Bott manifolds with vanishing Futaki invariants for all Kähler classes
We prove that the only Bott manifolds such that the Futaki invariant vanishes for any Kähler class are isomorphic to the products of the projective lines.
math.AG↗
arXiv subjects
Hajime Ono
Publications and source records attributed to Hajime Ono.
One-bit Submission for Locally Private Quasi-MLE: Its Asymptotic Normality and Limitation
Local differential privacy~(LDP) is an information-theoretic privacy definition suitable for statistical surveys that involve an untrusted data curator. An LDP version of quasi-maximum likelihood estimator~(QMLE) has been developed, but the existing method to build LDP QMLE is difficult to implement for a large-scale survey system in the real world due to long waiting time, expensive communication cost, and the boundedness assumption of derivative of a log-likelihood function. We provided an alternative LDP protocol without those issues, which is potentially much easily deployable to a large-scale survey. We also provided sufficient conditions for the consistency and asymptotic normality and limitations of our protocol. Our protocol is less burdensome for the users, and the theoretical guarantees cover more realistic cases than those for the existing method.
stat.ML↗
Differentially Private Variational Autoencoders with Term-wise Gradient Aggregation
This paper studies how to learn variational autoencoders with a variety of divergences under differential privacy constraints. We often build a VAE with an appropriate prior distribution to describe the desired properties of the learned representations and introduce a divergence as a regularization term to close the representations to the prior. Using differentially private SGD (DP-SGD), which randomizes a stochastic gradient by injecting a dedicated noise designed according to the gradient's sensitivity, we can easily build a differentially private model. However, we reveal that attaching several divergences increase the sensitivity from O(1) to O(B) in terms of batch size B. That results in injecting a vast amount of noise that makes it hard to learn. To solve the above issue, we propose term-wise DP-SGD that crafts randomized gradients in two different ways tailored to the compositions of the loss terms. The term-wise DP-SGD keeps the sensitivity at O(1) even when attaching the divergence. We can therefore reduce the amount of noise. In our experiments, we demonstrate that our method works well with two pairs of the prior distribution and the divergence.
cs.LG↗
Locally Private Distributed Reinforcement Learning
We study locally differentially private algorithms for reinforcement learning to obtain a robust policy that performs well across distributed private environments. Our algorithm protects the information of local agents' models from being exploited by adversarial reverse engineering. Since a local policy is strongly being affected by the individual environment, the output of the agent may release the private information unconsciously. In our proposed algorithm, local agents update the model in their environments and report noisy gradients designed to satisfy local differential privacy (LDP) that gives a rigorous local privacy guarantee. By utilizing a set of reported noisy gradients, a central aggregator updates its model and delivers it to different local agents. In our empirical evaluation, we demonstrate how our method performs well under LDP. To the best of our knowledge, this is the first work that actualizes distributed reinforcement learning under LDP. This work enables us to obtain a robust agent that performs well across distributed private environments.
cs.LG↗
Lightweight Lipschitz Margin Training for Certified Defense against Adversarial Examples
How can we make machine learning provably robust against adversarial examples in a scalable way? Since certified defense methods, which ensure $ε$-robust, consume huge resources, they can only achieve small degree of robustness in practice. Lipschitz margin training (LMT) is a scalable certified defense, but it can also only achieve small robustness due to over-regularization. How can we make certified defense more efficiently? We present LC-LMT, a light weight Lipschitz margin training which solves the above problem. Our method has the following properties; (a) efficient: it can achieve $ε$-robustness at early epoch, and (b) robust: it has a potential to get higher robustness than LMT. In the evaluation, we demonstrate the benefits of the proposed method. LC-LMT can achieve required robustness more than 30 epoch earlier than LMT in MNIST, and shows more than 90 $\%$ accuracy against both legitimate and adversarial inputs.
cs.LG↗
Cahen-Gutt moment map, closed Fedosov star product and structure of the automorphism group
We show that if a compact Kaehler manifold $M$ of non-negative Ricci curvature admits closed Fedosov star product then the reduced Lie algebra of holomorphic vector fields on $M$ is reductive. This comes in pair with the obstruction previously found by La Fuente-Gravy. More generally we consider the squared norm of Cahen-Gutt moment map as in the same spirit of Calabi functional for the scalar curvature in cscK problem, and prove a Cahen-Gutt version of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kaehler manifolds. The proof uses a Hessian formula for the squared norm of Cahen-Gutt moment map.
math.DG↗
Volume minimization and obstructions to solving some problems in Kähler geometry
There is an obstruction to the existence of Kähler -Einstein metrics which is used to define the GIT weight for K-stability, and it has been extended to various geometric problems. This survey paper considers such extended obstructions to the existence problem of Kähler -Ricci solitons, Sasaki-Einstein metrics and (conformally) Einstein-Maxwell Kähler metrics. These three cases have a common feature that the obstructions are parametrized by a space of vector fields. We see, in these three cases, the obstructions are obtained as the derivative of suitable volume functionals. This tells us for which vector fields we should try to solve the existence problems.
math.DG↗
On the existence problem of Einstein-Maxwell Kähler metrics
In this expository paper we review on the existence problem of Einstein-Maxwell Kähler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell Kähler metrics, and give extensions of volume minimization principle, the notion of toric K-stability and other related results to the general set-up. Secondly, we consider the toric case when the manifold is the one point blow-up of the complex project plane and the Kähler class $Ω$ is chosen so that the area of the exceptional curve is sufficiently close to the area of the rational curve of self-intersection number 1. We observe by numerical analysis that there should be a Killing vector field $K$ which gives a toric K-stable pair $(Ω, K)$ in the sense of Apostolov-Maschler.
