Intersection complex via residues

1.   Introduction

Intersection homology theory is a generalization of singular homology for singular algebraic varieties. In Ref. [1], Sheng and Zhang established a positive characteristic analog of an intersection cohomology theory for polarised variations of Hodge structures and proposed an algebraic definition of the intersection complex, but with the help of coordinate systems. Here, we provide an intrinsic definition of the intersection complex via residues and provide a geometric description of it.

The remainder of this paper is organized as follows. Section 2 establishes notations and presents key definitions. Section 3 provides the main theorem and its proof. Finally, in Section 4, an explicit computation following the spirit of proof in surface case is made, and a counterexample is discussed.

2.   Intersection complex

Let $\left(X, D \right)$ be a smooth scheme over a regular locally Noetherian scheme S with a reduced smooth normal crossing divisor $D = \displaystyle \sum\limits_{i \in {\cal I}} {{D_i}}$, where ${\cal{I}}$ is a finite index set, and $\varepsilon$ be a locally free coherent sheaf with an integrable logarithmic λ-connection $\nabla$ along ${D}$.

We introduce some natural morphisms of log-differential sheaves before providing our definitions.

Suppose X is of relative dimension n over S. Owing to smoothness of $X$ and the definition of simple normal crossing divisors, for any $x \in X$, there exists a neighborhood $U$ of $x$ such that we can find a coordinate system $\left( t_{1}, \cdots ,t_{r} ;\; t_{r+1},\cdots , t_{n} \right)$ such that $D \cap U$ is defined by the equation $t_{1} \cdot t_{2} \cdots t_{r} = 0$. As an immediate result, $\varOmega_{U/S}^{1}({\rm log}\,D)$ admits an ${\cal{O}}_{U}$ basis

Moreover, it induces a free system of generators for $\varOmega_{U/S}^{a}({\rm log}\,D)$.

For $1 \leqslant i,j \leqslant r$ and $a \geqslant 1$, we define

where $\phi^{\prime}$ lies in span $\omega_{I}$ with $i \notin I$.

One can consider $\beta_{i}^{a}$ as taking the residual part of a log differential form along $D_{i}$, and $\gamma_{j}^{a}$ is the restriction of the $D_{i}$ regular log differential forms to $D_{i}$. Obviously, $\beta_{i}^{a}$ and $\gamma_{j}^{a}$ are surjective and independent of the coordinate system, respectively. For simplicity, we omit the upper symbol $a$.

Clearly, for any log connection $\nabla$, the composite map $(\beta_{i} \otimes Id) \circ \nabla$ factors through $\gamma_{i}$

We call the second map the residue map of $\nabla$ along $D_{i}$, and denote it as ${\rm Res}_{i}\left( \nabla\right)$.

We can generalize morphisms above to the multi-indices case as follows. For a subset $I = \left\lbrace j_{1},\cdots , j_{a}\right\rbrace \subseteq \left\lbrace 1,2,\cdots, r\right\rbrace$ with $j_{1} < j_{2} < \cdots < j_{a}$, set $D_{I} = \cap_{i\in I} D_{i}$, and define the residue Res_I of the connection $\nabla$ along $D_{I}$ as follows:

We define $\beta_{I}$ and $\gamma_{I}$ in a similar manner.

The following diagram naturally commutes.

where $l_{i}:\varepsilon \rightarrow \varepsilon|_{D_{i}}$ is the canonical restriction map.

Now we can define the intersection complex. Set

then the following descending chain gives rise to a stratification of $X$:

And let $j_s:U_s:=X-X_s\to U_{s-1}=X-X_{s-1}$ be the natural inclusion for $n-|{\mathcal{I}}|\leqslant s \leqslant n-1$.

Definition 2.1.　Notations as above. We inductively define res-intersection complex $IC_{r}$ as follows:

• $IC_r^*\left( {\varepsilon ,\nabla } \right){|_{{U_{n - 1}}}} = IC_r^*\left( {\varepsilon ,\nabla } \right){|_{{U_{n - 1}}}}$;

• Assume $IC_{r}^{*}(H,\nabla)(U_s)$ is defined. A section $\beta \in {j_{s}}_{*} IC_{r}^{*}(H,\nabla) (U_s)$ belongs to $IC_{r}^{*}(\varepsilon,\nabla)(U_{s-1})$ when the following two conditions are satisfied:

① $\beta$ has log pole along $D|_{U_{s-1}}$;

② ${{\rm{Res}}_{{D_J}}}\beta \in {\rm{Im}}({{\rm{Res}}_{{D_J}}}\nabla :\varepsilon {|_{{D_J}}} \to \varepsilon {|_{{D_J}}}) \otimes \varOmega _{{D_J}/S}^{l - n + s},\; \forall J \subset I$ with $\left| J \right| = n - s.$

