# Do Poincaré residue and integrable log connection commute?

Here are some basic notations and definitions: (ignore this part if familiar)

1.Let $$(X,omega)$$ be a compact Kähler manifold of dimension $$n$$, and let $$D=sum_{i=1}^r D_i$$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $$D_i$$ intersecting each other transversally in $$X$$.
Let $$mathcal{V}$$ be a locally free coherent sheaf on $$X$$ and let
$$nabla:mathcal{V}to Omega^1_X(log D)otimesmathcal{V}$$ be a $$mathbb{C}$$-linear map satisfying
begin{align}nabla(fcdot e)=fcdotnabla e+dfotimes e.end{align}
One defines
$$nabla_a:Omega^a_X(log D)otimes mathcal{V}to Omega^{a+1}_X(log D)otimes mathcal{V}$$
by the rule
$$nabla_a (omegaotimes e)=domegaotimes e+(-1)^a omegawedge nabla e.$$
We assume that $$nabla_{a+1}circnabla_a=0$$ for all $$a$$. Such $$nabla$$ will be called an integrable logarithmic connection along $$D$$, or just a connection.
When $$omegain A^{0,q}(X,Omega_X^p(log D))$$, by taking $$diamondsuit=nabla+bar{partial}_mathcal{V}$$, we have
$$nabla(omegaotimes e)=partialomegaotimes e+{(-1)}^{p+q}omegawedgenabla(e),$$
$$barpartial_mathcal{V}(omegaotimes e)=barpartialomegaotimes e.$$
2.For any $$alphain A^{0,q}(X,Omega^p_X(log D))$$, we can write
begin{align}label{expression of alpha} alpha=alpha_1+alpha_2wedgefrac{dz^1}{z^1} end{align}
on $$U$$, where $$alpha_1$$ does not contain $$dz^1$$ and $$alpha_2in A^{0,q}(U,Omega^{p-1}_X(log sum_{i=2}^rD_i))$$. Denoting by
$$iota_{D_i}:D_ihookrightarrow X$$
the natural inclusion, without loss of generality, we may assume that $${z^1=0}=D_1cap U,$$ and set
$$label{defnRes} text{Res}_{D_1}(alpha)=iota^*_{D_1}(alpha_2)$$
on $$D_1cap U$$. Then $$text{Res}_{D_1}(alpha)$$ is globally well-defined and
$$text{Res}_{D_1}(alpha)in A^{0,q}(D_1,Omega^{p-1}_{D_1}(log sum_{i=2}^r D_icap D_1)).$$
Set the so-called Poincaré residue as
$$text{Res}(alpha)=sum_{i=1}^rtext{Res}_{D_i}(alpha).$$
3.For an integrable log connection $$nabla$$, we also define the residue map along $$D_1$$ to be the composed map:

$$text{Res}_{D_1}(nabla):mathcal Vxrightarrow{nabla} Omega_X^1(log D)otimesmathcal Vxrightarrow{text{Res}_{D_1}otimestext{id}_mathcal V}mathcal O_{D_1}otimesmathcal V$$

Question:
Then now, for $$betain A^{0,q}(X,Omega_X^p(log D)otimesmathcal V),$$
we first have the composed map acting on $$beta:$$
$$text{Res}_{D_1}(nabla):A^{0,q}(X,Omega_X^p(log D)otimesmathcal V)xrightarrow{nabla} A^{0,q}(X,Omega_X^{p+1}(log D)otimesmathcal V)xrightarrow{text{Res}_{D_1}otimestext{id}_mathcal V} A^{0,q}(D_1,Omega_{D_1}^{p}(log(sum_{i=2}^{r} ({D_i}cap {D_1}))otimesmathcal V)$$

Naturally, we also can let $$text{Res}_{D_1}otimestext{id}_mathcal V$$ first act on $$beta$$, as $$gamma=(text{Res}_{D_1}otimestext{id}_mathcal V)(beta)in A^{0,q}(D_1,Omega_{D_1}^{p-1}(log(sum_{i=2}^{r} ({D_i}cap {D_1}))otimesmathcal V).$$

Then my question is, can we find some induced integrable log connection $$nabla^prime$$ acting on $$gamma$$ satisfying $$nabla^prime(gamma)=text{Res}_{D_1}(nabla)(beta),$$
i.e., do $$text{Res}otimestext{id}_mathcal V$$ and $$nabla$$ commute in some sense?

Any suggestion and references will be appreciated! Thanks a lot!

MathOverflow Asked on November 22, 2021

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