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The same thing is true in positive characteristic, the degree of $c_n$ is equal to the Euler characteristic (except if you consider de Rham cohomology where it only is the Euler characteristic mod $p$ …

Yes, it is true, though an algebraic proof seems (there may be a simpler proof however) somewhat tricky. Such a line bundle lies in $\mathrm{Pic}^\tau(X)$. This is a general fact as a line bundle l …

You have to be careful with what you mean here. As your equations involve complex conjugation they do not define a complex variety. They do define a real algebraic variety. However, then you have to b …

I would be a little bit nervous about things when the group scheme is not smooth (there may not be any problems though) but you are interested in a smooth case anyway. To me it seems that the most nat …

Let me try an argument different from Christian's: $\sigma$ does not act freely as $\chi(\mathcal O_X)=1$ and hence not divisible by $2$. At a fixed point $x$, $\sigma$ acts by $\pm1$ on the fibre of …

If we have a central extension of group schemes $1\rightarrow B \rightarrow C\rightarrow A\rightarrow1$ with $A$ abelian, then we get a commutator mapping $\Lambda^2A\rightarrow B$ (of sheaves as $\La …

This is not an answer to the questions but some general comments. One should be aware that the relation between vector fields and Hasse derivations in characteristic $p$ is not at all analogous to the …

This is an attempt to realise Sándor's program of getting an example based on Kodaira vanishing or non-vanishing varying in a family. It will be done by keeping the surface fixed but varying the line …

We have that $\mathcal F^\ast$ is, by the pairing induced by the exterior algebra, canonically isomorphic to $\Lambda^{d-1}\mathcal F\bigotimes(\Lambda^d\mathcal F)^{-1}$. Now, in general if $\mathcal …

One example is given by Enriques surfaces in characteristic $2$. There are three types depending on the value of $\mathrm{Pic}^\tau$ (as a group scheme) which can be either $\mathbb Z/2$, $\mu_2$ or $ …

Certainly the fact that the ring of differential operators is non-Noetherian is an inconvenience but it is not clear if it is more than that. For instance one can define the notion of holonomic module …

I think that $\mathcal F$ is indeed locally free of rank $n$: Pick a point $x\in X$. It will be enough to show that there is a neighbourhood of $x$ on which $\mathcal F$ is free of rank $n$. Now, the …

I am not going to add any new examples but suggest a systematic way of looking at examples. If one looks at special phenomena in characteristic $2$ one can classify them as follows (though this divisi …

a): An element of $C^\times$ can be thought of as a pair $(a,b)$ of elements of $C$ with $ab=1$. This gives a) by applying of existence extension to $a$ and $b$ and unicity to $ab$ and $1$. b): The r …

If by "Grothendieck spectral sequence" you mean the spectral sequence associated to the composite of functors (fulfilling the Grothendieck condition) then I am skeptical as to whether this is possible …