Then we provide a geometric description of res-intersection complex in the sequel of this section. For any subset $I$ of $\mathcal{I}$, let $D_{I} = \cap_{i\in I} D_{i}$, $D_{I}^{*}=D_{I} - \cup_{j\notin I}\left( D_{i}\cap D_{I} \right)$ and let $D_{\emptyset}^{*}= X-D$. Set theoretically, we have $X ={\coprod}_{I \subset {\mathcal{I}}} {D_I^{*}}$. Each $D_{I}^{*}$ is a locally closed subspace of $X$, and thus we can endow $D_{I}^{*}$ with reduced subscheme structure.

Proposition 2.1. If ${{\text{Res}}_i}(\nabla)$ are bundle morphisms for all $i \in \left\lbrace 1,2,\cdots,r\right\rbrace$, then the res-intersection complex is a complex of locally free sheaves if it is restricted to each stratum $D_{I}^{*}$, where $I$ is an index subset of $\left\lbrace 1,2,\cdots,r\right\rbrace$ and $D_{I}^{*}$ is endowed with a reduced subscheme structure.

We employ the following lemma to prove Proposition 2.1, see Ref. [2] for details.

Lemma 2.1. Let $X$ be a reduced Noetherian scheme, and let ${\cal{F}}$ be a coherent sheaf on $X$. Consider the function

where $k(x) = {\cal{O}}_{x}/m_{x}$ is the residue field at point $x$. If $\phi$ is constant, then ${\cal{F}}$ is locally free.

Proof of Proposition 2.1. Consider the reduced scheme $D_{I}^{*}$ and its associated coherent sheaf $IC^{i}_{I} = IC_{r}^{i}\left( \mathit{X},\varepsilon \right)|_{D_{I}^{*}}$. Because of the assumption the divisors are reduced, Lemma 2.1, the proposition is proven if we can show that the dimension of the fibre of sheaf, which is $\phi_{IC^{i}_{I}}(x)$, is constant over $D_{I}^{*}$.

For each $x \in D_{I}^{*}$, $IC_{r}^{*}\left( X,\varepsilon \right)(x)$ is an $\varOmega_{X,x}^{*}$ module spanned by basis

where $\overline{\rm{Res}}_{J}(\nabla)$ represents the restriction of ${\rm {Res}}_{J}(\nabla)$ on fibre. Due to that ${\rm Res}_i(\nabla) \;\; \text{are bundle morphisms,}\;\; \phi_{IC^{i}_{I}}$ is constant if we restrict it to each degree i and stratum $D_{I}^{*}$. Therefore, $IC_{r}^{*}\left( X,\varepsilon \right)|_{{D_{I}}^{*}}$ is a complex of locally free sheaves.

Proof of Lemma 2.1. It is a local problem, we may assume $X = {\rm{Spec}} \, A$ and ${\cal{F}} = {\tilde{M}}$, where $A$ is a reduced commutative local ring with maximal ideal $\mathfrak{m}$ and M is a finite A-module.

We only have to show M is a free A-module. Assume that $k({\mathfrak{m}})$ vector space $M/{\mathfrak{m}}M$ has dimension n. We use Nakayama’s lemma to lift the basis for $M / {\mathfrak{m}}M$ into a set of generators ${{m}_{1}},{{m}_{2}},\cdots ,{{m}_{n}}$. It is sufficient to demonstrate that m_i is linearly independent. Suppose that ${\sum}_{i} a_{i} m_{i} = 0$, where $a_{i} \in A$. In addition, a_i must lie in ${\mathfrak{m}}$ for all i, because the generators $m_{i}$ form the basis of the fibre $M / {\mathfrak{m}}M$. Choose ${\mathfrak{q}} \in {\rm{ Spec }}\, A$ arbitrarily; then, the images of $m_{i}$ in $M_{\mathfrak{q}} /{\mathfrak{q}}M_{\mathfrak{q}}$ generate vector space. In addition, $\phi$ is constant, implying that they are, in fact, a basis, similarly to $a_{i} \in {\mathfrak{q}}$ for all $i$.

Therefore, $a_{i}$ lies in the intersection of the prime ideals of $A$, which is the nilradical of $A$, and thus $a_{i} = 0$ because $A$ is assumed to be reduced. This completes this proof.

