arXiv subjects

Tuyen Trung Truong

Publications and source records attributed to Tuyen Trung Truong.

At least 55 records · Page 3Linked to original sources

arXiv2019

The Monge-Ampere operator of some singular (1,1) currents coming from pseudo-isomorphisms in dimension $3$

A wide and natural class of closed currents - which are differences of positive closed currents - can be constructed by pulling back smooth closed forms using rational maps. These currents are very singular in general, and hence defining intersections between them is challenging. In this paper, we use our previous results to investigate this question in the case where the rational maps in question are pseudo-isomorphisms (i.e. bimeromorphic maps which, along with their inverses, have no exceptional divisors) in dimension $3$. Our main result, to be described in a more concrete form later in the paper, is as follows. {\bf Theorem.} Let $X,Y$ be compact Kähler manifolds of dimension $3$, and $f:X\dashrightarrow Y$ be a pseudo-isomorphism. Let $α_2,α_3$ be smooth closed $(1,1)$ forms on $Y$, and $T_1$ a difference of two positive closed $(1,1)$ currents on $X$. Then, whether the intersection of the currents $T_1$, $f^*(α_2)$ and $f^*(α_3)$ satisfies a Bedford-Taylor's type monotone convergence depends only on the cohomology classes of $α_2,α_3$. Special attention is given to the case where $T_1=f^*(α_1)$ where $α_1$ is a smooth closed $(1,1)$ form on $Y$. It is then shown that satisfying the above mentioned Bedford-Taylor's type monotone convergence is asymmetric in $α_1$, $α_2$ and $α_3$, but in contrast the resulting signed measure is symmetric in $α_1$, $α_2$ and $α_3$. We relate this Bedford-Taylor's type monotone convergence to the least-negative intersection we defined previously. These results can be extended to the case where $α_1$, $α_2$, $α_3$ are more singular. Dynamics of pseudo-isomorphisms in dimension $3$ are essential in proving these results.

math.CV↗

arXiv2018

Bounded birationality and isomorphism problems are computable

Let $X,Y$ be two irreducible subvarieties of the projective space $\mathbb{P}^n$, and $d\geq 1$ an integer number. The main result of this paper is an algorithm to construct {\bf explicitly}, in terms of $d$ and the ideals defining $X$ and $Y$, a quasi-affine algebraic variety parametrising the set of all birational maps $f$ from $X$ onto $Y$ which can be extended to a self-rational map of $\mathbb{P}^n$ of degree $\leq d$. Based on this result, we propose an approach towards the rationality problem (see Section 3 below), solve it for some simple cases (varieties of general type or curves), and state a rough strategy for reducing it to some simpler cases via Iitaka's fibrations. We also prove similar results for the case $f$ is a dominant rational map, regular morphism, isomorphism or regular embedding. Similar results are valid for varieties over an arbitrary algebraically closed field, and also for maps on non-projective varieties.

math.AG↗

arXiv2017

On subelliptic manifolds

A smooth complex quasi-affine algebraic variety $Y$ is flexible if its special group $\SAut (Y)$ of automorphisms (generated by the elements of one-dimensional unipotent subgroups of $\Aut (Y)$) acts transitively on $Y$. An irreducible algebraic manifold $X$ is locally stably flexible if it is the union $\bigcup X_i$ of a finite number of Zariski open sets, each $X_i$ being quasi-affine, so that there is a positive integer $N$ for which $X_i\times \mathbb{C}^N$ is flexible for every $i$. The main result of this paper is that the blowup of a locally stably flexible manifold at a smooth algebraic submanifold (not necessarily equi-dimensional or connected) is subelliptic, and hence Oka. This result is proven as a corollary of some general results concerning the so-called $k$-flexible manifolds.

math.AG↗

arXiv2016

Algebraic subellipticity and dominability of blow-ups of affine spaces

Little is known about the behaviour of the Oka property of a complex manifold with respect to blowing up a submanifold. A manifold is of Class $\mathscr A$ if it is the complement of an algebraic subvariety of codimension at least $2$ in an algebraic manifold that is Zariski-locally isomorphic to $\mathbb C^n$. A manifold of Class $\mathscr A$ is algebraically subelliptic and hence Oka, and a manifold of Class $\mathscr A$ blown up at finitely many points is of Class $\mathscr A$. Our main result is that a manifold of Class $\mathscr A$ blown up along an arbitrary algebraic submanifold (not necessarily connected) is algebraically subelliptic. For algebraic manifolds in general, we prove that strong algebraic dominability, a weakening of algebraic subellipticity, is preserved by an arbitrary blow-up with a smooth centre. We use the main result to confirm a prediction of Forster's famous conjecture that every open Riemann surface may be properly holomorphically embedded into $\mathbb C^2$.

math.AG↗

arXiv2016

Relations between dynamical degrees, Weil's Riemann hypothesis and the standard conjectures

