The Viggo Brun Prize 2020 is awarded to John Christian Ottem for his deep and original contributions to algebraic geometry, especially to the theory for birational varieties of higher dimension and to questions concerning positivity and existence of cycles.

John Christian Ottem (born 1985) is a Norwegian mathematician with a bachelor and a master degree from the University of Oslo. His PhD degree is from the University of Cambridge in 2013, where his advisor was Burt Totaro. Ottem held a post doc position at the University of Cambridge from 2013 to 2016. In 2016 he became an associate professor – and in 2019 a professor – at the University of Oslo.

Ottem’s research is within algebraic geometry. His work shows large breadth and originality. He obtains deep results, both alone and in collaboration with others, and publishes in very good, general mathematical journals. The prob- lems he chooses are often classical, but he attacks and solves them by using sophisticated modern techniques.

Ottem has contributed to three of “hottest” problems in algebraic geometry: higher dimensional birational geometry, the Hodge conjecture, and Hartshorne’s conjecture.

The study of cycles on projective varieties is a recurring theme in Ottem’s research, in particular in connection with properties like ampleness and positiv- ity. His two first papers, wwritten when he was a master student, were about Cox rings of certain algebraic surfaces. In his PhD thesis he studied ample sub- varieties and line bundles on projective varieties and the relationship between ampleness and positivity. In an article published in the Journal of the European Mathematical Society Ottem studies subvarieties with ample normal bundle in a smooth projective variety. In codimension 1 these are positive divisors and well understood, but in higher codimension there are many open questions. Ottem shows that the cohomology classes of curves with ample normal bun- dle are “big” and movable. This has potentially important consequences for Hartshorne’s famous conjecture about complete intersections.

Together with Jørgen Vold Rennemo, Ottem gave the first counterexamples to the socalled “birational Torelli problem” for Calabi–Yau threefolds. This is a well known conjecture about Calabi–Yau threefolds going back to the 1980’s. The article has led to much acitivity in the area of derived equivalent Calabi–Yau manifolds and the Grothendieck ring of varieties.

Among other highlights is a joint work with Olivier Benoist, where they show that the integral version of the Hodge conjecture does not hold for three- folds with Kodaira dimension 0. In another paper they consider Grothendieck’s “coniveau” and “strong coniveau” filtrations on the cohomology groups of a variety and give the first examples where these two filtrations are different.

A joint work with Stefan Schreieder has as its starting point Mori’s question from 1975 on whether smooth deformations of hypersurfaces of prime degree also are hypersurfaces. Ottem and Schreieder give a positive answer to this question for quintic hypersurfaces of arbitrary dimension and septic hypersurfaces of dimension three. A corollary of their results is a famous theorem of Horikawa concerning deformations of quintic surfaces.

In collaboration with Johannes Nicaise, Ottem considers the classical prob- lem of determining which hypersurfaces are rational, i.e., birationally equivalent to a projective space. Nicaise and Ottem use motivic obstruction to show the existence of several new classes of stably irrational hypersurfaces and complete intersections. An important ingredient is tropical degeneration techniques. In another joint work they construct a refinement of motivic volume, which general- izes the version of motivic volume due to Nicaise and Schinder and the birational version due to Kontsevitch and Tschinkel. They show how their techniques give rise to explicit examples of obstructions to stable rationality.