Log de Rham cohomology

Let p be a prime number. Let k be a perfect field of characteristic p. Let U be a smooth variety over k. Choose a compactification U ⊂ X over k such that X is smooth over k and such that the divisor D = X – U is a strict normal crossings divisor D = D_1 ∪ … ∪ D_n. Then we can define the log de Rham complex Ω_X^*(log D) and try to define

H^*_{dR, log}(U) = H^*_{dR}(X, Ω_X^*(log D)

I would like to know is whether there is a published/online proof of the independence of the choice of the compactification provided one has a sufficiently strong form of resolution of singularities (RS). I did the calculation myself on a napkin (see explanation below), but it’d be great if somebody can point to a more honest writeup. Of course I searched the web for a while… also I think one of my students told me this calculation works and somebody else (maybe Illusie himself?) told me a student of Illusie worked on a better version of this a while ago? Does this ring a bell?

What do I mean by RS? Well, I mean that we can go from any compactification to any other one by a sequence of good blowing ups and good blowing downs. A good blowing up of an X as above is one which has an irreducible smooth center Z contained in the boundary (of course) such that for any I ⊂ {1, …, n} the intersection Z ∩ ⋂_{i in I} D_i is either all of Z or empty or a smooth closed subscheme of Z of codimension equal to the number of elements of I. (Aside: I think that embedded resolution of singularities will imply RS.)

Assuming RS the independence claimed above boils down to a local calculation. Think of affine r + s space A^{r + s} as the spectrum of k[x_1,…,x_r,y_1,…,y_s] with divisor D given by x_1…x_r = 0. A good blowing up looks etale locally like the blowing up of A^{r + s} in the ideal generated by x_1,…,x_{r’}, y_1, …, y_{s’} for some 1 ≤ r’ ≤ r and 0 ≤ s’ ≤ s. This blowing up is clearly equal to the blowing up of A^{r’ + s’} times the other factors. By a suitable Kunneth argument for logarithmic complexes this reduces us to the case r = r’ > 0 and s = s’. OK, so denote b : W → A^{r + s} this blowing up with exceptional divisor E isomorphic to P^{r + s – 1}. What I did was compute the cokernels of the maps

b^*Ω^i_{A^{r + 1}}(log D) → Ω^i_W(log b^{-1}D)

for all i. My napkin calculation for i = 1 showed the cokernel to be equal to Q = O_E(-1)^s. For notational convenience set S = O_E^r. Then for i = 2 my calculation gave a cokernel with a filtration having 3 graded pieces, namely

S ⊗ Q, ∧^2(Q)(1), ∧^2(Q).

For i = 3 we get graded pieces

∧^2(S) ⊗ Q, S ⊗ ∧^2(Q)(1), S ⊗ ∧^2(Q), ∧^3(Q)(2), ∧^3(Q)(1), ∧^3(Q).

And so on. If correct (caveat emptor), these cokernels have zero cohomology in all degrees. (Note that Q has rank s which is ≤ dim(E) because r > 0.) Hence the displayed arrow defines an isomorphism on cohomology and we get the desired isomorphism on logarithmic de Rham cohomology because Rb_* b^* = id on locally free coherent modules.

Looking forward to your comments!

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