Harder-Narasimhan Filtrations which are not split by the Frobenius maps

Details Harder-Narasimhan Filtrations which are not split by the Frobenius maps Harder-Narasimhan Filtrations which are not split by the Frobenius maps

2012-04-09

arxiv.org/abs/1204.1911v1

Mathematics

Algebraic Geometry

3 pages

Scientific paper

Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural question to ask is whether this still holds in higher dimension. Indranil Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that there is always a counterexample to this over any algebraically closed field of positive characteristic which is uncountable. However, we will produce a smooth projective variety over $\mathbb Z$ and a rank 2 vector bundle on it, which, restricted to each prime $p$ in a nonempty open subset of $\spec\mathbb Z$, constitutes a counterexample over $p$. Indeed, given any split semisimple simply connected algebraic group $G$ of semisimple rank $>1$ over $\mathbb Z$, we will show that there exists a smooth projective homogeneous space $X_Z$ over $\mathbb Z$ and a vector bundle $V$ on $X_Z$ of rank 2 such that for each prime $p$ in a nonempty open subset of $\spec\mathbb Z$, the restriction $V\otimes\mathbb F_p$ as a vector bundle over $X_Z\otimes\mathbb F_p$ is a counterexample. We only use the Borel-Weil-Bott theorem in characteristic 0 and Frobenius Splitting of $G/B$ in characteristic $p$.

Affiliated with

Also associated with

No associations

Rating

Harder-Narasimhan Filtrations which are not split by the Frobenius maps does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Harder-Narasimhan Filtrations which are not split by the Frobenius maps, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Harder-Narasimhan Filtrations which are not split by the Frobenius maps will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-652160