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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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