math.DG↗
Volume minimization and Conformally Kähler, Einstein-Maxwell geometry
Let $M$ be a compact complex manifold admitting a Kähler structure. A conformally Kähler, Einstein-Maxwell metric (cKEM metric for short) is a Hermitian metric $\tilde{g}$ on $M$ with constant scalar curvature such that there is a positive smooth function $f$ with $g = f^2 \tilde{g}$ being a Kähler metric and $f$ being a Killing Hamiltonian potential with respect to $g$. Fixing a Kähler class, we characterize such Killing vector fields whose Hamiltonian function $f$ with respect to some Kähler metric $g$ in the fixed Kähler class gives a cKEM metric $\tilde{g} = f^{-2}g$. The characterization is described in terms of critical points of certain volume functional. The conceptual idea is similar to the cases of Kähler-Ricci solitons and Sasaki-Einstein metrics in that the derivative of the volume functional gives rise to a natural obstruction to the existence of cKEM metrics. However, unlike the Kähler-Ricci soliton case and Sasaki-Einstein case, the functional is neither convex nor proper in general, and often has more than one critical points. The last observation matches well with the ambitoric examples studied earlier by LeBrun and Apostolov-Maschler.
math.DG↗
Conformally Einstein-Maxwell Kähler metrics and structure of the automorphism group
Let $(M,g)$ be a compact Kähler manifold and $f$ a positive smooth function such that its Hamiltonian vector field $K = J\mathrm{grad}_g f$ for the Kähler form $ω_g$ is a holomorphic Killing vector field. We say that the pair $(g,f)$ is conformally Einstein-Maxwell Kähler metric if the conformal metric $\tilde g = f^{-2}g$ has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kähler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature Kähler manifolds. More generally we consider extensions of Calabi functional and extremal Kähler metrics, and prove an extension of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kähler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture.
math.DG↗
Almost all standard Lagrangian tori in C^n are not Hamiltonian volume minimizing
In 1993, Y.-G. Oh proposed a problem whether standard Lagrangian tori in C^n are volume minimizing under Hamiltonian isotopies of C^n. In this article, we prove that most of them do not have such property if the dimension n is greater than two. We also discuss the existence of Hamiltonian non-volume minimizing Lagrangian torus orbits of compact toric Kahler manifolds.
math.DG↗
Almost all Lagrangian torus orbits in ${\mathbb C}P^n$ are not Hamiltonian volume minimizing
All principal orbits of the standard Hamiltonian $T^n$-action on the complex projective space ${\mathbb C}P^n$ are Lagrangian tori.In this article, we prove that most of them are not volume minimizing under Hamiltonian isotopies of ${\mathbb C}P^n$ if the complex dimension $n$ is greater than two, although they are Hamiltonian minimal and Hamiltonian stable.
math.DG↗
An example of asymptotically Chow unstable manifolds with constant scalar curvature
Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature Kähler metrics in $c_1(L)$ and its automorphism group Aut$(M,L)$ is discrete, $(V,L)$ is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case when Aut$(V,L)$ is not discrete.
math.DG↗
Algebro-geometric semistability of polarized toric manifolds
Let $Δ\subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_Δ$ and the very ample $(\mathbb{C}^\times)^n$-equivariant line bundle $L_Δ$ on $X_Δ$ associated with $Δ$. In the present paper, we give a necessary and sufficient condition for Chow semistability of $(X_Δ,L_Δ^i)$ for a maximal torus action. We then see that asymptotic (relative) Chow semistability implies (relative) K-semistability for toric degenerations, which is proved by Ross and Thomas, without any knowledge of Riemann-Roch theorem and test configurations.
math.AG↗
Hilbert series and obstructions to asymptotic semistability
Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kähler (cscK) metric. A natural question is whether or not the other obstructions are linearly dependent on the obstruction to the existence of a cscK metric. The purpose of this paper is to see that this is not the case by exhibiting toric Fano threefolds in which these obstructions span at least two dimension. To see this we show that on toric Fano manifolds these obstructions are obtained as derivatives of the Hilbert series. This last observation should be regarded as an extension of the volume minimization of Martelli, Sparks and Yau.
math.DG↗
A necessary condition for Chow semistability of polarized toric manifolds
Let Δ\subset \mathbb{R}^n be an n-dimensional Delzant polytope. It is well-known that there exist the n-dimensional compact toric manifold X_Δand the very ample (\mathbb{C}^\times)^n-equivariant line bundle L_Δon X_Δassociated with Δ. In the present paper, we show that if (X_Δ,L_Δ^i) is Chow semistable then the sum of integer points in iΔis the constant multiple of the barycenter of Δ. Using this result we get a necessary condition for the polarized toric manifold (X_Δ,L_Δ) being asymptotically Chow semistable. Moreover we can generalize the result of Futaki, Sano and the author to the case when X_Δis not necessarily Fano.
math.AG↗
Einstein metrics and GIT stability
In this expository article we review the problem of finding Einstein metrics on compact Kähler manifolds and Sasaki manifolds. In the former half of this article we see that, in the Kähler case, the problem fits better with the notion of stability in Geometric Invariant Theory if we extend the problem to that of finding extremal Kähler metrics or constant scalar curvature Kähler (cscK) metrics. In the latter half of this paper we see that most of ideas in Kähler geometry extend to Sasaki geometry as transverse Kähler geometry. We also summarize recent results about the existence of toric Sasaki-Einstein metrics.
math.DG↗
Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds
In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant $f_1$ for the first Chern class case becomes an obstruction to the existence of transverse Kähler metric of constant scalar curvature. We prove the existence of transverse Kähler-Ricci solitons (or {\it Sasaki-Ricci soliton}) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if $S$ is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on $S$. As an application we obtain irregular toric Sasaki-Einstein metrics on the unit circle bundles of the powers of the canonical bundle of the two-point blow-up of the complex projective plane.
math.DG↗