It is interesting to investigate the case where $(\varepsilon,\nabla)$ comes from the polarized variation of Hodge structures. Let us consider a quick recall of this (cf. Ref. [3]). If $X$ is a complex variety, and $E$ is a local system over $X-D$ underlies a polarized variation of Hodge structures, then we obtain a vector bundle $(\varepsilon,\nabla)$ equipped with a flat connection via a Riemann-Hilbert correspondence over $X-D$. There is a canonical extension of $\varepsilon$ to a vector bundle with a logarithmic flat connection over $X$, with the residue of the connection along divisor $D_{i}$ being the log of the monodromy of the divisor (up to a scalar), which we denote as $N_{i}$. It can be observed that $N_{i}$ is topologically defined.

In Refs. [4, 5], the intermediate extension complex can be fibre-wisely expressed as follows: for $x \in D_I^*$ and a set of coordinates $z_{i}$, the fibre of intermediate extension complex at $x$ is an $\varOmega _{{X,x}}^{*}$ sub-module generated by the sections $\tilde v{ \wedge _{j \in J}}\dfrac{{{\rm{d}}{z_j}}}{{{z_j}}}$ for $v \in {N_J} \varepsilon$ and $J \subset I$. The differential map of the complex at fibre is defined as

Note that the residue of the connection $\nabla_{I}$ is exactly (up to a scalar) the endomorphism $N_{I}$ if it is restricted to the stratum $D_{I}$. It can be easily seen that the res-intersection complex coincides with the intermediate extension complex. From this perspective, we provide an algebraic definition of the intermediate extension complex.

3.   Main theorem

In the following, we show that the res-intersection subcomplex above coincides with the intersection subcomplex defined in Ref. [1].

Let $X, D, \varepsilon$ be as in the previous section. Given a coordinate system

of $U$, locally we can write

due to that the set $\left\lbrace {\rm dlog}\,t_{i} ,{\rm d}t_{k}|i \leqslant r,r+1 \leqslant k \leqslant n\right\rbrace$ forms a basis of the log sheaf. For subset $I =\left\lbrace j_{1},\cdots , j_{a}\right\rbrace \subseteq \cal{I}$ with $j_{1} < j_{2} < \cdots < j_{a}$, let $\nabla_{I}= \nabla_{j_{1}} \circ \nabla_{j_{2}} \circ \cdots \circ \nabla_{j_{a}}$. We can generalize diagram (1) as follows:

In Ref. [1], the intersection complex is defined as follows:

Definition 3.1. $IC^{*}\left( X,{\varepsilon} \right)$ is an $\varOmega^{*}_{U}$ graded submodule of ${\varOmega}_{X}^{*}\left( {\rm log} \, D \right) \otimes {\varepsilon}$ generated by the abelian subsheaf

where U is an open subset of $X$, and $M = \left\lbrace 1,\;2,\;\cdots,r \right\rbrace$.

Our main theorem is as as follows:

Theorem 3.1.　If ${\rm Res}_{i}(\nabla): \varepsilon|_{D_{i}} \rightarrow \varepsilon|_{D_{i}}$ are bundle morphisms for all $i \in \cal{I}$, then $IC_r^*\left( {{{X}},\varepsilon } \right) = I{C^*}\left( {{{X}},\varepsilon } \right)$.

This proof makes essential use of the weight filtration of the log complex.

Definition 3.2. Weight filtration $\mathit{W}_{\cdot}$ of the logarithmic complex is defined as follows:

We establish the following lemma for weight filtration to prove the theorem.

Lemma 3.1. ① One has exact sequence:

where $D_{(m)}$ is the disjoint union over subset $K \subset I$ of cardinal $m$ of $D_{K}: = \cap_{i\in K}D_{i}$, and $\alpha_{m}$ is the natural immersion $D_{(m)} \rightarrow X$.

② The following diagram is commutative.

where $l_{i}:\varepsilon \rightarrow \varepsilon|_{D_{i}}$ is the canonical restriction map. And if ${\rm{Re}}{{\rm{s}}_I}$ is a bundle morphism, then

Proof. It is a basic fact of weight filtration, for a rigorous proof of this lemma the reader is referred to Refs. [6, 7] .

Firstly, one have to verify that the upper arrow is well defined. That is, one have to show $\nabla_{I}\omega_{I}(\varepsilon\otimes \varOmega_{X}^{a-|I|})$ is contained in $W^{\left| I \right|}\varOmega_{X}^{a}\left( {\rm log}\,D \right) \otimes \varepsilon$. It is straightforward because the source the map $\varepsilon \otimes \varOmega_{X} ^{a-|I|}$ is weight zero and the map $\nabla_{I} \omega_{I}$ is of weight $|I|$.