Let $\mathbb{K}$ be an algebraically closed field, $X$ a smooth projective variety over $\mathbb{K}$ and $f:X\rightarrow X$ a dominant regular morphism. Let $N^i(X)$ be the group of algebraic cycles modulo numerical equivalence. Let $χ(f)$ be the spectral radius of the pullback $f^*:H^*(X,\mathbb{Q}_l)\rightarrow H^*(X,\mathbb{Q}_l)$ on $l$-adic cohomology groups, and $λ(f)$ the spectral radius of the pullback $f^*:N^*(X)\rightarrow N^*(X)$. We prove in this paper, by using consequences of Deligne's proof of Weil's Riemann hypothesis, that $χ(f)=λ(f)$. This answers affirmatively a question posed by Esnault and Srinivas. Consequently, the algebraic entropy $\log χ(f)$ of an endomorphism is both a birational invariant and étale invariant. More general results are proven if either $\mathbb{K}=\overline{\mathbb{F}_p}$ or the Fundamental Conjecture D (numerical equivalence vs homological equivalence) holds. Among other results in the paper, we show that if some properties of dynamical degrees, known in the case $\mathbb{K}=\mathbb{C}$, hold in positive characteristics, then simple proofs of Weil's Riemann hypothesis follow.

math.AG↗

arXiv2016

Étale dynamical systems and topological entropy

In this paper, we consider two questions about topological entropy of dynamical systems. We propose to resolve these questions by the same approach of using étale analogs of topological and algebraic dynamical systems. The first question is to define topological entropy for a topological dynamical system $(f,X,Ω)$. The main idea is to make use - in addition to invariant compact subspaces of $(X,Ω)$ - of compactifications of étale covers $π:(f',X',Ω')\rightarrow (f,X,Ω)$, that is $π\circ f'=f\circ π$ and the fibers of $π$ are all finite. We prove some basic results and propose a conjecture, whose validity allows us to prove further results. The second question is to define topological entropy for algebraic dynamical systems, with the requirement that it should be as close to the pullback on cohomology groups as possible. To this end, we develop an étale analog of algebraic dynamical systems.

math.DS↗

arXiv2016

Comments on Sampson's approach toward Hodge conjecture on Abelian varieties

Let $A$ be an Abelian variety of dimension $n$. For $0<p<2n$ an odd integer, Sampson constructed a surjective homomorphism $π:J^p(A)\rightarrow A$, where $J^p(A)$ is the higher Weil Jacobian variety of $A$. Let $\widehatω$ be a fixed form in $H^{1,1}(J^p(A),\mathbb{Q})$, and $N=\dim (J^p(A))$. He observes that if the map $π_*(\widehat{ω}^{N-p-1}\wedge .): H^{1,1}(J^p(A),\mathbb{Q})\rightarrow H^{n-p,n-p}(A,\mathbb{Q})$ is injective, then the Hodge conjecture is true for $A$ in bidegree $(p,p)$. In this paper, we give some clarification of the approach and show that the map above is {not injective} except some special cases where the Hodge conjecture is already known. We propose a modified approach.

math.AG↗

arXiv2016

Relative dynamical degrees of correspondences over a field of arbitrary characteristic

Let $K$ be an algebraically closed field of arbitrary characteristic, $X$ an irreducible variety and $Y$ an irreducible projective variety over $K$, both are not necessarily smooth. Let $f:X\rightarrow X$ and $g:Y\rightarrow Y$ be dominant correspondences, and $π:X\rightarrow Y$ a dominant rational map such that $π\circ f=g\circ π$. We define relative dynamical degrees $λ_p(f|π)$ ($p=0,\ldots ,\dim (X)-\dim (Y)$). These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when $Y$ is smooth and $g$ is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy $(φ,ψ)$ from $(X_2,f_2)\rightarrow (Y_2,g_2)$ to $(X_1,f_1)\rightarrow (Y_1,g_1)$ we have $λ_p(f_1|π_1)\geq λ_p(f_2|π_2)$ for all $p$. Many of our results are new even when $K=\mathbb{C}$. We make use of de Jong's alterations and Roberts' version of Chow's moving lemma. In the lack of resolution of singularities, the consideration of correspondences is necessary even when $f,g$ are rational maps. The case $K$ is not algebraically closed further requires working with correspondences over reducible varieties.

math.AG↗

arXiv2016

Growth of the number of periodic points for meromorphic maps

We show that any dominant meromorphic self-map f of a compact Kaehler manifold X is an Artin-Mazur map. More precisely, if P_n(f) is the number of its isolated periodic points of period n (counted with multiplicity), then P_n(f) grows at most exponentially fast with respect to n and the exponential rate is at most equal to the algebraic entropy of f. Further estimates are given when X is a surface. Among the techniques introduced in this paper, the h-dimension of the density between two arbitrary positive closed currents on a compact Kaehler surface is obtained.

