1} s_{2}$, so that $r_{2}r_{1} \alpha =\partial {\overline{\partial }}A^{-2} s_{2}+\partial {\overline{\partial }}^{*}A^{-2}s_{2}$. The form $\partial {\overline{\partial }}^{*}A^{-2}s_{2}=r_{2}r_{1} \alpha -\partial {\overline{\partial }}A^{-2} s_{2}$ is ${\overline{\partial }}$-closed and ${\overline{\partial }}^{*}$-exact for the metric is Kähler hence $\partial {\overline{\partial }}^{*}A^{-2}=-{\overline{\partial }}^{*}\partial A^{-2}$. It must vanishes, therefore $r_{2}r_{1} \alpha =\partial {\overline{\partial }}A^{-2} s_{2}$ $\square$

1.3 Appendix 1.3: Examples

We prove that an infinite covering ${\mathbb {R}}^{n}/{\mathbb {Z}}^{n-r''}\rightarrow {\mathbb {R}}^{n}/{\mathbb {Z}}^{n}$ of a compact torus has vanishing $l^{2}$-cohomology and reprove in this context [21, Lemma 2.15] using Fourier transform. We apply then the analysis to complex tori. Laplace operator and Sobolev spaces will be defined as in Appendix 1.1 with respect to the flat Euclidean structure.

We first identify the action of the Von Neumann algebra on square integrable functions on the covering. Let $n=r'+r''$. The group ${\mathbb {Z}}^{r''}$ acts on ${\mathbb {R}}^{r''}$ by translation in the usual way. The isomorphism $l^{2}({\mathbb {Z}}^{r''})\rightarrow l^{2}({\mathbb {R}}^{r''}/{\mathbb {Z}}^{r''})$ identifies the Von Neumann algebra $\text {M}({\mathbb {Z}}^{r''})$ with $L^{\infty }({\mathbb {R}}^{r''}/{\mathbb {Z}}^{r''})$. Let $\tau _{v''}$ be the translation operator on $L^{2}({\mathbb {R}}^{r''}_{x''})$ defined by $v''\in {\mathbb {Z}}^{r''}$. The Fourier transform $f\mapsto {\mathcal {F}}{f}$ from $L^{2}({\mathbb {R}}^{r''}_{x''})$ to $L^{2}({\mathbb {R}}^{r''}_{\zeta ''})$ identifies $\tau _{v''}$ to the multiplication operator by the function $e^{iv''.\zeta ''}$. The action of $r\in \text {M}({\mathbb {Z}}^{r''})$ on $L^{2}({\mathbb {R}}^{r''}_{x''})$ is given by ${\mathcal {F}}{r.f}=r(e^{i\zeta ''_{1}},\ldots ,e^{i\zeta ''_{r''}}).{\mathcal {F}}{f}(\zeta '')$. Hence r is injective iff $r(e^{i\zeta ''_{1}},\ldots ,e^{i\zeta ''_{r''}})\not =0$ for almost every $\zeta ''$.

A constant elliptic system E of order one on ${\mathbb {R}}^{n}/{\mathbb {Z}}^{n}$ is a first-order operator $E=\sum \limits _{k=1}^{n}E_{k}(-i\partial _{x_{k}})$ with constant matrix coefficients acting on the section of the trivial Hermitian bundle ${\mathbb {R}}^{n}/{\mathbb {Z}}^{n}\times {\mathbb {C}}^{N}$ such that $\forall \zeta =(\zeta _{1},\ldots ,\zeta _{n})\in {\mathbb {R}}^{n}\setminus \{0\}$,

$$\begin{aligned} \hat{E}(\zeta ):=\sum _{k=1}^{n}E_{k}\zeta _{k}\in \mathrm {Gl}({\mathbb {C}}^{N}) . \end{aligned}$$

Let $r',r''$ be positive integers and $n=r'+r''$.