Note that we have $\beta_{I}( W_{\left| I \right|}\varOmega_{X}^{a}\left( {\rm log}\, D \right)) = \varOmega_{D_{I}}^{a-|I|}$, hence the vertical arrow on the right is well-defined. The commutativity of the diagram follows from restricting diagram (1) on subbundle $\varepsilon\otimes \varOmega_{X}^{a-|I|} \subset \varepsilon \otimes \varOmega_{D_{I}/S}({\rm log }((D-\sum_{i\in I}D_{i})|_{D_{1}})$. It remains to show Eq. (6). It is easy to see the sheaf on right side is contained in left side. By the commutativity of diagram (1) again, one has the left side of Eq. (6) is contained in

Therefore, Eq. (6) follows from the following claim.

Claim. If ${\rm Res}_{I}$ is a bundle morphism, then we have the following equation:

For the $"\supseteq"$ direction, it is obvious. For the other direction, the sheaf ${\rm Res}_{I}(\nabla)(\varepsilon|_{D_{I}})$ is locally free due to the assumption that ${\rm Res}_{I}$ is a bundle morphism, thus it has no torsion along $D_{j} \cap D_{I}$, where $j\in {\cal{I}} - I$. This completes the proof of the claim.

Remark 3.1. The first sublemma illustrates that weight equals to the number of times of the irreducible components of the divisor intersect (in other words, the depth of the stratification). The second provides a local description of the weight filtration along divisor in terms of the coordinates.

We now return to the proof of the main theorem.

Proof of Theorem 3.1. Without loss of generality, we assume that $r = n$.

Clearly, we have $IC^{*}\left( \mathit{X},\varepsilon \right) \subset IC_{r}^{*}\left( \mathit{X},\varepsilon \right)$, because the restriction of $\nabla_{i}$ on $D_{i}$ is exactly the residue map ${\rm Res}_{i}$.

Conversely, we consider any $s \in IC_{r}^{*} \left( \mathit{X},\varepsilon \right) \left( U\right) \bigcap \varOmega_{X}^{a}\left( {\rm log}\, D\right) \otimes \varepsilon\left( U\right)$. Taking $a = |I|$ in the diagram (5), we will obtain a section $e \in \varepsilon (U)$ such that ${\rm Res}_{I} \circ \gamma_{I} (e) = \beta_{I}(s)$, by the comutitivity of the diagram we have $s-\nabla_{I}(e)\otimes \omega_{I} \in \varepsilon \otimes \ker \beta_{I}$, and we denote it as $s_{1}$. Taking $m = a$ in the exact sequence (4), we know that $\ker\beta_{I}$ is equal to $W^{a-1}(\varOmega_{X}^{a}\left( {\rm log}\, D \right) )$; hence, $s_{1} \in \varepsilon \otimes W^{a-1}(\varOmega_{X}^{a}\left( {\rm log}\, D \right) )$.

Replacing $s$ with $s_{1}$, by the definition of res-intersection complex, one has $l_{I}\otimes\beta_{I}(s_{1}) \in {\rm Im}({\rm Res}_{D_I}\nabla:\varepsilon|_{D_I}\to \varepsilon|_{D_I}) \otimes \varOmega_{D_{I}/S}^{a} \left({\rm log }\left( D-\sum_{i \in I}D_{i} \right)\vert_{D_{I}} \right)$ and one has $s_{1} \in \varepsilon \otimes W^{a-1}(\varOmega_{X}^{a}\left( {\rm log}\, D \right) )$ by the construction. Then we can chase in the diagram (5) for index set $I$ with $|I|=a-1$, by Eq. (6), we obtain a section $e_{I} \in \varepsilon \otimes \varOmega_{X/S}^{a-1}$ such that $s_{1}-\nabla_{I}(e_{I})\omega_{I} \in \varepsilon \otimes \ker \beta_{I}$, therefore, we have

where $I$ is a of cardinal $(a-1)$. Therefore, we have $s_{2} \in W^{a-2}(\varOmega_{X}^{a}\left( {\rm log}\, D \right) )\otimes \varepsilon$ by (4).

Repeat the processes above $a$ times, we obtain $s_{a} \in \varepsilon \otimes W^{0}=\varOmega_{X}^{a} \otimes E$ and $e_{I} \in \varepsilon \otimes \varOmega_{X/S}^{a-|I|}$ with $I \subseteq \cal{I}$, satisfying

which implies $s \in IC(X,\varepsilon) \bigcap \varOmega_{X}^{a}\left( {\rm log}\, D\right) \otimes \varepsilon$. This completes our proof.

4.   Surface case

In this section, we provide an explicit calculation for the surface case and present an example to the main theorem without bundle morphism conditions.

