Solves Lempert's question on Nakano semi-positivity preservation.
problem Preserving Nakano semi-positivity under limits of metrics.
method Establishes L2 extension theorem and uses multiplier submodule sheaves. result Affirmative solution to Lempert's question on Nakano semi-positivity.
Defines complex structure for families of Hilbert spaces with reasonable curvature.
problem Curvature of families of Hilbert spaces not forming a holomorphic bundle.
method Defines a new complex analytic structure and curvature for families of Hilbert spaces.
result New proof of Berndtsson's theorem on curvature of direct images of semi-positively twisted relative canonical bundles.
Proves Skoda's Division Theorem using degeneration and positivity of direct image bundles.
problem Division Theorem in Skoda's context
method Degeneration approach inspired by B. Berndtsson and L. Lempert's L2 extension theorem result Simplified and extended proof of L2 extension theorem We prove that elliptic tubes over properly convex domains of the real projective space are C-convex and complete Kobayashi-hyperbolic. We also study a natural construction of complexification of convex real projective manifolds.
In the paper "Direct Images, Fields of Hilbert Spaces, and Geometric Quantization", Lempert and Szőke proved that any flat analytic Hilbert field will induce a hermitian Hilbert bundle and gave an example of a flat Hilbert field that does not induce any Hilbert bundle. In this paper, we will provide an example of an an…
Proves generic surjectivity of vector bundles via degeneration.
problem Understanding generic surjectivity of vector bundles.
method Uses degeneration argument, Berndtsson's theorem, and Lempert's proof.
result Generalizes previous work on L2 division theorem. This paper provides an explicit formula for complex structures in embeddings of manifolds.
problem Finding explicit formulas for complex structures in embeddings of manifolds.
method Recursive expression and fiberwise Taylor expansion of the canonical complex structure.
result Evidence of canonical vanishing of integrability equations in general settings.
Advances M-polyfolds for complex geometry applications.
problem Complex geometry challenges in differential geometry.
method Introduces and proves geometric structures within M-polyfolds.
result Establishes M-polyfolds as useful differential geometric objects.
The paper defines positivity for singular metrics on vector bundles and proves related theorems.
problem Positivity of singular Hermitian metrics for holomorphic vector bundles.
method The method of Berndtsson and Lempert, along with a Berndtsson-type positivity theorem for holomorphic vector bundles.
result Sharp L2 extension theorem for holomorphic vector bundles. The Bergman-Szegő kernel is analyzed for weakly pseudoconvex CR manifolds of finite type.
problem Analyzing the Bergman-Szegő kernel for specific CR manifolds.
method Constructing a parametrix for the Szegő kernel, extending earlier results.
result Extending Fefferman's boundary asymptotics to weakly pseudoconvex domains in \(\mathbb{C}^{2}\).
We give a simple interpretation of the adapted complex structure of Lempert-Szoke and Guillemin-Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of X in TTX, where X is a real-analytic…
Sasaki manifolds have isometric spaces of potentials implying similar geometric properties.
problem Comparing Sasaki manifolds through isometric spaces of potentials.
method Analyzing regular and non-regular Sasaki manifolds, proving isometry implications.
result Isometric spaces of potentials imply similar universal covering spaces for regular Sasaki manifolds.
New characterization of Riemannian metric positivity and L2 estimates for d operator.
problem Characterize positivity of Riemannian metrics and L2 estimates for d operator. method Apply L2 technique developed by Deng-Ning-Wang-Zhou, new characterizations given. result Prove new results parallel to Liu-Yang-Zhou's answer to Lempert's question.
The abstract presents a new theorem using Ross-Witt Nyström correspondence and Berndtsson's theorem.
problem The abstract tackles the Ohsawa-Takegoshi extension theorem and its applications.
method The approach uses Ross-Witt Nyström correspondence and Berndtsson's theorem in \(\mathbb{C}^*\)-degeneration.
result The approach provides a quick proof of the Ohsawa-Takegoshi extension theorem without limits or singular weights.
Given a complex manifold X, any Kähler class defines an affine bundle over X, and any Kähler form in the given class defines a totally real embedding of X into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity…
Study of geometric properties of almost calibrated forms on Kähler manifolds.
problem Understanding the geometry of almost calibrated (1,1) forms on compact Kähler manifolds. method Investigates the infinite dimensional Riemannian manifold structure, CAT(0) geodesic metric space, and geodesics of the space of almost calibrated forms.
result The space of almost calibrated forms is an infinite dimensional Riemannian manifold with non-positive sectional curvature and CAT(0) geodesic metric space.
In the first part of the paper, comprising section 1 through 6, we introduce a sequence of functions in the tangent bundle TM of any smooth two-dimensional manifold M with smooth Riemannian metric g that correspond to the higher order Schwarzians of the linearized geodesic flow. With these functions and a classical the…