3.1 Logarithmic poles and residues
We recall several classical properties of “logarithmic poles” of holomorphic differential forms. The reader will find proofs in (Deligne 1970, II, (3.1) to (3.7)), for example.
A divisor Y in a smooth complex analytic variety X is said to be of normal crossing if the inclusion of Y into X is locally isomorphic to the inclusion of a union of coordinate hyperplanes in \mathbb{C}^n; this does not imply that Y is the union of smooth divisors. Let Y be a normal crossing divisor in X, and j the inclusion of X^*=X\setminus Y into X. We denote by \Omega_X^1\langle Y\rangle the locally free sub-\mathcal{O}-module of j_*\Omega_{X^*}^1 generated by \Omega_X^1 and the \mathrm{d}z_i/z_i for z_i local coordinates of a local irreducible component of Y. The sheaf \Omega_X^p\langle Y\rangle of differential p-forms on X with logarithmic poles along Y is by the definition the locally free sub-sheaf \wedge^p\Omega_X^1\langle Y\rangle of j_*\Omega_{X^*}^p.
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- A section \alpha of j_*\Omega_{X^*}^p belongs to \Omega_X^p\langle Y\rangle if and only if \alpha and \mathrm{d}\alpha have at worst simple poles along the divisor Y.
- The \Omega_X^p\langle Y\rangle form the smallest subcomplex of j_*\Omega_{X^*}^\bullet that is stable under exterior product and that contains \Omega_X^\bullet and the logarithmic differential \mathrm{d}f/f of every local section of j_*\Omega_{X^*}^p meromorphic along Y.
We call \Omega_X^\bullet\langle Y\rangle the logarithmic de Rham complex of X along Y. By (ii) of (3.1.3), this complex is contravariant in the pair (X,X^*).
Locally on X, Y is the union of various smooth divisors Y_i, and we denote by Y^n (resp. \widetilde{Y}^n) the union (resp. disjoint sum) of the n-fold intersections of the Y_i. The Y^n glue to give a subspace Y^n of X, and the \widetilde{Y}^n glue to give the normalisation variety of Y^n. We have \widetilde{Y}^0=Y^0=X, and we set \widetilde{Y}=\widetilde{Y}^1.
We define the two-element set of orientations of an n-element set E as the set of generators of \wedge^n\mathbb{Z}^E. For n\geqslant2, this set is that of the conjugation classes under the alternating group of total orders on E.
If, to each point y of \widetilde{Y}^n, we associate the set of n local components of Y that contain the image in X of a neighbourhood of y in \widetilde{Y}^n, then we define on \widetilde{Y}^n a local system E_n of n-element sets. The local system of orientations of these sets is a \mathbb{Z}/(2)-torsor. This torsor defines, via the inclusion of \mathbb{Z}/(2) into \mathbb{C}^*, a complex local system \varepsilon^n of rank 1 on \widetilde{Y}^n, endowed with an isomorphism (\varepsilon^n){\otimes2}\simeq\mathbb{C}. We have \varepsilon^n \simeq \bigwedge^n \mathbb{C}^{E_n}.
Locally on \widetilde{Y}^n, \varepsilon^n is endowed with two opposite isomorphisms \pm\alpha\colon\varepsilon^n\xrightarrow{\sim}\mathbb{C}. We set \varepsilon_\mathbb{Z}^n = \alpha^{-1}((2\pi i)^{-n}\mathbb{Z}). There is a way of seeing \varepsilon^n, endowed with \varepsilon_\mathbb{Z}^n, as a twisted version of \mathbb{Z}(-n) over \widetilde{Y}^n.
We denote by \varepsilon_X^n (resp. by (\varepsilon_X^n)}) the direct image of \varepsilon^n (resp. of \varepsilon_\mathbb{Z}^n) under the mat from \widetilde{Y}^n to X. We have \varepsilon_X^n \simeq \bigwedge^n\varepsilon_X^1 \qquad\text{for }n\geqslant0. \tag{3.1.4.1}
If Y is the union of distinct smooth divisors (Y_i)_{i\in I}, then the choice of a total order on I trivialises the \varepsilon^n.
