Demailly's mathematical works primarily concerned
complex analytic geometry, using techniques from
complex geometry with applications to
algebraic geometry and
number theory. Such analytic results have had many applications to
algebraic geometry. In particular, Boucksom, Demailly, Păun, and Peternell showed that a
smooth complex projective variety X is
uniruled if and only if its
canonical bundle K_X is not pseudo-effective.
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. This led to the first effective criteria for a line bundle on a complex projective variety X of any dimension n to be
very ample, that is, to have enough global sections to give an embedding of X into
projective space. For example, Demailly showed in 1993 that 2K_X+ 12n^nL 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.
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 X of
degree at least 21 in projective space \mathbb{CP}^3 is hyperbolic; equivalently, every
holomorphic map \Complex \to X is constant. For any variety X of
general type, Demailly showed that every holomorphic map \Complex \to X satisfies some (in fact, many)
algebraic differential equations. ==Awards and honors==