Berndtsson's first results concern
zero sets of
holomorphic functions, and in 1981 he showed that any divisor with finite area in the
unit ball in the two-dimensional complex space is defined by a bounded holomorphic function (which is not true in higher dimensions). In the 1980s he also developed (together with
Mats Andersson) a formalism to generate weighted integral representation formulas for holomorphic functions and solutions to the so-called \bar\partial-equation, which is the higher-dimensional generalization of the
Cauchy–Riemann equations in the plane. This formalism led to new results concerning division and interpolation of holomorphic functions. In the 1990s Berndtsson started to work with L^2 methods that had been introduced by
Lars Hörmander,
Joseph J. Kohn and others in the 1960s and he modified these methods to obtain uniform estimates for the \bar\partial-equation. At this time he also achieved results about interpolation and sampling in
Hilbert spaces of holomorphic functions using L^2-estimates. More recently Berndtsson has worked on global problems on
complex manifolds. In a series of papers starting in 2005 he has obtained positivity results for the curvature of holomorphic
vector bundles naturally associated to holomorphic
fibrations. These vector bundles arise as the zeroth direct images of the adjoint of an
ample line bundle over the fibration. The case of a trivial line bundle was considered in earlier work by
Phillip Griffiths in connection to variations of Hodge structures and by Fujita,
Kawamata and
Eckart Viehweg in
algebraic geometry. Berndtsson has also explored applications of these positivity results in Kähler geometry (e.g., to
geodesics in the space of Kähler metrics) and algebraic geometry (e.g., a new proof of the Kawamata subadjunction formula in a collaboration with Mihai Păun). ==Further activities==