Lemma 6.3

Let $({\mathbb {R}}^{r'}/{\mathbb {Z}}^{r'})\times {\mathbb {R}}^{r''}\rightarrow ({\mathbb {R}}^{r'}/{\mathbb {Z}}^{r'})\times ({\mathbb {R}}^{r''}/{\mathbb {Z}}^{r''})\simeq {\mathbb {R}}^{n}/{\mathbb {Z}}^{n}$ be a non-compact covering of a compact torus. Let E be a constant elliptic system of order one on ${\mathbb {R}}^{n}/{\mathbb {Z}}^{n}$. Then the kernel of E acting on $H^{s}({\mathbb {R}}^{n}/{\mathbb {Z}}^{r'},{\mathbb {C}}^{N})$, $(s\in {\mathbb {R}})$, is trivial. Moreover, there exists $r\in \text {M}({\mathbb {Z}}^{r''})$ injective such that for any section $f\in H^{s}({\mathbb {R}}^{n}/{\mathbb {Z}}^{r'},{\mathbb {C}}^{N})$, r.f belongs to the range of E acting on $H^{s+1}({\mathbb {R}}^{n}/{\mathbb {Z}}^{r'},{\mathbb {C}}^{N})$.

$l^{2}$-homology fits into this framework: cells define currents with compact support and under this duality, the exterior derivative is given by the boundary operator. An example is given by the covering ${\mathbb {R}}\rightarrow S^{1}$. The 0-cell $\delta _{0}$ is in the closure of the range of the boundary operator $\mathrm {b}$:

$$\begin{aligned} \delta _{0}-\frac{1}{n}\sum _{i=1}^{n}\delta _{i}=\mathrm {b} \left( \frac{1}{n}\sum _{i=1}^{n}1_{[0,i]} \right) \text { and } ||\frac{1}{n}\sum _{i=1}^{n}\delta _{i}||^{2}=\frac{1}{n^{2}}\sum _{i=1}^{n}||\delta _{i}||^{2}=\frac{1}{n}. \end{aligned}$$

Then $(\tau _{1}-\tau _{0})\delta _{0}=\delta _{1}-\delta _{0}=\mathrm {b} 1_{[0,1]}$ is already in the range of the boundary operator. Moreover, as current, $\delta _{0}\in H^{-1-\epsilon }$ and $1_{[0,1]}\in H^{-\epsilon }$ for any $\epsilon >0$.

Proof

A section f which belongs to $H^{s}({\mathbb {R}}^{n}/{\mathbb {Z}}^{r'},{\mathbb {C}}^{N})$ has a Fourier series decomposition

$$\begin{aligned} f(x',x'')=\sum _{n'\in {\mathbb {Z}}^{r'}}f_{n'}(x'')e^{in'.x'} \end{aligned}$$

such that $(n',x'')\rightarrow (1+||n'||^{2})^{\frac{s}{2}}f_{n'}(x'')$ belongs to $L^{2}({\mathbb {Z}}^{n'}, H^{s}({\mathbb {R}}^{n''},{\mathbb {C}}^{N}))$. Let $s\in {\mathbb {R}}$. The operator $(1+\Delta )^{\frac{s}{2}}$ commutes with E and gives an isomorphism from $H^{s}({\mathbb {R}}^{n}/{\mathbb {Z}}^{r'},{\mathbb {C}}^{N})$ to $L^{2}({\mathbb {R}}^{n}/{\mathbb {Z}}^{r'},{\mathbb {C}}^{N}$). Hence it is enough to prove the lemma for $s=0$.

The partial Fourier transform $\mathcal {F}$ in the variable $x''$ gives

$$\begin{aligned} {\mathcal {F}}{Ef}(x',\zeta '')= \sum _{n'\in {\mathbb {Z}}^{r'}}\hat{E}(n',\zeta '') {\mathcal {F}}{f_{n'}}(\zeta '')e^{in'.x'}. \end{aligned}$$

Assume f belongs to the Kernel of E. Then ${\mathcal {F}}{f_{n'}}$ is vanishing almost everywhere for $(n',\zeta '')$ is non-vanishing on ${\mathbb {R}}^{r''}_{\zeta ''}\setminus \{0\}$ hence $\hat{E}(n',\zeta '')$ is injective there.