math.DS↗

arXiv2015

Some approaches toward the Jacobian conjecture

In this paper, we study a so-called Condition C1 and a weaker Condition C2. For Druzkowski maps Condition C2 is equivalent to the Jacobian conjecture. Main results obtained: - Stating new equivalent formulations of the Jacobian conjecture. - Formulating some generalisations of the Jacobian conjecture and giving both theoretical and experimental evidences to support them. - Showing Condition C1 holds for a generic matrix of any given rank, is an invariant for a certain group action, and Condition C2 is an invariant for cubic similarity matrices. - Giving one heuristic argument for the truth of the Jacobian Conjecture. - Giving an effective (time saving) method to check whether a given Druzkowski map satisfies the Jacobian conjecture, explaining theoretically and checking on many examples including those previously considered by other authors. - Proposing approaches toward resolving the Jacobian conjecture. Showing that a generic Druzkowski map satisfies the criteria of some of these approaches (see Theorem 1.12), and hence expecting to be able to check these approaches for a given Druzkowski map very quickly. -As an application, proposing a strategy to use cubic similarity to check that Druzkowski maps of dimension $\leq 9$ satisfy the Jacobian conjecture.

math.AG↗

arXiv2015

Automorphisms of blowups of threefolds being Fano or having Picard number $1$

Let $X_0$ be a smooth projective threefold which is Fano or which has Picard number $1$. Let $π:X\rightarrow X_0$ be a finite composition of blowups along smooth centers. We show that for "almost all" of such $X$, if $f\in Aut(X)$ then its first and second dynamical degrees are the same. We also construct many examples of finite blowups $X\rightarrow X_0$, on which any automorphism is of zero entropy. The main idea is that because of the log-concavity of dynamical systems and the invariance of Chern classes under holomorphic automorphisms, there are some constraints on the nef cohomology classes. We will also discuss a possible application of these results to a threefold constructed by Kenji Ueno.

math.AG↗

arXiv2015

(Relative) dynamical degrees of rational maps over an algebraic closed field

The main purpose of this paper is to define dynamical degrees for rational maps over an algebraic closed field of characteristic zero and prove some basic properties (such as log-concavity) and give some applications. We also define relative dynamical degrees and prove a "product formula" for dynamical degrees of semi-conjugate rational maps in the algebraic setting. The main tools are the Chow's moving lemma and a formula for the degree of the cone over a subvariety of $\mathbb{P}^N$. The proofs of these results are valid as long as resolution of singularities are available (or more generally if appropriate birational models of the maps under consideration are available). This observation is applied for the cases of surfaces and threefolds over a field of positive characteristic.

math.AG↗

arXiv2014

Curves with decomposable normal vector bundles and automorphism groups

If a smooth projective threefold $X$ satisfies a certain Property A (see below for definition), then any automorphism of $X$ has zero entropy. Let $Y$ be a smooth projective threefold satisfying Property A. Let $π:X\rightarrow Y$ be a blowup at either a point or at a smooth curve $C\subset Y$ with the following two properties: i) $c_1(Y).C$ is an odd number, and ii) the normal vector bundle $N_{C/Y} $ is decomposable. Then we show that $X$ also satisfies Property A. As a further application of Property A we prove the following result. Let $X_1$ be the blowup of $X_0=\mathbb{P}^3$ at a finite number of points, and let $X=X_2$ be the blowup of $X_1$ at a finite number of pairwise disjoint smooth curves (here the images of these curves in $X_0$ may intersect). Then any automorphism of $X$ has the same first and second dynamical degrees. Under some further conditions, then any automorphism of $X$ has zero entropy. The result is also valid for threefolds $X_0$ satisfying a certain condition on the second Chern class. Some explicit examples are given.

math.AG↗

arXiv2014

A mathematical model for measurements in Quantum Mechanics

Let $V=\mathbb{C}^N$, and $H$ (an observable) a Hermitian linear operator on $V$. Let $v_1,..., v_n$ be an orthonormal basis for $V$. Let $\mathcal{M}$ be a measurement apparatus prepared to measure a state of an observed system and collapses the state to one of the $v_j$'s. Here we propose a simple model which explains the Born rule and is compatible with entanglement.

math-ph↗

arXiv2014

Time and Space separation in General Relativity

Let $(M,g)$ be a spacetime. That is, $M$ is a real manifold of dimension $4$ equipped with a Lorentzian metric $g$. We show that any separation of time and space in $M$ is equivalent to introducing a (non-smooth) Riemann metric $h$. If $h$ is smooth, it induces a smooth line bundle $T_p\rightarrow M$, whose any fiber is generated by a time-like vector, called the time bundle. Whether $(M,g,h)$ is time orientable or not corresponds to whether this line bundle is trivial or not. As well-known, the last condition is characterized by the first Stiefel-Whitney class $w^1(T_p)\in H^1(M,\mathbb{Z}/2)$. We then define a partial time orientation of $M$ as a section of the line bundle $T\rightarrow M$. As applications, we discuss time and space differentiations on $M$.

