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Let $X$ be a smooth projective complex variety, and $D=\cup_{j=1}^m D_j$ a simple normal crossings divisor on $X$. Then we have an exact sequence $$0\to \Omega_X^1\to \Omega_X^1(\log D)\to \oplus_{j=1}^m i^*_j\mathcal{O}_{D_j}\to 0,$$ where $i_j:D_j\to X$ is the inclusion. Note that $H^0(i_j^*\mathcal{O}_{D_j})=\mathbb{C}$. So now I'm curious about which global residues $(r_1,\dots,r_m)\in \oplus_{j=1}^m i^*_j\mathcal{O}_{D_j}$ can come from a global section $\omega\in H^0(\Omega_X^1(\log D))$, so in other words, I want to know what the image of the map $H^0(\Omega_X^1(\log D))\to \mathbb{C}^m$ is.

This seems like a very natural question, but I cannot find any information about this. Hence some pointers to literature would be appreciated.

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    $\begingroup$ Can you please clarify your notation? By my computation, your short exact sequence is correct only if you define the closed subsets $D_j$ to be the connected components of $D$ rather than the irreducible components. Anyway, the connecting map $H^0(D,\mathcal{O}_D) \to H^1(X,\Omega^1_X)$ is the map that sends the generator of each connected component $D_i$ to its cycle class in de Rham cohomology. So the kernel of the connecting map is generated by homological equivalences among the components $D_j$ of $D$. $\endgroup$
    – Jason Starr
    Commented Jun 30, 2018 at 10:57

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Jason has given an essentially complete answer, which I'm just repeating it here, so that the question can be considered answered. (I think the sequence is OK, however. E.g. for $D=\{xy=0\}$ locally, $Res$ sends $fdx/x+gdy/y\to (f(0,y), g(x,0))$, and this clearly surjects.) The image of $H^0(\Omega_X^1(\log D)\to \mathbb{C}^m$ is the kernel of the map $\mathbb{C}^m\to H^2(X,\mathbb{C})$ sending $(r_i)\to \sum r_i[D_i]$

Added Regarding your comment, let me consider a more general situation $\omega\in H^0(\Omega^1(\log D))$ on a surface $X$. Suppose $D_1$ and $D_2$ meet at $p$. Let $\omega$ have residues $r_i$ along $D_i$. Let $\pi:Y\to X$ be the blow up at $p$, with exceptional divisor $E$. Then, according to my calculations, $\pi^*\omega$ has residues $r_i$ along the strict transforms $D_i'$ of $D_i$ and $r_1+r_2$ along $E$. So the cohomological condition still holds for $\pi^*\omega$.

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  • $\begingroup$ Could you explain the following: consider e.g. $X=\mathbb{P}^2$, $D=\{x=0\}\cup \{z=0\}$, and the logarithmic form $\omega$ with local equation $dx/x$ on $\mathbb{P}^2\setminus V(z)$. This satisfies your criterion, since it has residues $1$ and $-1$. But now blowup $Y$ in the point $[0:0:1]$, and take the pullback of $\omega$ along the blowup map. This is again a logarithmic form, but it has residues $1,-1$ along the strict transforms of $V(x)$ resp. $V(z)$, but also a residue of $1$ along the exceptional divisor. So the image in $H^2(Y)$ seems to be non-zero? $\endgroup$
    – user2520938
    Commented Jul 1, 2018 at 11:55
  • $\begingroup$ (too long to fit in 1 comment) specifically, we consider the pullback as living in $H^0(Ω^1_Y(\log D))$, where $D$ is given by the strict transforms of $V(z)$ and $V(x)$, and the exceptional divisor $E$. The image of this form under the composition $H^0(Ω^1_Y(\log D))\to \mathbb{C}^3\to H^2(Y,\mathbb{Z})$ is equal to $[E]$, but this is not zero. $\endgroup$
    – user2520938
    Commented Jul 1, 2018 at 12:18
  • $\begingroup$ Thanks for your edit, and sorry to bother you again. However, I consider the situation where $p$ lies only on $D_1$, in which case I believe the residue along $E$ would be $r_1$, and then I don't see why the cohomological condition would be satisfied? $\endgroup$
    – user2520938
    Commented Jul 1, 2018 at 22:00
  • $\begingroup$ That case is even easier, $\pi*\omega$ has residue $r_1$ along $D_1'$ and $E$. So if $r_1[D_1] +\ldots =0$, then $r_1[D_1]+r_1[E] + r_2[D_2] \ldots = \pi^*\sum r_i[D_i]= 0$ $\endgroup$
    – Donu Arapura
    Commented Jul 1, 2018 at 22:57

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