We denote by W_n(\Omega_X^p\langle Y\rangle) the sub-module of \Omega_X^p\langle Y\rangle consisting of linear combinations of products \alpha \wedge \frac{\mathrm{d}t_{i(1)}}{t_{i(1)}} \ldots \wedge \frac{\mathrm{d}t_{i(m)}}{t_{i(m)}} \qquad\text{for }m\leqslant n with \alpha holomorphic and the t_{i(j)} local coordinates of the distinct local components Y_j of Y. We define the weight filtration of \Omega_X^\bullet\langle Y\rangle to be the increasing filtration by the subcomplexes W_n(\Omega_X^\bullet\langle Y\rangle). We have W_n(\Omega_X^p\langle Y\rangle) \wedge W_m(\Omega_X^p\langle Y\rangle) \subset W_{n+m}(\Omega_X^{p+q}\langle Y\rangle). \tag{3.1.5.1}
If we denote by i_n the map from \widetilde{Y}^n to X, then we can show that the mapping \alpha \wedge \frac{\mathrm{d}t_{i(1)}}{t_{i(1)}} \ldots \wedge \frac{\mathrm{d}t_{i(m)}}{t_{i(m)}} \longmapsto (\alpha|Y_{i(1)}\cap\ldots\cap Y_{i(n)})\otimes(\text{orientation }i(1)\ldots i(n)) defines an isomorphism of complexes \operatorname{Res}\colon\operatorname{Gr}_n^W(\Omega_X^\bullet\langle Y\rangle) \approx (i_n)_*\Omega_{\widetilde{Y}^n}^\bullet(\varepsilon^n)[-n] \tag{3.1.5.2} (the Poincaré residue).
The interpretation that follows, in terms of (3.1.5.2), of the Leray spectral sequence for the inclusion of X^* into X, was pointed out to me by N. Katz. It will allow us to prove a point that I had initially considered as evident (the first part of (ii) of (3.2.5)).
Every point of X admits a fundamental system of Stein open neighbourhoods whose intersections on X^* are again Stein. For a coherent analytic sheaf \mathscr{F} on X^*, we thus have \mathrm{R}^i j_*\mathscr{F}=0 for i>0. The de Rham complex \Omega_{X^*}^\bullet is thus a resolution of the constant sheaf \mathbb{C} by sheaves acyclic for the functor j_*. Then \operatorname{H}^\bullet(X^*,\mathbb{C}) \xrightarrow{\sim}\operatorname{\mathbb{H}}^\bullet(X^*,\Omega_{X^*}^\bullet) \xleftarrow{\sim}\operatorname{\mathbb{H}}^\bullet(X,j_*\Omega_{X^*}^\bullet) and the Leray spectral sequence for the morphism j can be identified with the hypercohomology spectral sequence for j_*\Omega_{X^*}^\bullet corresponding to the filtration \tau by the subcomplexes \tau_{\leqslant -n}(j_*\Omega_{X^*}^\bullet) (1.4.6).
The morphisms of filtered complexes (\Omega_X^\bullet\langle Y\rangle, W) \xleftarrow{\alpha} (\Omega_X^\bullet\langle Y\rangle, \tau) \xhookrightarrow{\beta} (j_*\Omega_{X^*}^\bullet, \tau) are filtered quasi-isomorphisms. They define an isomorphism between the Leray spectral sequence for j in complex cohomology and the hypercohomology spectral sequence of the filtered complex (\Omega_X^\bullet\langle Y\rangle, W) on X.
Proof. By (1.4.5) and (3.1.7), it suffices to prove the first claim. In (Deligne 1970, II, (6.9)) or (Atiyah and Hodge 1955), one can find a proof of the fact that \beta is a quasi-isomorphism, and thus a filtered quasi-isomorphism. We can also directly calculate the cohomology sheaves of the two sides: those of \Omega_X^\bullet\langle Y\rangle are determined by (3.1.5.2), while those of j_*\Omega_{X^*}^\bullet are the \mathrm{R}^ij_*\mathbb{C}, which can be calculated by topological methods.
For n\geqslant p, we have W_n(\Omega_X^p\langle Y\rangle) = \Omega_X^p\langle Y\rangle and so \alpha is a morphism from \Omega_X^\bullet\langle Y\rangle, endowed with \tau, to \Omega_X^\bullet\langle Y\rangle, endowed with the decreasing filtration associated to W (1.1.3). By (3.1.5.2), we have \operatorname{\mathscr{H}}^i(\operatorname{Gr}_n^W(\Omega_X^\bullet\langle Y\rangle)) = \begin{cases} 0 &\text{for }i\neq n \\\varepsilon_X^n &\text{for }i=n \end{cases} \tag{3.1.8.1} and we deduce from the first line of this equation that \alpha is a filtered quasi-isomorphism. This proves (3.1.8).