math-ph↗

arXiv2014

Least negative intersections of positive closed currents on compact Kähler manifolds

Let $X$ be a compact Kähler manifold of dimension $k$. Let $R$ be a positive closed $(p,p)$ current on $X$, and $T_1,\ldots ,T_{k-p}$ be positive closed $(1,1)$ currents on $X$. We define a so-called least negative intersection of the currents $T_1,T_2,\ldots ,T_{k-p}$ and $R$, as a sublinear bounded operator \begin{eqnarray*} \bigwedge (T_1,\ldots ,T_{k-p},R):~C^0(X)\rightarrow \mathbb{R}. \end{eqnarray*} This operator is {\bf symmetric} in $T_1,\ldots ,T_{k-p}$. It is {\bf independent} of the choice of a quasi-potential $u_i$ of $T_i$, of the choice of a smooth closed $(1,1)$ form $θ_i$ in the cohomology class of $T_i$, and of the choice of a Kähler form on $X$. Its total mass $<\bigwedge (T_1,\ldots ,T_{k-p},R),1>$ is the intersection in cohomology $\{T_1\}\{T_2\}\ldots \{T_{k-p}\}.\{R\}$. It has a semi-continuous property concerning approximating $T_i$ by appropriate smooth closed $(1,1)$ forms, plus some other good properties. If $p=0$ and $T_1=\ldots =T_k=T$, we have a least negative Monge-Ampere operator $MA(T)=\bigwedge (T,\ldots ,T)$. If the set where $T$ has positive Lelong numbers does not contain any curve, then $MA(T)$ is positive. Several examples are given.

math.CV↗

arXiv2014

Pseudo-isomorphisms in dimension $3$ and applications to complex Monge-Ampere equation

Let $X$ and $Y$ be compact Kähler manifolds of dimension $3$. A bimeromorphic map $f:X\rightarrow Y$ is pseudo-isomorphic if $f:X-I(f)\rightarrow Y-I(f^{-1})$ is an isomorphism. In this paper we investigate some properties of pseudo-isomorphisms. As an application, we associate to any pseudo-isomorphism in dimension $3$ and a smooth closed $(3,3)$ form $δ$ on $X\times X$ representing the cohomology class of the diagonal $Δ_X$, a Monge-Ampere operator $MA(f^*(θ),δ)=f^*(θ)\wedge f^*(θ)\wedge f^*(θ)$, here $θ$ is a smooth closed $(1,1)$ form on $Y$. We show that this Monge-Ampere operator is independent of the choice of $δ$, if the following cohomologous condition is satisfied: {\bf Condition.} For any curve $C\subset I(f^{-1})$, we have $\{θ\}.\{C\}=0$ in cohomology. We conclude the paper examining a simple pseudo-isomorphism in dimension $3$.

math.CV↗

arXiv2014

Complex Monge-Ampere operators via pseudo-isomorphisms: the well-defined cases

Let $X$ and $Y$ be compact Kähler manifolds of dimension $3$. A bimeromorphic map $f:X\rightarrow Y$ is pseudo-isomorphic if $f:X-I(f)\rightarrow Y-I(f^{-1})$ is an isomorphism. Let $T=T^+-T^-$ be a current on $Y$, where $T^{\pm}$ are positive closed $(1,1)$ currents which are smooth outside a finite number of points. We assume that the following condition is satisfied: {\bf Condition 1.} For every curve $C$ in $I(f^{-1})$, then in cohomology $\{T\}.\{C\}=0$. Then, we define a natural push-forward $f_*(φdd^cu\wedge f^*(T))$ for a quasi-psh function $u$ and a smooth function $φ$ on $Y$. We show that this pushforward satisfies a Bedford-Taylor's monotone convergence type. Assume moreover that the following two conditions are satisfied {\bf Condition 2.} The signed measure $T\wedge T\wedge T$ has no mass on $I(f^{-1})$. {\bf Condition 3.} For every curve $C$ in $I(f^{-1})$, the measure $T\wedge [C]$ has no Dirac mass. Then, we define a Monge-Ampere operator $MA(f^*(T))=f^*(T)\wedge f^*(T)\wedge f^*(T)$ for $f^*(T)$. We show that this Monge-Ampere operator satisfies several continuous properties, including a Bedford-Taylor's monotone convergence type when $T$ is positive. The measures $MA(f^*(T))$ are in general quite singular. Also, note that it may be not possible to define $f^*(T^{\pm})\wedge f^*(T^{\pm})\wedge f^*(T^{\pm})$.

math.CV↗