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I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:

  • $H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^{\bullet}_{\text{dR}}$ is a Weil cohomology theory (in characteristic zero), which implies the result if $X$ is a smooth, projective curve over $\text{char}(k)=0$.

  • Before taking cohomology, we also have the vanishing $\Omega^i_{X/k}=0$ for $i>1$.

What I would like to know is under what conditions these vanishing statements remain true when allowing one or more of the following hypothesis relaxations:

  • Smooth $\to$ singular (or nonsmooth) (even reducible, but still connected)

  • Projective $\to$ quasiprojective

  • Characteristic $0$ $\to$ arbitrary characteristic.

Unfortunately I've been unsuccesful in finding proofs or references, or in proving statements myself. So I would appreciate some help.

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    $\begingroup$ For $X$ a curve with one ordinary double point $p$, $\Omega ^2_{X/k}$ is nonzero (it is isomorphic to $k(p)$). $\endgroup$ – abx Aug 23 '19 at 16:16
  • $\begingroup$ @abx ok, thank you. My calculations suggested this, but I was unsure of myself. And how about the cohomology? Is there anywhere I can see these calculations carried out? $\endgroup$ – Somatic Custard Aug 23 '19 at 16:19
  • $\begingroup$ See this MO question. You won't find many references, because the approach in the smooth case clearly does not work in the singular case. For a correct approach, see Hartshorne's paper in Publ. IHES 45 (1975). $\endgroup$ – abx Aug 23 '19 at 16:38
  • $\begingroup$ I am aware of that MO question and that paper, but so far I haven't been able to extract the desired information from them. $\endgroup$ – Somatic Custard Aug 23 '19 at 17:08

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