Intersection complex via residue

A normal crossing divisor gives rise to a stratification of a smooth scheme, and a logarithmic connection of a vector bundle along the divisor induces residue maps along each stratums. Abstract: We provide an intrinsic algebraic definition of the intersection complex for a variety.


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.

Intersection complex
Let be a smooth scheme over a regular locally Noetherian scheme S with a reduced smooth normal crossing divisor , where is a finite index set, and be a locally free coherent sheaf with an integrable logarithmic λconnection along . We introduce some natural morphisms of log-differential sheaves before providing our definitions.

X
x ∈ X U x (t 1 , · · · , t r ; t r+1 , · · · , t n ) D ∩ U t 1 · t 2 · · · · · t r = 0 Ω 1 U/S (log D) O U Suppose X is of relative dimension n over S. Owing to smoothness of and the definition of simple normal crossing divisors, for any , there exists a neighborhood of such that we can find a coordinate system such that is defined by the equation . As an immediate result, admits an basis .
Moreover, it induces a free system of generators for .
One can consider as taking the residual part of a log differential form along , and is the restriction of the regular log differential forms to . Obviously, and are surjective and independent of the coordinate system, respectively. For simplicity, we omit the upper symbol .
Clearly, for any log connection , the composite map factors through We call the second map the residue map of along , and denote it as .
We can generalize morphisms above to the multi-indices case as follows. For a subset with , set , and define the residue Res I of the connection along as follows: • Res j2 (∇) • · · · • Res ja (∇) . β I γ I We define and in a similar manner.
The following diagram naturally commutes.
where is the canonical restriction map. Now we can define the intersection complex. Set X n−s = ∪ J⊂I,|J|=k ∩ j∈J D j , 1 ⩽ s ⩽ |I|, X then the following descending chain gives rise to a stratification of : And let be the natural inclusion for .
IC r Definition 2.1. Notations as above. We inductively define res-intersection complex as follows: • Assume is defined. A section belongs to when the following two conditions are satisfied: (1) has log pole along , and (2) Then we provide a geometric description of res-intersection complex in the sequel of this section. For any subset of , let , and let . Set theoretically, we have the equation . Each is a locally closed subspace of , and thus we can endow with reduced subscheme structure. Res If are bundle morphisms for all , then the res-intersection complex is a complex of locally free sheaves if it is restricted to each stratum , where is an index subset of and is endowed with a reduced subscheme structure.
We employ the following lemma to prove Proposition 2.1 [2] .
Let be a reduced Noetherian scheme, and let be a coherent sheaf on . Consider the function where is the residue field at point . If is constant, then is locally free.
Proof of Proposition 2.1. Consider the reduced scheme and its associated coherent sheaf . 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 , is constant over .
where represents the restriction of on fibre. Due to that is constant if we restrict it to each degree and stratum . Therefore, is a complex of locally free sheaves.
Proof of Lemma 2.1. It is a local problem, we may assume and , where is a reduced commutative local ring with maximal ideal and M is a finite Amodule.

M/mM
We only have to show M is a free A-module. Assume that k(m) vector space has dimension n. We use Na- kayama's lemma to lift the basis for into a set of generators . It is sufficient to demonstrate that m i is linearly independent. Suppose that , where . In addition, a i must lie in for all i, because the generators form the basis of the fibre . Choose arbitrarily; then, the images of in generate vector space. In addition, is constant, implying that they are, in fact, a basis, similarly to for all .
Therefore, lies in the intersection of the prime ideals of , which is the nilradical of , and thus because is assumed to be reduced. This completes this proof.
It is interesting to investigate the case where comes from the polarized variation of Hodge structures. Let us consider a quick recall of this (cf. Ref. [3]). If is a complex variety, and is a local system over underlies a polarized variation of Hodge structures, then we obtain a vector bundle equipped with a flat connection via a Riemann-Hilbert correspondence over . There is a canonical extension of to a vector bundle with a logarithmic flat connection over , with the residue of the connection along divisor being the log of the monodromy of the divisor (up to a scalar), which we denote as . It can be observed that is topologically defined.
In Refs. [4,5], the intermediate extension complex can be fibre-wisely expressed as follows: for and a set of coordinates , the fibre of intermediate extension complex at is an sub-module generated by the sections for and . The differential map of the complex at fibre is defined as Note that the residue of the connection is exactly (up to a scalar) the endomorphism if it is restricted to the stratum . 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.

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

X, D, ε
Let be as in the previous section. Given a coordinate system ) , In Ref. [1], the intersection complex is defined as follows: is an graded submodule of generated by the abelian subsheaf ∑ is an open subset of , and . Our main theorem is as as follows: If are bundle morphisms for all , then . This proof makes essential use of the weight filtration of the log complex. , , for 0 ⩽ m ⩽ p We establish the following lemma for weight filtration to prove the theorem. Lemma 3.1. ① One has exact sequence: where is the disjoint union over subset of cardinal of , and is the natural immersion . ② The following diagram is commutative.
where is the canonical restriction map. And if is a bundle morphism, then It is a basic fact of weight filtration, for a rigorous proof of this lemma the reader is referred to [6] and [7] .
Firstly, one have to verify that the upper arrow is well defined. That is, one have to show is contained in . It is straightforward because the source the map is weight zero and the map is of weight .
Note that we have , hence the vertical arrow on the right is well-defined. The commutativity of the diagram follows from restricting the diagram 1 on subbundle . It remais to show the equation 6. It is easy to see the sheaf on right side is contained in left side. By the commutativity of the diagram 1 again, one has the left side of the equation 6 is contained in Therefore, the equation 6 follows from the following claim.

Claim:
If is a bundle morphism, then we have the equation For the direction, it is obvious. For the other direction, the sheaf is locally free due to the assumption that is a bundle morphism, thus it has no torsion along , where . This completes the proof of the claim.
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 a loss of generality, we as- Clearly, , since the restriction of on is exactly the residue map .
Conversely, we consider any . Taking in the diagram (5), we will obtain a section such that , by the comutitivity of the diagram we have , and we denote it as . Taking in the exact sequence (4), we know that is equal to ; hence, . s s Replacing with , by the definition of res-intersection complex, one has and one has by the construction. Then we can chase in the diagram (5) for index set with , by the equation 6, we obtain a section such that , therefore, we have where is a of cardinal . Therefore, we have by (4).
Repeat the processes above times, we obtain and with , satisfying . This completes our proof.
Repeat the processes a times, finally we obtain , which implies

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 condition.