Positivity and geometry of higher codimension subvarieties (completed)

About the project

This is a project in algebraic geometry, a subject whose overlying goal is to classify and study algebraic varieties.

The project revolves around several central questions related to algebraic cycles, birational geometry, and Hodge theory. The main theme of the project is to explore how geometric properties of an algebraic variety are reflected in its special subvarieties. One example is given by so-called `positive subvarieties', e.g., subvarieties having non-negative intersection numbers with all other subvarieties.

Outcomes

1. A refinement of the motivic volume, and specialization of birational types (with J. Nicaise) (2020)

2. Two coniveau filtrations (with O. Benoist) (2020)

3. Tropical degenerations and stable rationality (Updated version: July 2020) (with J. Nicaise) (2019).

4. A pencil of Enriques surfaces with non-algebraic integral Hodge classes (with F. Suzuki) Mathematische Annalen 377 (2020), 183–197

5. On deformations of quintic and septic hypersurfaces (with S. Schreieder) Journal de Mathématiques Pures et Appliquées 135 (2020), 140-158.

6. Remarks on the positivity of the cotangent bundle of a K3 surface (with F. Gounelas) Épijournal de Géométrie Algébrique 4 (2020).
7. Curve classes on irreducible holomorphic symplectic varieties (with G. Mongardi) Communications in Contemporary Mathematics (2020)
8. Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero (with O. Benoist). Commentarii Mathematici Helvetici 95 (2020) 27-35.
9. A counterexample to the birational Torelli problem for Calabi-Yau 3-folds (with J. V. Rennemo). Journal of the London Mathematical Society 97 (2018), 427-440
10. Positivity of the diagonal (with B. Lehmann). Advances in Mathematics 335 (2018), 664-695.
11. Effective cones of cycles on blow-ups of projective space (with I. Coskun, J. Lesieutre). Algebra & Number Theory 10-9 (2016).
12. Nef cycles on some hyperkahler fourfolds. (2016)
13. Huybrechts, D.; Rennemo, J. V. Hochschild Cohomology versus the Jacobian Ring and the Torelli Theorem for Cubic Fourfolds. Algebr. Geom. 2019, 6 (1), 76–99.
14. Automorphisms of Hilbert schemes of points on surfaces (P. Belmans, G. Oberdieck, J. V. Rennemo)
15. A proof of the Donaldson-Thomas crepant resolution conjecture (S. Beentjes, J. Calabrese, J.V. Rennemo)
16. Some ways to reconstruct a sheaf from its tautological image on a Hilbert scheme of points A. Krug, J. V. Rennemo
17. Singularities of Restriction Varieties in OG(k,n): S. Adalı.

Financing

Research Council of Norway, Ground-breaking research - Young research talent. Project number 250104, total budget 9,2 mill NOK.