The simplicity of the first spectral radius of a meromorphic map

Truong, Tuyen Trung

doi:10.1307/mmj/1409932635

Cited by 12 publications

(19 citation statements)

References 21 publications

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“…The latter, when combined with the proof of Lemma 4 in [45] and our assumption (ii), implies that {g * θ}. {C} = 0 for all π-exceptional curve C. The proof is completed.…”

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confidence: 59%

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Some dynamical properties of pseudo-automorphisms in dimension 3

Truong

2014

Trans. Amer. Math. Soc.

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Abstract. Let X be a compact Kähler manifold of dimension 3 and let f : X → X be a pseudo-automorphism. Under the mild condition that λ 1 (f ) 2 > λ 2 (f ), we prove the existence of invariant positive closed (1, 1) and (2, 2) currents, and we also discuss the (still open) problem of intersection of such currents. We prove a weak equidistribution result (which is essentially known in the literature) for Green (1, 1) currents of meromorphic selfmaps, not necessarily 1-algebraic stable, of a compact Kähler manifold of arbitrary dimension; and discuss how a stronger equidistribution result may be proved for pseudo-automorphisms in dimension 3. As a byproduct, we show that the intersection of some dynamically related currents are well-defined with respect to our definition here, even though not obviously to be seen so using the usual criteria.

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“…The latter, when combined with the proof of Lemma 4 in [45] and our assumption (ii), implies that {g * θ}. {C} = 0 for all π-exceptional curve C. The proof is completed.…”

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confidence: 59%

“…, D m ⊂ X be the irreducible subvarieties of dimension dim(X) − 2 in the images of the exceptional divisors of π : Z → X. Then by Lemma 4 in [45], we have in cohomology…”

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confidence: 99%

Some dynamical properties of pseudo-automorphisms in dimension 3

Truong

2014

Trans. Amer. Math. Soc.

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“…Let F 1 (which is isomorphic to P 1 ) be a fibre of the restriction of the blowup π 1 to the exceptional divisor π −1 1 (D 1 ). From the results in [18] we have that…”

Section: Proofs Of Main Theorems and Generalisationsmentioning

confidence: 95%

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

Truong¹

2018

Preprint

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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.Theorem. Let X, Y be compact Kähler manifolds of dimension 3, and f : X Y be a pseudo-isomorphism. Let α2, α3 be smooth closed (1, 1) forms on Y , and T1 a difference of two positive closed (1, 1) currents on X. Then, whether the intersection of the currents T1, f * (α2) and f * (α3) satisfies a Bedford-Taylor's type monotone convergence depends only on the cohomology classes of α2, α3.Special attention is given to the case where T1 = f * (α1) where α1 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 α1, α2 and α3, but in contrast the resulting signed measure is symmetric in α1, α2 and α3. 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 α1, α2, α3 are more singular. The use of global approximations (instead of local approximations) of singular currents and dynamics of pseudo-isomorphisms in dimension 3 are essential in proving these results. At the end of the paper we discuss the intersection of currents which are differences of positive closed (1, 1) currents in general.

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“…Some comments are in order. It is a theorem due to [Tru14] that the property (1) implies the existence of a unique class θ 1 (ϕ) in the group of classes of divisors N 1 (X) which satisfies the property ϕ * θ 1 (ϕ) = λ 1 (ϕ)θ 1 (ϕ). Property (2) implies that the class θ 1 (ϕ) is not nef and the last property is required to avoid transformations obtained after flopping a curve with infinite orbit (see Section 5).…”

Section: Introductionmentioning

confidence: 99%

Regularizations of positive entropy pseudo-automorphisms

Кузнецова¹

2022

Preprint

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We study positive entropy birational automorphisms of threefolds. We identify some conditions which imply that such an automorphism is non-regularizable. We show that this criterion applies in the example of a positive entropy birational automorphism of P 3 constructed in [Bla13], thus showing that for a general choice of parameters it is non-regularizable. Additionally, we establish a criterion which proves that the automorphism in this example does not preserve a structure of a fibration over a surface.

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The simplicity of the first spectral radius of a meromorphic map

Cited by 12 publications

References 21 publications

Some dynamical properties of pseudo-automorphisms in dimension 3

Some dynamical properties of pseudo-automorphisms in dimension 3

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

Regularizations of positive entropy pseudo-automorphisms

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