Acta Mathematica Vietnamica
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Issue 1, Volume 45, March 2020
2020
Special Issue: Nevanlinna theory and Complex Geometry
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Le Van Thiem—the Founder of Contemporary Mathematics in Vietnam Ha Huy Khoai PDF3-10
▸ Abstract
We give a brief biography of Le Van Thiem and a survey on his contributions in mathematics and for the development of mathematics in Vietnam.
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Normal Projective Varieties Admitting Polarized or Int-amplified Endomorphisms Sheng Meng , De-Qi Zhang PDF11-26
▸ Abstract
Let X be a normal projective variety admitting a polarized or int-amplified endomorphism f. We list up characteristic properties of such an endomorphism and classify such a variety from the aspects of its singularity, anti-canonical divisor, and Kodaira dimension. Then, we run the equivariant minimal model program with respect to not just the single f but also the monoid SEnd(X) of all surjective endomorphisms of X, up to finite-index. Several applications are given. We also give both algebraic and geometric characterizations of toric varieties via polarized endomorphisms.
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Prolongations of Infinitesimal Automorphisms of Cubic Hypersurfaces with Nonzero Hessian Jun-Muk Hwang PDF27-47
▸ Abstract
We study the connected component of the automorphism group of a cubic hypersurface over complex numbers. When the cubic hypersurface has nonzero Hessian, this group is usually small. But there are examples with unusually large automorphism groups: the secants of Severi varieties. Can we characterize them by the property of having unusually large automorphism groups? We study this question from the viewpoint of prolongations of the Lie algebras. Our result characterizes the secants of Severi varieties, among cubic hypersurfaces with nonzero Hessian and smooth singular locus, in terms of prolongations of a certain type.
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Infinite Entropy for Transcendental Entire Functions with an Omitted Value Anna Miriam Benini , John Erik Fornæss , Han Peters PDF49-52
▸ Abstract
We prove that entire transcendental holomorphic functions with an omitted value have infinite entropy. A proof for general transcendental entire functions will be given in an upcoming paper.
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$L^p$ Metric Geometry of Big and Nef Cohomology Classes Eleonora Di Nezza , Chinh H. Lu PDF53-69
▸ Abstract
Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$, and let $\theta$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p$, $p\ge 1$, on the finite energy space $\mathcal {E}^{p}(X,\theta )$, making it a complete geodesic metric space.
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Uniqueness Results for Holomorphic Mappings on the Disc Min Ru , Richard Walden PDF71-81
▸ Abstract
This paper derives several unicity results for a class of holomorphic mappings from the disc into compact Riemann surfaces as well as into the complex projective space $\mathbb P^n(\mathbb C)$. This is done by using the Nevanlinna theory for holomorphic maps where the source is a disc developed by Ru-Sibony (to appear).
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A Remark on the Continuous Subsolution Problem for the Complex Monge-Ampère Equation Sławomir Kołodziej , Ngoc Cuong Nguyen PDF83-91
▸ Abstract
We prove that if the modulus of continuity of a plurisubharmonic subsolution satisfies a Dini-type condition then the Dirichlet problem for the complex Monge-Ampère equation has the continuous solution. The modulus of continuity of the solution also given if the right hand side is locally dominated by capacity.
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Complex Monge-Ampère Equation in Strictly Pseudoconvex Domains Hoang-Son Do , Thai Duong Do , Hoang Hiep Pham PDF93-101
▸ Abstract
We study the complex Monge-Ampère equation $(dd^c u)^n=\mu$ in a strictly pseudoconvex domain $\Omega$ with the boundary condition $u=\varphi$, where $\varphi\in C(\partial\Omega)$. We provide a non-trivial sufficient condition for continuity of the solution $u$ outside “small sets”.
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Some Open Problems on Holomorphic Foliation Theory Tien-Cuong Dinh , Nessim Sibony PDF103-112
▸ Abstract
We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible.
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Torus Equivariant Szegő Kernel Asymptotics on Strongly Pseudoconvex CR Manifolds Hendrik Herrmann , Chin-Yu Hsiao , Xiaoshan Li PDF113-135
▸ Abstract
Let (X,T^{1,0}X) be a compact strongly pseudoconvex CR manifold of dimension $2n+1$. Assume that a $d$-dimensional torus $T^d$ acts on $X$. In this work, we study the behavior of torus equivariant Szegő kernels and prove that the weighted torus equivariant Szegő kernels admit asymptotic expansions.
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On the Homeomorphism Type of Smooth Projective Fourfolds Keiji Oguiso , Thomas Peternell PDF137-160
▸ Abstract
In this paper, we study smooth complex projective 4-folds which are topologically equivalent. First we show that Fano fourfolds are never oriented homeomorphic to Ricci-flat projective fourfolds and that Calabi-Yau manifolds and hyperkähler manifolds in dimension $\ge 4$ are never oriented homeomorphic. Finally, we give a coarse classification of smooth projective fourfolds which are oriented homeomorphic to a hyperkähler fourfold which is deformation equivalent to the Hilbert scheme $S^{[2]} $ of two points of a projective $K_3$ surface $S$.
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Semicontinuity of Isometry Groups and Isomorphism Groups: a Survey Robert E. Greene , Kang-Tae Kim PDF161-169
▸ Abstract
Isomorphisms that preserve a certain geometric structure are easily destroyed by an arbitrary small deformation of the structure, but restoring them requires a definite amount of change. Such an intuitive understanding has actually been established as precise theorems, via Lie groups, Riemannian geometry, and several complex variables. The aim of this paper is to present a concise but comprehensive exposition.
