No Arabic abstract
The main purpose of this paper is to make Nakayamas theorem more accessible. We give a proof of Nakayamas theorem based on the negative definiteness of intersection matrices of exceptional curves. In this paper, we treat Nakayamas theorem on algebraic varieties over any algebraically closed field of arbitrary characteristic although Nakayamas original statement is formulated for complex analytic spaces.
We give a new proof of Bradens theorem ([Br]) about emph{hyperbolic restrictions} of constructible sheaves/D-modules. The main geometric ingredient in the proof is a 1-parameter family that degenerates a given scheme Z equipped with a G_m-action to the product of the attractor and repeller loci.
For a proper semistable curve $X$ over a DVR of mixed characteristics we reprove the invariant cycles theorem with trivial coefficients (see Chiarellotto, 1999) i.e. that the group of elements annihilated by the monodromy operator on the first de Rham cohomology group of the generic fiber of $X$ coincides with the first rigid cohomology group of its special fiber, without the hypothesis that the residue field of $cal V$ is finite. This is done using the explicit description of the monodromy operator on the de Rham cohomology of the generic fiber of $X$ with coefficients convergent $F$-isocrystals given in Coleman and Iovita (2010). We apply these ideas to the case where the coefficients are unipotent convergent $F$-isocrystals defined on the special fiber (without log-structure): we show that the invariant cycles theorem does not hold in general in this setting. Moreover we give a sufficient condition for the non exactness.
Let G be the Tate module of a p-divisble group H over a perfect field k of characteristic p. A theorem of Scholze-Weinstein describes G (and therefore H itself) in terms of the Dieudonne module of H; more precisely, it describes G(C) for good semiperfect k-algebras C (which is enough to reconstruct G). In these notes we give a self-contained proof of this theorem and explain the relation with the classical descriptions of the Dieudonne functor from Dieudonne modules to p-divisible groups.
In this paper, we show Langtons type theorem on separatedness and properness of moduli functor of torsion free semistable sheaves on algebraic orbifolds over an algebraically closed field k
In this note I provide two extensions of a particular case of the classical Poncelet theorem.