To invert E, one notices that $\hat{E}(\zeta )^{-1}=||\zeta ||^{-1}\hat{E}(\frac{\zeta }{||\zeta ||})^{-1}$ and the set $\{\hat{E}(\zeta )^{-1},\,||\zeta ||=1\}$ is a compact subset of $\mathrm {Gl}({\mathbb {C}}^{N})$. Let C be a bound for the operator norm of its elements. A square integrable section $f=\sum \limits _{n'\in {\mathbb {Z}}^{r}}f_{n'}e^{in'.x'}$ belongs to the range of E iff

$$\begin{aligned} (n',\zeta '')\mapsto & {} {\hat{E}\left( \frac{(n',\zeta '')}{||(n',\zeta '')||}\right) }^{-1}\frac{1}{||(n',\zeta '')||}{\mathcal {F}}{f_{n'}}(\zeta '') e^{in'.x'} \end{aligned}$$

is square integrable on ${\mathbb {Z}}^{r'}\times {\mathbb {R}}^{r''}$ iff, using the uniform bound C,

$$\begin{aligned} (n',\zeta '')\mapsto \frac{1}{||(n',\zeta '')||}{\mathcal {F}}{f_{n'}}(\zeta '') \end{aligned}$$

is square integrable. Let $v''\in {\mathbb {Z}}^{r''}\setminus \{0\}$ be fixed. The bounded function $\zeta ''\mapsto (e^{i v''.\zeta ''}-1)$ is non-vanishing almost everywhere and $\zeta ''\mapsto \frac{(e^{i v''.\zeta ''}-1)}{||\zeta ''||}$ is bounded by $||v''||$. Hence the operator $(\tau _{v''}-\mathrm{Id})\in \text {M}({\mathbb {Z}}^{r''})$ is injective on $L^{2}({\mathbb {R}}^{n}/{\mathbb {Z}}^{r'},{\mathbb {C}}^{N})$ for

$$\begin{aligned} {\mathcal {F}}{(\tau _{v''}-\mathrm{Id}) f}(\zeta '')=(e^{i v''.\zeta ''}-1) {\mathcal {F}}{f}(x',\zeta ''). \end{aligned}$$

Then $(\tau _{v''}-\mathrm{Id}).f$ belongs to the range of E for

$$\begin{aligned} \sum _{n'\in {\mathbb {Z}}^{r}}\int _{{\mathbb {R}}^{r''}}||{\hat{E}(n',\zeta '')}^{-1}(e^{i n''.\zeta ''}-1) {\mathcal {F}}{f_{n'}}(\zeta '')||^{2}dV(\zeta '')\le & {} (C\max (2,||v''||) . ||f||)^{2} \end{aligned}$$

Elliptic regularity or above estimate prove that $E^{-1}((\tau _{v''}-\mathrm{Id})f)$ belongs to $H^{1}({\mathbb {R}}^{n}/{\mathbb {Z}}^{r'},{\mathbb {C}}^{N})$. $\square$.

This lemma applies to a non-compact covering of a compact complex torus. Let $\Lambda$ be a lattice of maximal rank in ${\mathbb {C}}^{n}$. Fix a linear isomorphism $L:{\mathbb {R}}^{2n}\rightarrow {\mathbb {C}}^{n}$ such that $L:{\mathbb {Z}}^{2n}\rightarrow \Lambda$ is an isomorphism. Let $0\le r'\le 2n$ and consider the infinite covering ${\mathbb {C}}^{n}/\Lambda '\rightarrow {\mathbb {C}}^{n}/\Lambda$ which is the pullback of the covering ${\mathbb {R}}^{2n}/{\mathbb {Z}}^{r'}\rightarrow {\mathbb {R}}^{2n}/{\mathbb {Z}}^{2n}$ through the isomorphism L. The standard hermitian metric on ${\mathbb {C}}^{n}$ is Kählerian hence the operators $\partial +\partial ^{*}$ and ${\overline{\partial }}+{\overline{\partial }}^{*}$ anticommute with square $(\partial +\partial ^{*})^{2}=({\overline{\partial }}+{\overline{\partial }}^{*})^{2}=\frac{1}{2}\Delta$ where $\Delta$ is the Euclidean Laplace operator.

