be diffeomorphic to the connected sum of two oriented 4-manifolds only if one of them has negative-definite intersection form . This was an early application of the Donaldson invariant (or instanton invariants). Any compact symplectic manifold admits a symplectic Lefschetz pencil . Donaldson's recent work centers on a problem in complex differential geometry concerning a conjectural relationship between algebro-geometric "stability" conditions for smooth projective varieties and the existence of "extremal" Kähler metrics, typically those with constant scalar curvature (see for example cscK metric). Donaldson obtained results in the toric case of the problem (see for example ). He then solved the Kähler–Einstein case of the problem in 2012, in collaboration with Chen and Sun. This latest spectacular achievement involved a number of difficult and technical papers. The first of these was the paper of on Gromov–Hausdorff limits. The summary of the existence proof for Kähler–Einstein metrics appears in . Full details of the proofs appear in . Conjecture on Fano manifolds and Veblen Prize In 2019, Donaldson was awarded the Oswald Veblen Prize in Geometry, together with Xiuxiong Chen and Song Sun, for proving a long-standing conjecture on Fano manifolds, which states "that a Fano manifold admits a Kähler–Einstein metric if and only if it is K-stable". It had been one of the most actively investigated topics in geometry since its proposal in the 1980s by Shing-Tung Yau after he proved the Calabi conjecture. It was later generalized by Gang Tian and Donaldson. The solution by Chen, Donaldson and Sun was published in the Journal of the American Mathematical Society in 2015 as a three-article series, "Kähler–Einstein metrics on Fano manifolds, I, II and III". Selected publications Books References External links Home page at Imperial College (Plenary Lecture 1) 1957 births Living people 20th-century English mathematicians 21st-century English mathematicians Differential geometers Algebraic geometers Fellows of the Royal Society Foreign associates of the National Academy of Sciences Foreign Members of the Russian Academy of Sciences Members of the Royal Swedish Academy of Sciences Members of the French Academy of Sciences Institute for Advanced Study visiting scholars Fields Medalists Wallis Professors of Mathematics Fellows of All Souls College, Oxford Academics of Imperial College London Alumni of Pembroke College, Cambridge Alumni of Worcester College, Oxford Royal Medal winners Whitehead Prize winners Knights Bachelor Fellows of the American Mathematical Society