Let $X ={\rm{ Spec }}(k[t_{1},t_{2}])$ be a surface and let $D=D_{1}+D_{2}$ defined by $t_{1}t_{2}=0$. The divisor gives rise to a stratification the surface as $X=D_{\emptyset}^{*} \coprod D_{1}^{*} \coprod D_{2}^{*} \coprod D_{12}^{*}$. With the help of the coordinates $t_{i}$, we can write $\nabla=\nabla_{1} {\rm dlog} \, t_{1} + \nabla_{2} {\rm dlog} \, t_{2}$.

① $IC_{r}^{0}\left( X,\varepsilon \right) = IC^{0}\left( X,\varepsilon \right)$, because both are equal to $\varepsilon$.

② In the one-degree term, note that the sections of the sheaf $IC^{1}\left( X,\varepsilon \right)$ are of the form:

Let $s \in IC\left( X,\varepsilon \right)^{1}(X)$. To verify $s\in IC_{c}^{1}\left( X,\varepsilon \right)$, we aim to find $e_{1}$ and $e_{2}$. By definition, $s$ satisfies $\beta_{i}(s) \in {\rm Im}\left( {\rm Res}_{D_{i}} \right)$: Consider a commutative diagram

The first vertical arrow is surjective, so we can find a section $e_{1} \in \varepsilon$ such that $s-\nabla(e_{1}){\rm dlog}\,t_1$ in the kernel of the second vertical, which is $\varepsilon \otimes \varOmega_{X}^{1} \left( {\rm log} \left( D_{2} \right) \right)$, replacing $1$ with $2$, and we obtain a section $e_{2}$ of $\varepsilon$, by the exact sequence (4), $s_{1} = s-( \nabla_{1}(e_{1}) {\rm dlog} \, t_{1}+\nabla_{2}(e_{2}) {\rm dlog} \,t_{2})$ is of weight zero. In other words, it is regular, which allows us to write:

where $e_{1},e_{2},f_{1},f_{2} \in \varepsilon(\mathit{U})$.

③ Using the same pattern, a section $\omega$ in $IC^{2}\left( X,\varepsilon \right)(X)$ is of the form

by definition. For any section $s \in IC_{r}^2(X, \varepsilon )(X)$, we aim to get section $e_{12}, e_{1}, e_{2}, f$. By (5), we have the following commutative diagram:

where both vertical arrows are surjective, we obtain $e_{12} \in \varepsilon$ such that $s_{1}: = s - ( \nabla(e_{12}) ) \omega_{12} \in \ker \varepsilon\otimes \beta_{12}$, which is of weight one by short exact sequence (4).

Then consider diagram (5) and set $a = 2$, $I = \left\lbrace1,2 \right\rbrace$. Using the same argument as above, we obtain $\tilde{e}_{1} \text{ and } \tilde{e}_{1} \in \varepsilon \otimes \varOmega_{X}^{1}$, such that $s_{1}- \nabla_{1}(e_{1}) {\rm dlog}\,t_{1} \wedge {\rm d}t_{2}-\nabla_{2}(e_{2}) {\rm dlog}\,t_{2}\wedge {\rm d}t_{1}$ in $\ker ( \beta_{2} \;\oplus\; \beta_{2})$, which is exactly $\varOmega_{X}^{2}\otimes \varepsilon$ by (4). Therefore, $e_{12},e_{1},e_{2},f$ are the section we want. This completes our proof.

In the sequel of the section we will present an example, which is provided by Zebao Zhang, to show that the main theorem will be wrong if the residue morphisms are not assumed to be bundle morphisms. Let $k$ be a perfect field of character $p$, $(X,D)$ is as above. Define logarithmic connection over

One can verify that $\nabla$ is integral, and the residue morphisms are as follows:

One can see that ${\rm{Res}}_{1}(\nabla)$ has torsion at $t_{2}=0$, hence it is not a bundle morphism. By definition, we have $IC^{2}=(t_{2}^{p} \cdot {\rm{dlog}}\, t_{1} \wedge {\rm{d}}t_{2}) {\cal{O}}_{X} + ({\rm{d}}t_{1} \wedge {\rm{d}}t_{2})\cal{O}_{X}$. Consider the section $t_{2}^{p} {\rm dlog }\,t_{1} \wedge {\rm dlog }\,t_{2}$, it is a section in $IC_{r}^{2}$ but not in $IC^{2}$.

Acknowledgements

I would especially like to thank Prof. Mao Sheng for his suggestions. And I would also like to thank Dr. Zebao Zhang for correcting the article.

Conflict of interest

The author declares that he has no conflict of interest.