The orthonormal flat basis of the bundle $\Lambda T^{*}{\mathbb {C}}^{n}$ given by $\{\epsilon _{I,J}dZ^{I}\wedge d\bar{Z}^{J}\}_{I,J}$ (the $\epsilon _{I,J}$ are suitable normalization constants) and linear coordinates $(x_{1},\ldots ,x_{2n})$ on ${\mathbb {R}}^{2n}$ allow to reduce the analysis of the operators $\partial +\partial ^{*}$, ${\overline{\partial }}+{\overline{\partial }}^{*}$ (and $d+d^{*}$) to anticommuting first-order constant elliptic systems on the trivial hermitian vector bundle $\Lambda ({\mathbb {C}}^{n})^{*}$ over ${\mathbb {R}}^{2n}/{\mathbb {Z}}^{r}$. For any square integrable form f on ${\mathbb {C}}^{n}/\Lambda '$, there exists $r\in \text {M}(\Lambda /\Lambda ')$ and a square integrable form g such that $r.f=(\partial +\partial ^{*})({\overline{\partial }}+{\overline{\partial }}^{*}) (g)$. As we saw in the proof of the $\partial {\overline{\partial }}$-lemma, the Kähler condition and the hypothesis that f is a closed (p, q)-form imply that $(\partial +\partial ^{*})({\overline{\partial }}+{\overline{\partial }}^{*})(g)=\partial {\overline{\partial }}g$.

This section contrasts with results in [32] where it is proved that the $\partial {\overline{\partial }}$-lemma for a d-exact (p, q)-form (no growth condition) does not hold for some (wild) Toroidal group ${\mathbb {C}}^{n}/\Lambda '$.

Appendix 2

In [21], we put a mixed Hodge structure on the $l^{2}$-cohomology groups of infinite covering using the notion of a quotient category. This was needed to ensure degeneracy of the weight and the Hodge spectral sequences. In this section, we show that the quotient categories involved are equivalent to the category of modules over a ring of operators.

More precisely the functor of reduction with respect to a torsion theory is isomorphic to the functor of tensorization by a localization of a ring. This was proved by Gabriel [24, Chap. V, Sect. 2].

This gives the analogy of tensorization by ${\mathbb {Q}}$ which kills the torsion in ${\mathbb {Z}}$-modules. The ring used is the left Von Neumann algebra $\text {M}(\Gamma )$ of the Galois group $\Gamma$ of the covering $p:X\rightarrow Y$. Its ring of quotients will be the algebra ${\mathcal {U}}(\Gamma )$ of operators affiliated to it, that is, densely defined operators on $l^{2}(\Gamma )$ which are invariant with respect to the right action of $\Gamma$ on $l^{2}(\Gamma )$. An element $u\in {\mathcal {U}}(\Gamma )$ may be written as $u=a. i^{-1}$ with $a,i\in \text {M}(\Gamma )$, i is injective with dense range and $i^{-1}$ is its densely defined inverse (see [36, Chap. XVI] or [35, p. 322]).

Interpretation of statements about elements in $H\otimes _{\text {M}(\Gamma )}{\mathcal {U}}(\Gamma )$ seems more commonly used than in $H \mod \tau _{{\mathcal {U}}(g)}$, an object of the quotient category, where even the notion of elements is not defined. The isomorphism of functors granted, statements in [21, Sects. 4 and 5] may be restated in the category of ${\mathcal {U}}(\Gamma )$-modules at least when $\tau =\tau _{{\mathcal {U}}(\Gamma )}$. As examples, we refer to section 4.3 of the present article and to the following classical corollary of mixed Hodge theory.

1.1 Appendix 2.1: Notations

Let ${\text {Mod}}(\text {M}(\Gamma ))$ be the category of $\text {M}(\Gamma )$-modules and ${\text {Mod}}({\mathcal {U}}(\Gamma ))$ be the category of ${\mathcal {U}}(\Gamma )$-modules.

