arXiv research

The paper defines positivity for singular metrics on vector bundles and proves related theorems.

Defines complex structure for families of Hilbert spaces with reasonable curvature.

The abstract presents a new theorem using Ross-Witt Nyström correspondence and Berndtsson's theorem.

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…

Extends construction of Kähler-Einstein metrics to noncompact manifolds.

Proves generic surjectivity of vector bundles via degeneration.

Proves Skoda's Division Theorem using degeneration and positivity of direct image bundles.

New curvature assumptions prove Nakano positivity for complex vector bundles.

Generalized Nakano positivity for certain singular cases.

We extend to any simply connected Kähler manifold with non-positive sectional curvature some conditions for interpolation in $\mathbb{C}$ 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.

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 $L^{2}$ extension implies Berndtsson…

We prove that the solution of a Wess-Zumino-Witten type equation from a domain $D$ in $\mathbb{C}^m$ to the space of Kähler potentials can be approximated uniformly by Hermitian-Yang-Mills metrics on certain vector bundles. The key is a new version of Berndtsson's theorem on the positivity of direct image bundles.

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 $C^{1,1}$ geodesics between two smooth non-degenerate Kähler potentials to the case where the metrics on the end points may have singularities on some a…

We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.

Generalizes Nakano-positivity to Hilbert space fields.

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 $L^2$ metrics and the valuative equivalence of plurisubh…

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 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…

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.

Uniqueness of weighted extremal metrics on Kähler manifolds proven.

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 $Θ^h$ as a matrix of currents. We …

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds of dimension $n$ and a relatively ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ 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.

Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.

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 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…

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^…

The paper studies curvature properties of direct image bundles.

This paper characterizes mu-cscK metrics using Perelman's W-entropy.

Improved iterative methods for risk parity portfolio weights.

We describe a novel optimization method for finite sums (such as empirical risk minimization problems) building on the recently introduced SAGA method. Our method achieves an accelerated convergence rate on strongly convex smooth problems. Our method has only one parameter (a step size), and is radically simpler than o…

A new method combines Laplace and Variational Bayes for scalable inference.

Unified framework for model explanation methods based on feature removal.

This work reviews and evaluates methods for predicting prediction intervals in regression problems.

Derives kernel PCA with Nyström method for scalability.

In this paper, the author considers the numerical computation of CVA for large systems by Mote Carlo methods. He introduces two types of stochastic mesh methods for the computations of CVA. In the first method, stochastic mesh method is used to obtain the future value of the derivative contracts. In the second method, …

New method combines spectral and sparse methods for Gaussian processes.

A comprehensive benchmark of 15 scRNA-seq imputation methods across various datasets and analyses.

New methods using natural gradient for structured optimization.

Recently, {\it stochastic momentum} methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of t…

We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference me…

A new method speeds up deep neural network training.

We propose a new stochastic dual coordinate ascent technique that can be applied to a wide range of regularized learning problems. Our method is based on Alternating Direction Multiplier Method (ADMM) to deal with complex regularization functions such as structured regularizations. Although the original ADMM is a batch…

NCG methods improve shape optimization efficiency.

Geometric methods study 3-manifold splittings.