KASS seminar

Abstract: In the 90's Okounkov showed how one can associate convex bodies to ample line bundles, generalizing the classical correspondence in toric geometry between polarized toric manifolds and Delzant polytopes. In 2008 Lazarsfeld-Mustata and independently Kaveh-Khovanskii showed that Okounkovs construction works more generally for e.g. big line bundles. Just as in the toric case, the Euclidean volume of the Okounkov body is equal to the volume of the line bundle divided by n! (n being the complex dimension), which means that results in convex analysis can be used in the study of the volume of line bundles. In their paper Lazarsfeld-Mustata asked if one could similarly define the Okounkov body of a big (1,1)-cohomology class on a compact Kähler manifold, and if then the Euclidean volume of the Okounkov body would be equal to the volume of the class divided by n! In 2015 Deng Ya proposed how to define the Okounkov body of a big class, but only managed to prove the conjectured volume equality in dimension one and two. In a recent preprint Tamas Darvas, Remi Reboulet, Mingchen Xia, Kewei Zhang and myself prove the volume equality for all dimensions. In my talk I will explain the construction of the Okounkov body, both in the line bundle case and in the "transcendental" case of a cohomology class, and maybe give some ideas of the proof of the volume equality.