Kählerian geometry

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]

Multiplier ideals

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]

Kobayashi hyperbolicity

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]