By (3.1.7), the isomorphism (3.1.8.1) defines an isomorphism \mathrm{R}^n j_*\mathbb{C} \simeq \operatorname{\mathscr{H}}^n(j_*\Omega_{X^*}^\bullet) \simeq \operatorname{\mathscr{H}}^n(\Omega_X^\bullet\langle Y\rangle) \simeq \varepsilon_X^n. \tag{3.1.8.2}
The isomorphisms (3.1.4.1) correspond, via (3.1.8.1), to the cup product.
The canonical morphism from \mathrm{R}^nj_*\mathbb{Z} to \mathrm{R}^nj_*\mathbb{C} identifies, via (3.1.8.2), the sheaf \mathrm{R}^nj_*\mathbb{Z} with (\varepsilon_X^n)} (3.1.4).
Proof. The question is local on X. We can thus suppose that X is an open polycylinder D^m, with D = \{z\in\mathbb{C}\mid |z|<1\} and also that Y=\bigcup_{k=1}^\ell Y_k with Y_k=\operatorname{pr}_k^{-1}(0). The fibre at 0 of \mathrm{R}^nj_*\mathbb{Z} is then the integer cohomology of X^*=(D^*)^\ell\times D^{m-\ell}, with D^* = \{z\in\mathbb{C}\mid 0<|z|<1\}. The space X^* has the homotopy type of a torus; its cohomology is thus torsion free, and the cup product defines isomorphisms \bigwedge^n(\mathrm{R}^1j_*\mathbb{Z}) \xrightarrow{\sim}(\mathrm{R}^nj_*\mathbb{Z})_0. It thus suffices to prove (3.1.9) for n=1.
The integer homology \operatorname{H}_1(X^*) is generated by the loops \gamma_k that go around the various Y_k. We have \oint_{\gamma_k} \frac{\mathrm{d}z_k}{z_k} = \pm2\pi i and so the integer cohomology is generated by the \frac{1}{2\pi i}\frac{\mathrm{d}z_k}{z_k}, and this proves (3.1.9).
Let \mathscr{F} be a coherent analytic sheaf on X^*, given as the restriction to X^* of a coherent analytic sheaf \mathscr{F}' on X. We define the meromorphic direct image j_*^\mathrm{m}\mathscr{F} of \mathscr{F} to be the inductive limit j_*^\mathrm{m}\mathscr{F} = \varinjlim\mathscr{F}'(nY).
Locally on X, Y is the sum of a finite family (Y_i){iI}} of smooth divisors, and we define the pole-order filtration P on j_*^\mathrm{m}\mathcal{O}_X^* by the equation P^p(j_*^\mathrm{m}\mathcal{O}_{X^*}) = \sum_{n\in A_p} \mathcal{O}_X\left(\sum(n_i+1)Y_i\right) \tag{3.1.10.1} where A_p = \left\{(n_i)_{i\in I} \mid \sum_i n_i\leqslant -p \text{ and } n_i\geqslant0\text{ for all } i\right\}. This construction globalises by endowing j_*^\mathrm{m}\mathcal{O}_{X^*} with an exhaustive filtration such that P^p=0 for p>0.
We define the pole-order filtration of the complex j_*^\mathrm{m}\Omega_{X^*}^\bullet=j_*^\mathrm{m}\mathcal{O}_{X^*}\otimes\Omega_X^\bullet to be the filtration P^p(j_*^\mathrm{m}\Omega_{X^*}^k) = P^{p-k}(j_*^\mathrm{m}\mathcal{O}_X)\otimes\Omega_X^k. \tag{3.1.10.2}
The filtration P induces, on the subcomplex \Omega_X^\bullet\langle Y\rangle of j_*^\mathrm{m}\Omega_{X^*}^\bullet, the stupid filtration by the \sigma_{\geqslant p}(\Omega_X^\bullet\langle Y\rangle), which we also call the Hodge filtration F.
The inclusion morphism (\Omega_X^\bullet\langle Y\rangle, F) \to (j_*^\mathrm{m}\Omega_{X_*}^\bullet, P) is a filtered quasi-isomorphism.
Proof. This statement was suggested to me by (Griffiths 1969). A proof can be found in (Deligne 1970, II, (3.13)).