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Helge Øystein Maakestad

Publications and source records attributed to Helge Øystein Maakestad.

29 records · Page 2Linked to original sources
arXiv2020

On the birational invariance of the arithmetic genus and Euler characteristic

The aim of this note is to use elementary methods to give a large class of examples of projective varieties $ Y \subseteq \mathbb{P}^d_k$ over a field $k$ with the property that $Y$ is not isomorphic to a hypersurface $H\subseteq \mathbb{P}^N_k$ in projective space $\mathbb{P}^N_k$ with $N:=dim(Y)+1$. We apply this construction to the study of the arithmetic genus $p_a(Y)$ of $Y$ and the problem of determining if $p_a(Y)$ is a birational invariant of $Y$ in general. We give an infinite number of examples of pairs of projective varieties $(Y, Y')$ in any dimension $dim(Y)=dim(Y')\geq 4$ where $Y$ is birational to $Y'$, but where $p_a(Y)\neq p_a(Y')$. The arithmetic genus is by Hodge theory known to be a birational invariant for smooth projective varieties over an algebraically closed field of characteristic zero. In each dimension $d\geq 4$ we give positive dimensional families of pairs of projective varieties $(Y,Y')$ that are birational but where the arithmetic genus differ. We prove a similar result on the Euler characteristic.

math.AG↗
arXiv2020

Classification of Lie algebras of differential operators

In a previous paper we introduced the notion of a D-Lie algebra $\tilde{L}$. A D-Lie algebra $\tilde{L}$ is an $A/k$-Lie-Rinehart algebra with a right $A$-module structure and a canonical central element $D$ satisfying several conditions. We used this notion to define the universal enveloping algebra of the category of $\tilde{L}$-connections and to define the cohomology and homology of an arbitrary connection. In this note we introduce the canonical quotient $L$ of a D-Lie algebra $\tilde{L}$ and use this to classify D-Lie algebras where $L$ is projective as $A$-module. We define for any 2-cocycle $f\in \operatorname{Z}^2(\operatorname{Der}_k(A),A)$ a functor $F_{f}(-)$ from the category of $A/k$-Lie-Rinehart algebras to the category of D-Lie algebras and classify D-Lie algebras with projective canoncial quotient using the functor $F_{f}(-)$. We prove a similar classification for non-abelian extensions of D-Lie algebras. We classify $\tilde{L}$-connections in the case when the canonical quotient $L$ of $\tilde{L}$ is projective as $A$-module. Any $\tilde{L}$-connection is determined by a 2-cocycle $f\in \operatorname{Z}^2(\operatorname{Der}_k(A),A)$ and an $L$-connection $(E,\nabla)$. We introduce the correspondence and Chow-operator of an $\tilde{L}$-connection. The aim of this construction is to relate connections on D-Lie algebras to algebraic cycles an the category of correspondences. The Chow-operator cannot be defined for an ordinary connection on an $A/k$-Lie-Rinehart algebra. It depends in a non-trivial way on the right $A$-module structure on $\tilde{L}$ and the canonical quotient $A/k$-Lie-Rinehart algebra $L$ has no such structure.

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arXiv2020

Modules of principal parts on the projective line

We study the splitting-type of the bi-modules of principal parts (Grothendiecks analogue of jet-bundles in algebraic geometry) as left and right O-module on the projective line in positive characteristic, and obtain explicit examples where the splitting-type as left O-module differs from the splitting-type as right O-module. Hence the principal parts are the first examples of a sheaf of abelian groups equipped with two non-isomorphic structures as locally free O-module. We also give explicit examples showing that in some cases the splitting-type changes with the characteristic of the base-field.

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arXiv2020

A Note on Principal Parts on Projective Space and Linear Representations

Let H be a closed subgroup of a linear algebraic group G defined over a field F. There is an equivalence of categories between the category of linear finite-dimensional representations of H and the category of finite rank G-homogeneous vectorbundles on G/H. In this short note we study this correspondence for the sheaves of principal parts (=jetbundles) of homogeneous linebundles on projective space in characteristic zero. We describe the representation corresponding to the principal parts of a homogeneous linebundle and as a corollary we obtain known formulas on the splitting type of the principal parts on projective space in any dimension.

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arXiv2020

Chern classes and Lie-Rinehart algebras

Classically the Chern-classes of a locally free coherent A-module W are defined using the curvature of a connection. If we more generally consider the problem of defining Chern-classes where W is a coherent A-module, a connection might not exist. In this paper we use the linear Lie-algebroid of W, where W is any coherent A-module of finite presentation, to define the first Chern-class of W. We also do explicit calculations of Chern-classes for maximal Cohen-Macaulay modules on isolated hypersurface-singularities and 2-dimensional quotient-singularities.

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arXiv2020

Principal parts on the projective line over arbitrary rings

We develop techniques to split the principal parts on the projective line over an arbitrary ring and apply these techniques to give a complete classification of the principal parts on the projective line over any field of characteristic zero.

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arXiv2020

The Chern character for Lie-Rinehart algebras

Classical Chern-Weil theory proves existence of a ring homomorphism - the Chern character - from the Grothendieck ring of locally free finite rank A-modules to the algebraic de Rham cohomology of A. In this paper we prove existence of a ring homomorphism from the Grothendieck ring of locally free finite rank A-modules with an L-connection to the Lie-Rinehart cohomology of L, where L is a Lie-Rinehart algebra. As a corollary we get the classical Chern character.

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arXiv2020

Jet bundles on projective space

In this paper we state and prove some results on the structure of the jetbundles as left and right module over the structure sheaf on the projective line and projective space using elementary techniques involving diagonalization of matrices, multilinear algebra and sheaf cohomology.

math.AG↗
arXiv2020

Notes on the Chern-character

Notes for some talks given at the seminar on characteristic classes at NTNU in autumn 2006. In the paper a proof of the existence of a Chern-character from complex K-theory to any cohomology theory with values in graded Q-algebras is given. The Chern-character respects the Adams and Steenrod operations.

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arXiv2020

Conjectures on L-functions for flag bundles on Dedekind domains

Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $S:=\operatorname{Spec}(\mathcal{O}_K)$. Let $T_0,\ldots,T_n$ be regular schemes of finite type over $S$ and let $X$ be a scheme of finite type over $T_n$ with a stratification of closed subschemes (a generalized cellular decomposition) \[ \emptyset=X_{-1} \subseteq X_0 \subseteq \cdots \subseteq X_{n-1} \subseteq X_n:=X \] with $X_i-X_{i-1}=E_i$ where $E_i$ is a vector bundle of rank $d_i$ on $T_i$. We prove that if the Beilinson-Soule vanishing conjecture and Soule conjecture holds for $T_i$ it follows the same conjectures hold for $X$. We develop a criteria for the conjectures to hold in terms of an open cover and use this criteria to prove the Beilinson-Soule vanishing conjecture and Soule conjecture for the partial flag bundle $\mathbb{F}(d,E)$ of any coherent $\mathcal{O}_S$-module $E$ on $S$. Hence we get non-trivial examples where the conjectures hold in arbitrary dimension. As a special case we prove the conjectures for any affine or projective fibration of finite type over $S$. We moreover reduce the study of the Beilinson-Soule vanishing conjecture and the Soule conjecture on L-functions to the study of affine regular schemes of finite type over $\mathbb{Z}$. We also discuss the Beilinson conjecture on special values for partial flag bundles. We reduce the study of the Bloch-Kato conjecture on special values for flag bundles to the case of Dedekind domains.

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