# Positivity and geometry of higher codimension subvarieties

## 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' - subvarieties having non-negative intersection numbers with all other subvarieties. The prototype of such a subvariety is the hyperplane section H of a projective variety X; this case the Lefschetz hyperplane theorem relates the topology of X to that of H.

## Outcomes

- Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero (with O. Benoist). arXiv (2018) PDF
- A counterexample to the birational Torelli problem for Calabi-Yau 3-folds (with J. V. Rennemo). To appear in
*Journal of the London Mathematical Society*(2018) PDF - Positivity of the diagonal (with B. Lehmann) PDF
- Effective cones of cycles on blow-ups of projective space (with I. Coskun, J. Lesieutre).
*Algebra & Number Theory*10-9 (2016). PDF - Nef cycles on some hyperkahler fourfolds. PDF

## Financing

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