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Some Analytic Results for Kimura Diffusion Operators Charles L. Epstein , Jon Wilkening PDF171-181
▸ Abstract
In this note, we prove several analytical results about generalized Kimura diffusion operators, $L$, defined on compact manifolds with corners, $P$. It is shown that the $\mathcal C^{0}(P)$-graph closure of $L$ acting on $\mathcal C^{2}(P)$ always has a compact resolvent. In the $1 d$-case, where $P = [0,1]$, we also establish a gradient estimate $\|\partial _{x} f\|_{\mathcal C^{0}([0,1])}\leq C\| L f\|_{\mathcal C^{0}([0,1])}$, provided that $L$ has strictly positive weights at $\partial[0,1] = \{0,1\}$. This in turn leads to a precise characterization of the domain of the $\mathcal C^0$-graph closure in this case.
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Fiber Invariants of Projective Morphisms and Regularity of Powers of Ideals Sankhaneel Bisui , Huy Tài Hà , Abu Chackalamannil Thomas PDF183-198
▸ Abstract
We introduce an invariant, associated to a coherent sheaf of graded modules over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and $a^*$-invariant of powers of homogeneous ideals. Specifically, for an equigenerated homogeneous ideal $I$ in a standard graded algebra over a Noetherian ring, we give bounds for the smallest values of power $q$ starting from which $a^*(I^q)$ and $\mathrm{reg}(I^q)$ become linear functions.
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Recent Developments in the Theory of Anderson Modules Bruno Anglès , Tuan Ngo Dac , Floric Tavares Ribeiro PDF199-216
▸ Abstract
Let $K$ be a global function field over a finite field of characteristic $p$ and let $A$ be the ring of elements of $K$ which are regular outside a fixed place of $K$. This report presents recent developments in the arithmetic of special $L$-values of Anderson $A$-modules. Provided that $p$ does not divide the class number of $K$, we prove an “analytic class number formula” for Anderson $A$-modules with the help of a recent work of Debry. For tensor powers of the Carlitz module, we explain how to derive several log-algebraicity results from the class number formula for these Anderson modules.
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Analyticity, Regularity, and Generalized Polynomial Chaos Approximation of Stochastic, Parametric Parabolic Two-Scale Partial Differential Equations Viet Ha Hoang PDF217-247
▸ Abstract
We study two-scale parabolic partial differential equations whose coefficient is stochastic and depends linearly on a sequence of pairwise independent random variables which are uniformly distributed in a compact interval. We cast the problem into a deterministic two-scale parabolic problem which depends on a sequence of real parameters in a compact interval. Passing to the limit when the microscale tends to zero, using two-scale homogenization, we obtain the parametric two-scale homogenized problem. This problem contains the solution to the homogenized equation which describes the solution to the original two-scale parabolic problem macroscopically, and the corrector which encodes the microscopic information. The solution of this two-scale homogenized equation is represented as a generalized polynomial chaos (gpc) expansion according to a polynomial basis of the $L^2$ space of the parameters. We use a semidiscrete Galerkin approximation which projects the solution into a space of parametric functions which contain a finite number of pre-chosen gpc modes. Analyticity of the solution of the two-scale homogenized equation with respect to the parameters is established. Under mild assumptions, we show the summability of the coefficients of the solution’s gpc expansion. From this, an explicit error estimate for the semidiscrete Galerkin approximation in terms of the number of the chosen gpc modes is derived when these gpc modes are chosen as the $N$ best ones according to the norms of the gpc coefficients. Regularity of the gpc coefficients and summability of their norms in the regularity spaces are also established. Using the solution of the best $N$ term semidiscrete Galerkin approximation, we derive an approximation for the solution of the original two-scale problem, with an explicit convergence rate in terms of the microscopic scale and the number of gpc modes in the Galerkin approximation.
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On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values Nguyen Tu Cuong , Pham Hung Quy PDF249-260
▸ Abstract
Let $(R, \frak {m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$. A famous result of Northcott says that if $M$ is Cohen-Macaulay, then the index of reducibility of parameter ideals on $M$ is an invariant of the module. The aim of this paper is to extend Northcott’s theorem for any finitely generated $R$-module. We call this invariant the stable value of the indices of reducibility of parameter ideals of $M$. We also introduce the limit value of the indices of reducibility of parameter ideals of $M$.
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Approximate Cloaking Using Transformation Optics for Acoustic and Electromagnetic Waves Hoai-Minh Nguyen , Michael S. Vogelius PDF261-280
▸ Abstract
This is a survey on approximate cloaking using transformation optics on acoustic and electromagnetic waves. Both the time-harmonic and the time regimes are discussed.
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A Note on the Singularities of Residue Currents of Integrally Closed Ideals Elizabeth Wulcan PDF281-289
▸ Abstract
Given a free resolution of an ideal $\mathfrak{a}$ of holomorphic functions, there is an associated residue current $R$ that coincides with the classical Coleff-Herrera product if $\mathfrak{a}$ is a complete intersection ideal and whose annihilator ideal equals $\mathfrak{a}$. In the case when $\mathfrak{a}$ is an Artinian monomial ideal, we show that the singularities of R are small in a certain sense if and only if $\mathfrak{a}$ is integrally closed.
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Holomorphic Curves into Algebraic Varieties Intersecting Moving Hypersurface Targets Gerd Dethloff , Tran Van Tan PDF291-308
▸ Abstract
In Ru (Ann. Math. 169, 255–267 2009), Min Ru proved a second main theorem for algebraically nondegenerate holomorphic curves in complex projective varieties intersecting fixed hypersurface targets. In this paper, by introducing a new proof method for the case of projective varieties, we generalize this result to moving hypersurface targets.