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By demand I expand a little on my answer. The holomorphic Lefschetz fixed point formula (aka the Woods-Hole formula) considers an endomorphism $f\colon M \to M$ of a smooth and compact complex manifol …

No, any line bundle with a flat connection has a trivial rational Chern class. Now, take any smooth connected projective variety $X$ for which the Chern classes of line bundles form a group of rank $r …

It seems unlikely that there is something nice. An interpretation should preferably be invariant under linear coordinate transformations and a homogeneous component itself isn't, it is only invariant …

Serre's construction (or, I believe, a version of it which is enough here) takes a commutative ring $R$ and an $R$-category $C$ for which all idempotents have kernels. (An $R$-category is a category e …

No it is not possible (see for instance Thm II:2.4 of Shafarevich: Basic Algebraic Geometry) which says that the exceptional locus is always of codimension $1$ provided the target is smooth. The cruci …

Yes, assuming at least that the cone point is the only singular point. Hence any isomorphism will preserve the ideal of that point which is the ideal of elements of positive degree.(I think that the c …

I haven't checked the indexing completely but I think Thm 1.1 and Prop. 1.3 of SGA 7 II:Exp II does what you want (with possibly a small number of exceptions if I've got the indexing a little bit wron …

You most certainly need to assume that $S$ is proper (or something similar); even when $X$ is the spectrum of a field but $S$ is affine, say, you will not get what you want (unless you accept somethin …

Two simple examples where $Y=\mathrm{Spec}k$ a point: Consider $0\to\mathcal{O}(-2)\to\mathcal{O}(-1)^2\to\mathcal{O}\to 0$ on $\mathbb P^1$, where the two maps $\mathcal{O}(-1)\to\mathcal{O}$ are g …

It is not necessarily trivial in the Zariski topology. Consider for instance the plane quadric $\{x^2+sy^2+tz^2\}\subseteq \mathbb P^2\times\mathrm{Spec}\mathbb C[s,s^{-1},t,t^{-1}]$ as a family of $\ …

I think it would be difficult to give a general result that covers everything you want and can do but there is a collection of techniques (maybe better described as a dictionary) that works in many ca …

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 …

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 …

Qing Liu's example probably works, only we don't know if an abelian variety in positive characteric is determined by its class in $K_0(\mathrm{Var}_k)$. However, we do know that in characteristic zero …