Following [36, Chap. XVI] and [35, p. 327], we recall that $m\in \text {M}(\Gamma )$ is almost invertible iff it is injective with dense range. But $\text {M}(\Gamma )$ is finite, hence it is enough for m to be injective. Therefore the set S of almost invertible operator is a multiplicative subset of the ring $\text {M}(\Gamma )$. From [35, Def. 8.14, Th. 8.22], the Ore localization $\text {M}(\Gamma )S^{-1}$ of $\text {M}(\Gamma )$ with respect to this multiplicative system is isomorphic to ${\mathcal {U}}(\Gamma )$, the ring of affiliated operator to $\text {M}(\Gamma )$ (densely defined operator which commute with $\text {M}(\Gamma )'$).

Following [21, 2.5], [35, 8.4], [24, Chap. 5, Sect. 2], [51], we recall that

$$\begin{aligned} {\mathcal {T}}_{{\mathcal {U}}(\Gamma )}=\{M\in {\text {Mod}}(\text {M}(\Gamma )) \text { s.t. } \forall m\in M:\, \exists s\in S \text { s.t. } s.m=0\} \end{aligned}$$

defines a torsion theory $\tau _{{\mathcal {U}}(\Gamma )}$ on ${\text {Mod}}(\text {M}(\Gamma ))$.

1.2 Appendix 2.2: Isomorphism of Functors and Applications

Lemma 7.1

• (1) $\text {M}(\Gamma )\rightarrow {\mathcal {U}}(\Gamma )$ is a flat rings extension.

• (2) The localization functor ${\text {Mod}}( \text {M}(\Gamma ))\rightarrow {\text {Mod}}( \text {M}(\Gamma ))_{\tau _{{\mathcal {U}}(\Gamma )}}$ is isomorphic to the functor of scalars extension

  $$\begin{aligned}&{\text {Mod}}( \text {M}(\Gamma )) \rightarrow {\text {Mod}}({\mathcal {U}}(\Gamma ))\\&\quad M\mapsto M\otimes _{ \text {M}(\Gamma )}{\mathcal {U}}(\Gamma )\text { and } f\mapsto f\otimes id_{{\mathcal {U}}(\Gamma )}. \end{aligned}$$

Proof

• (1) see [35, Th. 8.22, p. 327].

• (2) From [35, Th. 8.22(1), p. 327], the multiplicative set S satisfies the Ore condition. Hence Proposition 5 of [24, Chap. 5, p. 415] applies. It contains the statement (2).$\square$

Note that by definition, if $m\in M$ and $s\in \text {M}(\Gamma )$, almost invertible, satisfy $s.m=0$, then in $M\otimes {\mathcal {U}}(\Gamma )$,

$$\begin{aligned} m\otimes 1= s.m\otimes s^{-1}=0\otimes s^{-1}=0. \end{aligned}$$

From the lemma, we deduce the localization with respect to ${\mathcal {T}}_{{\mathcal {U}}(\Gamma )}$ of a spectral sequence in ${\text {Mod}}( \text {M}(\Gamma ))$ is the tensor product by ${\mathcal {U}}(\Gamma )$ of the spectral sequence. As an example, we state the following classical consequence of the $E_{1}$-degeneration of the Hodge spectral sequence (see [19, Cor. 3.2.14]).

Corollary 7.2

Let D be a normal crossing divisor in Y. There exists a morphism

$$\begin{aligned} H^{0}(Y,{ p_{*(2)} }\Omega ^{.}_{Y}(log D))\rightarrow H^{.}(Y\setminus D,{ p_{*(2)} }{\mathbb {C}}) \end{aligned}$$

which is moreover injective.

