Jean-Pierre Demailly

One main topic of Demailly's research is Pierre Lelong's generalization of the notion of a Kähler form to allow forms with singularities, known as currents. In particular, for a compact complex manifold ${\displaystyle X}$, an element of the Dolbeault cohomology group ${\displaystyle H^{1,1}(X,\mathbb {R} )}$ is called pseudo-effective if it is represented by a closed positive (1,1)-current (where "positive" means "nonnegative" in this phrase), or big if it is represented by a strictly positive (1,1)-current; these definitions generalize the corresponding notions for holomorphic line bundles on projective varieties. Demailly's regularization theorem says, in particular, that any big class can be represented by a Kähler current with analytic singularities.[5]

Such analytic results have had many applications to algebraic geometry. In particular, Boucksom, Demailly, Păun, and Peternell showed that a smooth complex projective variety ${\displaystyle X}$ is uniruled if and only if its canonical bundle ${\displaystyle K_{X}}$ is not pseudo-effective.[6]

For a singular metric on a line bundle, Nadel, Demailly, and Yum-Tong Siu developed the concept of the multiplier ideal, which describes where the metric is most singular. There is an analog of the Kodaira vanishing theorem for such a metric, on compact or noncompact complex manifolds.[7] This led to the first effective criteria for a line bundle on a complex projective variety ${\displaystyle X}$ of any dimension ${\displaystyle n}$ to be very ample, that is, to have enough global sections to give an embedding of ${\displaystyle X}$ into projective space. For example, Demailly showed in 1993 that ${\displaystyle 2K_{X}+12n^{n}L}$ is very ample for any ample line bundle L, where addition denotes the tensor product of line bundles. The method has inspired later improvements in the direction of the Fujita conjecture.[8]

Demailly used the technique of jet differentials introduced by Green and Phillip Griffiths to prove Kobayashi hyperbolicity for various projective varieties. For example, Demailly and El Goul showed that a very general complex surface ${\displaystyle X}$ of degree at least 21 in projective space ${\displaystyle \mathbb {CP} ^{3}}$ is hyperbolic; equivalently, every holomorphic map ${\displaystyle \mathbb {C} \to X}$ is constant.[9] For any variety ${\displaystyle X}$ of general type, Demailly showed that every holomorphic map ${\displaystyle \mathbb {C} \to X}$ satisfies some (in fact, many) algebraic differential equations.[10]