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.
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. 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 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. 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.
We explain how the formal aspects of the theory of Kahler-Einstein metrics can be developed in the framework of moment maps. The central result we use is the Berndtsson convexity theorem, which is interpreted as defining a metric on the space of complex structures. We discuss some applications of these ideas to the Kah…
New curvature assumptions prove Nakano positivity for complex vector bundles.
problem Proving Nakano positivity for complex vector bundles under varying curvature assumptions.
method Using a variant of Hörmander's theorem, the authors show Nakano positivity under more general curvature conditions.
result Nakano positivity holds for complex vector bundles under different curvature assumptions.
We extend to any simply connected Kähler manifold with non-positive sectional curvature some conditions for interpolation in C and in the unit disk given by Berndtsson, Ortega-Cerdà and Seip. The main tool is a comparison theorem for the Hessian in Kähler geometry due to Greene, Wu and Siu, Yau.
Canonical bundle formula due to Kawamata and others has played fundamental roles in algebraic geometry. We show that the canonical bundle formula has analytic characterization in terms of fiberwise integration, which confirms a folklore conjecture. The proof uses L2 metrics and the valuative equivalence of plurisubh…
Motivated by the results of B. Berndtsson, in this memoir we use the new estimates developed by W. He to extend a theorem of the second author on the existence of weak C1,1 geodesics between two smooth non-degenerate Kähler potentials to the case where the metrics on the end points may have singularities on some a…
New equation approximates Kähler potentials using Hermitian-Yang-Mills metrics.
problem Approximating Kähler potentials in complex domains.
method New Wess-Zumino-Witten type equation and Berndtsson's theorem on direct image bundles.
result Approximation of Kähler potentials by Hermitian-Yang-Mills metrics.
Generalizes Nakano-positivity to Hilbert space fields.
problem Extending Nakano-positivity to Hilbert space fields.
method Exhaustion arguments to generalize the theorem.
result Log-plurisubharmonic variation results for Stein manifolds.
Extends construction of Kähler-Einstein metrics to noncompact manifolds.
problem Construct complete Kähler-Einstein metrics on noncompact manifolds.
method Iterative construction using Berndtsson's method.
result Induces semipositively curved metric on relative canonical bundle.
Generalized Nakano positivity for certain singular cases.
problem Nakano positivity of direct image sheaves for singular cases.
method Generalization of Berndtsson's result to singular cases.
result Nakano positivity for direct image sheaves in special singular cases.
We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kaehler manifold can be solved using a variational method independent of Yau's theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate…
In this note, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Kollár and Jonsson-Mustată implies the truth of twisted versions of the strong openness conjecture; our optimal L2 extension implies Berndtsson…
We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang motivated by Arakelov geometry is obtained; applications to SU(2)-invariant determ…
We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.
problem Proving uniqueness of solutions to complex Monge-Ampère equations.
method Local and global analysis of bounded hyperconvex domains and compact complex manifolds.
result Uniqueness of solutions confirmed for small temperature parameters.
Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct …
This paper is a sequel to \cite{Berndtsson}. In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth Kähler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Naka…
We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Gri…
Ehrhart's conjecture proposes a sharp upper bound on the volume of a convex body whose barycenter is its only interior lattice point. Recently, Berman and Berndtsson proved this conjecture for a class of rational polytopes including reflexive polytopes. In particular, they showed that the complex projective space has t…
Study weak geodesic lines in Kähler metric space, disproving a conjecture.
problem Disproving a conjecture about weak geodesic lines in Kähler metrics.
method Establish Ross-Witt Nyström correspondence, construct weak geodesic lines.
result Some weak geodesic lines are smooth, disproving a popular conjecture.
Uniqueness of weighted extremal metrics on Kähler manifolds proven.
problem Uniqueness of weighted extremal Kähler metrics on compact Kähler manifolds.
method Proof of uniqueness using modified Mabuchi energy and weighted K-semistability.
result Uniqueness of weighted extremal Kähler metrics up to automorphisms.
We introduce and study a notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and P{ă}un. We define what it means for such a metric to be curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the cuvature Θh as a matrix of currents. We …
Given a holomorphic family f:X→S of compact complex manifolds of dimension n and a relatively ample line bundle L→X, the higher direct images Rn−pf∗ΩX/Sp(L) carry a natural hermitian metric. We give an explicit formula for the curvature tensor of these direct images.…
Holomorphic families yield metrics with explicit curvature formulas.
problem Positivity of direct images with Poincaré type singularities.
method Analyzes holomorphic families and line bundles with Poincaré type singular metrics.
result Explicit formula for curvature of direct image metrics.
Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.
problem Existence of non-trivial holomorphic vector fields on compact complex manifolds.
method Vanishing result for measure preserving holomorphic vector fields, Gibbs stability, and log terminal singularities.
result No non-trivial holomorphic vector fields on compact complex manifolds with big anti-canonical line bundle.
We characterize the global maximizers of a certain non-local functional defined on the space of all positively curved metrics on an ample line bundle L over a Kahler manifold X. This functional is an adjoint version, introduced by Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian functional whose cr…
The paper studies curvature properties of direct image bundles.
problem Investigating curvature properties of direct image bundles.
method Using subharmonic metrics and mean curvature analysis.
result Direct image bundles carry metrics with positive mean curvature.
We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size $k^…
This paper characterizes mu-cscK metrics using Perelman's W-entropy.
problem Characterizing mu-cscK metrics and understanding their properties.
method Using Perelman's W-entropy as a functional on the tangent bundle of Kähler metrics, the paper characterizes mu-cscK metrics as critical points of this functional.
result The W-entropy is monotonic along geodesics and provides a lower bound for mu-entropy.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
Analyzes Saito vanishing theorem using L2 methods.
problem Proving the Saito vanishing theorem.
method Uses L2-methods to prove the theorem. result Analytic proof of the Saito vanishing theorem.
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.
Method upgrades limit theorems to mixing limit theorems for dynamical systems.
problem Improving limit theorems for dynamical systems.
method General method for upgrading limit theorems to mixing limit theorems.
result Mixing limit theorems for specific subbundles of the Kontsevich-Zorich cocycle.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…
The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.
problem Understanding the distinction between incidence theorems over division rings and fields.
method Extending the surface-graph approach to noncommutative settings, the paper analyzes the topological properties of graphs embedded on surfaces of different genera.
result Theorems associated with graphs on the sphere hold over any division ring, while those on surfaces of positive genus typically hold only if the ground ring is a field.