The 2019 Oswald Veblen Prize in Geometry will be awarded to Xiuxiong Chen, Simon Donaldson and Song Sun, for their three-part series, Kähler-Einstein metrics on Fano manifolds, I, II and III, published in 2015 in the Journal of the American Mathematical Society, in which they proved a long-standing conjecture in differential geometry.

In 1982 Shing-Tung Yau received the Fields Medal in part for his proof of the so-called Calabi Conjecture. He later conjectured that a solution in the case of Fano manifolds, i.e., those with positive first Chern class, would necessarily involve an algebro-geometric notion of stability. Seminal work of Gang Tian and then Donaldson clarified and generalized this idea. The resulting conjecture—that a Fano manifold admits a Kähler-Einstein metric if and only if it is K-stable—became one of the most active topics in geometry.

Chen, Donaldson, and Sun announced a complete solution of the conjecture for Fano manifolds in International Mathematics Research Notices in 2014, and full proofs followed in Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than $2\pi$, and Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches $2\pi$ and completion of the main proof, all published in 2015 in the Journal of the AMS.

As one nominator put it, This is perhaps the biggest breakthrough in differential geometry since Perelman’s work on the Poincaré conjecture. It is certainly the biggest result in Kähler geometry since Yau’s solution of the Calabi conjecture 35 years earlier. It is already having a huge impact that will only grow with time.

About the Veblen Prize

The Oswald Veblen Prize in Geometry is awarded every three years for a notable research work in geometry or topology that has appeared in the last six years. The work must be published in a recognized, peer-reviewed venue. The 2019 prize will be awarded Thursday, January 17 during the Joint Prize Session at the 2019 Joint Mathematics Meetings in Baltimore.