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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 $$\nabla:\mathcal{V}\to \Omega^1_X(\log D)\otimes\mathcal{V}$$ be a $\mathbb{C}$-linear map satisfying \begin{align}\nabla(f\cdot e)=f\cdot\nabla e+df\otimes 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 $$\nabla_a (\omega\otimes e)=d\omega\otimes e+(-1)^a \omega\wedge \nabla e.$$ We assume that $\nabla_{a+1}\circ\nabla_a=0$ for all $a$. Such $\nabla$ will be called an integrable logarithmic connection along $D$, or just a connection. When $\omega\in A^{0,q}(\Omega_X^p(\log D))$, by taking $\diamondsuit=\nabla+\bar{\partial}_\mathcal{V}$, we have $$\nabla(\omega\otimes e)=\partial\omega\otimes e+{(-1)}^{p+q}\omega\wedge\nabla(e),$$ $$\bar\partial_\mathcal{V}(\omega\otimes e)=\bar\partial\omega\otimes e.$$

According to Junjiro Noguchi, A short analytic proof of closedness of logarithmic forms, Kodai Math. J. 18, (1995), No. 2, 295-299, for any $\alpha\in A^{0,q}(X,\Omega^p_X(\log D)\otimes\mathcal V)$, there is a current $T_\alpha\in \mathcal{D}'^{p,q}(X,\mathcal V)$ associated with it, which is defined by $$ T_{\alpha}(\beta)=\int_X \alpha\wedge \beta,\quad \beta\in A^{n-p,n-q}(X,\mathcal V^*). $$

Now my question is, can the logarithmic connection operate on $T_{\alpha}$? How?

Here is my thought:

According to this post: Can logarithmic connection on holomorphic vector bundle induce logarithmic connection on dual bundle? , now $\nabla\beta\in A^{0,n-q}(X,\Omega_X^{n-p}(\log D)\otimes \mathcal V^*)$, we define $\nabla T_\alpha\in\mathcal{D}'^{p+1,q}(X,\mathcal V)$ by the following rule:

$$\langle\nabla T_\alpha,\beta\rangle={(-1)}^{p+q+1}\langle T_\alpha,\nabla\beta\rangle={(-1)}^{p+q+1}\int_X \alpha\wedge\nabla\beta,\quad \text{for } \beta\in A^{n-p-1,n-q}(X,\mathcal V^*). $$

So is my method right? Any advice will be appreciated. Thanks a lot.

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  • $\begingroup$ Do you mean action on the space of currents? Or currents which do come from differential forms by the formula you decribed? On all currents it is not possible, on differential form you will obtain a current yes. $\endgroup$
    – Lev Soukhanov
    Jun 18, 2020 at 13:50
  • $\begingroup$ @LevSoukhanov，I mean action on special currents those come from log forms, thanks for your help. $\endgroup$
    – Invariance
    Jun 18, 2020 at 14:04
  • $\begingroup$ So you ask whether they act on logarithmic forms? They act, yes, indeed (if you mean log poles on the same divisor as the divisor of the logartihmic connection). $\endgroup$
    – Lev Soukhanov
    Jun 20, 2020 at 11:17
  • $\begingroup$ Dear @LevSoukhanov, I feel a little confused. I ask can $\nabla$ act on $T_\alpha$ then obtain a new current $\nabla T_\alpha$? Here, $T_\alpha$ come from $\alpha\in A^{0,q}(X,\Omega_X^p(\log D)\otimes\mathcal V)$. Thanks for your help again! $\endgroup$
    – Invariance
    Jun 21, 2020 at 9:56
  • $\begingroup$ I do not understand why you insist on using currents. I think we can just try to make sense of $\nabla \alpha$ for any $\alpha \in A^{0,q}(X, \Omega_X^p(\text{log }D) \otimes \mathcal{V})$? $\endgroup$
    – Lev Soukhanov
    Jun 21, 2020 at 13:57

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