We introduce a new curvature constraint that provides an analog of the real bisectional curvature considered by Yang–Zheng [28] for the Aubin–Yau inequality. A unified perspective of the various forms of the Schwarz lemma is given, leading to novel Schwarz-type inequalities in both the Kähler and Hermitian categories.
In this paper, we study the stability of feedback particle filter (FPF) for linear filtering systems with Gaussian noises. We first provide some local contraction estimates of the exact linear FPF, whose conditional distribution is exactly the posterior distribution of the state as long as their initial values are equal. Then we study the convergence of the linear FPF formed by $N$ particles, and prove that the mean squared errors between the actual moments $(m_t, P_t)$ and their approximations $(m^{(N)}_t , P^{(N)}_t)$ by FPF are of order $\mathcal{O}(1/N)$ and decay exponentially fast as time $t$ goes to infinity.
We obtain an analogue of Nevanlinna theory of holomorphic mappings from a complete and stochastically complete Kähler manifold into a complex projective manifold. When certain curvature conditions are imposed, some Nevanlinna-type defect relations based on heat diffusion are established.
For each odd integer $r$ greater than one and not divisible by three we give explicit examples of infinite families of simply and tangentially homotopy equivalent but pairwise nonhomeomorphic $5$-dimensional closed homogeneous spaces with fundamental group isomorphic to $\mathbb{Z}/r$. As an application we construct the first examples of manifolds which possess infinitely many metrics of nonnegative sectional curvature with pairwise non-homeomorphic homogeneous souls of codimension three with trivial normal bundle, such that their curvatures and the diameters of the souls are uniformly bounded. As a by-product, if $L$ is a smooth and closed manifold homotopy equivalent to the standard $3$-dimensional lens space then the moduli space of complete smooth metrics of nonnegative sectional curvature on $L \times S^2 \times \mathbb{R}$ has infinitely many components. These manifolds are the first examples of manifolds fulfilling such geometric conditions and they serve as solutions to a problem posed by I. Belegradek, S. Kwasik and R. Schultz.
In this paper, we construct a Dieudonné theory for $\pi$-divisible $\mathcal{O}$-modules over a perfect field of characteristic $p$. Applying this theory, we prove the existence of slope filtration of $\pi$-divisible $\mathcal{O}$-modules over an integral normal Noetherian base. We also study minimal $\pi$-divisible $\mathcal{O}$-modules over an algebraically closed field of characteristic $p$ and prove that their isomorphism classes are determined by their $\pi$-torsion parts by introducing Oort’s filtration. Moreover, after a detailed study of deformations of $\pi$-divisible $\mathcal{O}$-modules via displays, we prove the generalized Traverso’s isogeny conjecture.
We study the deformed Hermitian–Yang–Mills equation on the blowup of complex projective space. Using symmetry, we express the equation as an ODE which can be solved using combinatorial methods if an algebraic stability condition is satisfied. This gives evidence towards a conjecture of the first author, T.C. Collins, and S.-T. Yau on general compact Kähler manifolds.
The primary objects of study in information geometry are statistical manifolds, which are parametrized families of probability measures, induced with the Fisher–Rao metric and a pair of torsion-free conjugate connections. In recent work [ZK20], the authors considered parametrized probability distributions as partially-flat statistical manifolds admitting torsion and showed that there is a complex-to-symplectic duality on the tangent bundles of such manifolds, based on the dualistic geometry of the underlying manifold.
In this paper, we explore this correspondence further in the context of Hessian manifolds, in which case the conjugate connections are both curvature- and torsion-free, and the associated dual pair of spaces are Kähler manifolds. We focus on several key examples and their geometric features. In particular, we show that the moduli space of univariate normal distributions gives rise to a correspondence between a Siegel domain and the Siegel–Jacobi space, which are spaces that appear in the context of automorphic forms.
Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realization in Cauchy–Riemann (CR) space. By introducing a new type of 3‑cell, we construct a different cell decomposition $\mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle admits a branched CR structure, whose branch locus is contained in the union of all edges of $\mathcal{D}_f$. Furthermore, we explicitly compute the ramification order around each component of the branch locus and analyse the corresponding holonomy representations.
In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau [7]. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang–Mills metrics on a given Kähler manifold. The goal of this paper is to give an $\varepsilon$-regularity theorem for the line bundle mean curvature flow. To establish the theorem, we provide a scale invariant monotone quantity. As a critical point of this quantity, we define self-shrinker solution of the line bundle mean curvature flow. The Liouville type theorem for self-shrinkers is also given. It plays an important role in the proof of the $\varepsilon$-regularity theorem.
We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded $3$-manifolds in complex surfaces. Topologically pseudoconvex (TPC) $3$-manifolds behave similarly to their smooth analogues, cutting out open domains of holomorphy (Stein surfaces), but they are much more common. We provide tools for constructing TPC embeddings, and show that every closed, oriented $3$-manifold $M$ has a TPC embedding in a compact, complex surface (without boundary) realizing any homotopy class of almost-complex structures (the analogue of the homotopy class of the contact plane field in the smooth case). We prove our tool theorems with invariants that classify almost-complex structures on any $4$-manifold homotopy equivalent to $M$. These invariants are amenable to computation and respected by homeomorphisms (not necessarily smooth). We study the two equivalence classes of smoothings on the product of a $3$-manifold with a line, and on collared ends. Both classes of smoothings are realized by holomorphic embeddings exhibiting any preassigned homotopy class of almost-complex structures. One class arises from TPC embedded $3$-manifolds, while the other likely does not.