Proof

(see the notations in [21, Th. 4.6(iv)]). The $E_{1}$ degeneration of the spectral sequence of the filtered complex $(R\Gamma ({ p_{*(2)} }\Omega ^{.}_{Y}(log D)),F)\otimes _{ \text {M}(\Gamma )}{\mathcal {U}}(\Gamma )$ and the edge homomorphism give

$$\begin{aligned} E^{i,0}_{\infty }(F)&\overset{\simeq }{\rightarrow }&E^{i,0}_{1}(F). \end{aligned}$$

(22)

The first term is

$$\begin{aligned} \mathrm {Ker}(d: H^{0}(Y,{ p_{*(2)} }\Omega ^{i}_{Y}(log D))\rightarrow H^{0}(Y,{ p_{*(2)} }\Omega ^{i+1}_{X}(log D))) \otimes _{ \text {M}(\Gamma )} {\mathcal {U}}(\Gamma ) \end{aligned}$$

(23)

while the second is

$$\begin{aligned} H^{0}(Y,{ p_{*(2)} }\Omega ^{i}_{X}(log D))\otimes _{ \text {M}(\Gamma )} {\mathcal {U}}(\Gamma ). \end{aligned}$$

(24)

Let $\alpha \in H^{0}(Y,{ p_{*(2)} }\Omega ^{i}_{Y}(log D))$ then $d(\alpha \otimes 1_{{\mathcal {U}}(\Gamma )})=0$, hence there exists $r\in \text {M}(\Gamma )$ injective such that $r d\alpha =0$. Lemma 2.8 implies that $d\alpha =0$.

Degeneracy at $E_{1}$ of the spectral sequence also implies that the map

$$\begin{aligned}&{\mathbb {H}}^{i}(Y,F^{i}({ p_{*(2)} }\Omega ^{.}_{Y}(log ,)d))\otimes _{ \text {M}(\Gamma )} {\mathcal {U}}(\Gamma ) \nonumber \\&\quad \rightarrow {\mathbb {H}}^{i}(Y,({ p_{*(2)} }\Omega ^{.}_{Y}(log ,)d))\otimes _{ \text {M}(\Gamma )} {\mathcal {U}}(\Gamma ) \end{aligned}$$

(25)

is injective hence

$$\begin{aligned} H^{0}(Y,{ p_{*(2)} }\Omega ^{i}_{Y}(log D))\otimes _{ \text {M}(\Gamma )} {\mathcal {U}}(\Gamma )\rightarrow & {} H^{i}(Y\setminus D,{ p_{*(2)} }{\mathbb {C}})\otimes _{ \text {M}(\Gamma )} {\mathcal {U}}(\Gamma ) \end{aligned}$$

(26)

is injective. We infer as above that $H^{0}(Y,{ p_{*(2)} }\Omega ^{i}_{Y}(log D))\rightarrow H^{0}(Y\setminus D,{ p_{*(2)} }{\mathbb {C}})$ is injective. $\square$

Remark 5.9

• (i) The monomorphism of differential sheaves $(\Omega _{Y}^{.},d)\!\rightarrow \! (\Omega ^{.}_{Y}(log D),d)$ is an isomorphism on $Y\setminus D$.

• (ii) The differential in the logarithmic complex does not coincide with the differential in the sense of currents: let $j: Y\setminus D\rightarrow Y$ be the canonical immersion and [a] be the current defined by a form $a\in L^{1}_{loc}$. Let $\alpha$ be a section of $\,\,\Omega ^{.}_{Y}(log D)$ then $[d\alpha ]= i_{*} d i^{*}[\alpha ]$.

• (iii) Let $\mathcal {A}^{.}$ be the sheaf of smooth differential forms on Y. An element of $H^{0}(Y\setminus D,{ p_{*(2)} }\mathcal {A}^{.})$ is a form on $X\setminus p^{-1}(D)$ which is square integrable on $p^{-1}(K)$ for any compact set $K\subset Y\setminus D$.

Corollary 1 is equivalently stated as:

Any holomorphic section $\alpha$ of ${ p_{*(2)} }\Omega ^{.}_{Y}(log D)$ over Y defines a closed form $\alpha _{|Y\setminus D}$ on

$X\setminus p^{-1}(D)$. Moreover if $\alpha _{|Y\setminus D}=du$ with $u\in H^{0}(Y\setminus D,{ p_{*(2)} }\mathcal {A}^{.})$, then $\alpha =0$.