Junhu Guo | Searcharxiv

arXiv2025

Normal forms for ordinary differential operators, III

In this paper, which is a follow-up of our first paper "Normal forms for ordinary differential operators, I", we extend the explicit parametrisation of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups obtained earlier to analogous parametrisation of torsion free sheaves of arbitrary rank with vanishing cohomology groups on projective irreducible curves. As an illustration of our theorem we calculate one explicit example of such parametrisation, namely for rank two sheaves on a Weierstrass cubic curve.

math.AG↗

arXiv2022

On some questions around Berest's conjecture

Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove the following two results. Assume there exists a non-zero polynomial $f(X,Y)\in K[X,Y]$, which has a non-trivial solution $(P,Q)\in A_{1}^{2}$ with $[P,Q]=0$, and the number of orbits under the group action of $Aut(A_1)$ on solutions of $f$ in $A_{1}^{2}$ is finite. Then the Dixmier conjecture holds, i.e $\forall \varphi\in End(A_{1})-\{0\}$, $\varphi$ is an automorphism. Assume $\varphi$ is an endomorphism of monomial type (in particular, it is not an automorphism, see theorem 4.1). Then it has no non-trivial fixed point, i.e. there are no $P\in A_1$, $P\notin K$, s.t. $\varphi (P)=P$.

math.AG↗