Viktor Balch Barth

$Image of Viktor Balch Barth$

Email viktorbb [at] math.uio.no

Room 1105

Username

Visiting address Moltke Moes vei 35 Niels Henrik Abels hus 0851 Oslo

Postal address Postboks 1053 Blindern 0316 Oslo

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Academic interests

I'm interested in complex geometry, algebraic Oka-theory and motivic homotopy theory.

Preprints

V. Balch Barth, W. Hornslien, G. Quick, G. M. Wilson: Making the motivic group structure on the endomorphisms of the projective line explicit (2023) [arXiv link]

V. Balch Barth: Surjective morphisms from affine space to its Zariski open subsets (2023) [arXiv link]

Background

MSc in Mathematical Sciences with specialization in topology, obtained in 2020 at NTNU.
BSc in Mathematical Sciences, obtained in 2019 at NTNU.

Courses taught

Fall 2018 (at NTNU): TMA4145 Linear Methods (Exercise sessions).

Awards

Abelstipendet 2019, granted by NMF.

Other Activities

Co-organizer of the 'What is..?'-seminar.

Tags: Mathematics, Several Complex Variables, Algebraic Geometry

Barth, Viktor Balch; Hornslien, William; Quick, Gereon & Wilson, Glen Matthew (2025). Making the motivic group structure on the endomorphisms of the projective line explicit. Advances in Mathematics. ISSN 0001-8708. 461, p. 1–65. doi: 10.1016/j.aim.2024.110080. Full text in Research Archive Show summary
We construct a group structure on the set of pointed naive homotopy classes of scheme morphisms from the Jouanolou device to the projective line. The group operation is defined via matrix multiplication on generating sections of line bundles and only requires basic algebraic geometry. In particular, it is completely independent of the construction of the motivic homotopy category. We show that a particular scheme morphism, which exhibits the Jouanolou device as an affine torsor bundle over the projective line, induces a monoid morphism from Cazanave’s monoid to this group. Moreover, we show that this monoid morphism is a group completion to a subgroup of the group of scheme morphisms from the Jouanolou device to the projective line. This subgroup is generated by a set of morphisms that are very simple to describe.
Barth, Viktor Balch & Truong, Trung Tuyen (2024). Images of dominant endomorphisms of affine space. Communications in Algebra. ISSN 0092-7872. doi: 10.1080/00927872.2024.2418374. Full text in Research Archive
Barth, Viktor Balch (2023). Surjective morphisms from affine space to its Zariski open subsets. International Journal of Mathematics. ISSN 0129-167X. 34(12). doi: 10.1142/S0129167X23500751. Full text in Research Archive Show summary
We prove constructively the existence of surjective morphisms from affine space onto certain open subvarieties of affine space of the same dimension. For any algebraic set [Formula: see text], we construct an endomorphism of [Formula: see text] with [Formula: see text] as its image. By Noether’s normalization lemma, these results extend to give surjective maps from any [Formula: see text]-dimensional affine variety [Formula: see text] to [Formula: see text].

View all works in NVA

Barth, Viktor Balch (2022). Surjective algebraic maps onto algebraically subelliptic manifolds. Full text in Research Archive
Barth, Viktor Balch (2021). Surjective algebraic maps onto algebraically subelliptic manifolds. Full text in Research Archive