Pub Date : 2019-10-09DOI: 10.4310/AJM.2021.v25.n3.a3
Guolei Zhong
Let $X$ be a normal compact Kahler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int-amplified if $f^*xi-xi=eta$ for some Kahler classes $xi$ and $eta$. First, we show that this definition generalizes the notation in the projective setting. Second, we prove that for the cases of $X$ being smooth, a surface or a threefold with mild singularities, if $X$ admits an int-amplified endomorphism with pseudo-effective canonical divisor, then it is a $Q$-torus. Finally, we consider a normal compact Kahler threefold $Y$ with only terminal singularities and show that, replacing $f$ by a positive power, we can run the minimal model program (MMP) $f$-equivariantly for such $Y$ and reach either a $Q$-torus or a Fano (projective) variety of Picard number one.
{"title":"Int-amplified endomorphisms of compact Kähler spaces","authors":"Guolei Zhong","doi":"10.4310/AJM.2021.v25.n3.a3","DOIUrl":"https://doi.org/10.4310/AJM.2021.v25.n3.a3","url":null,"abstract":"Let $X$ be a normal compact Kahler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int-amplified if $f^*xi-xi=eta$ for some Kahler classes $xi$ and $eta$. First, we show that this definition generalizes the notation in the projective setting. Second, we prove that for the cases of $X$ being smooth, a surface or a threefold with mild singularities, if $X$ admits an int-amplified endomorphism with pseudo-effective canonical divisor, then it is a $Q$-torus. Finally, we consider a normal compact Kahler threefold $Y$ with only terminal singularities and show that, replacing $f$ by a positive power, we can run the minimal model program (MMP) $f$-equivariantly for such $Y$ and reach either a $Q$-torus or a Fano (projective) variety of Picard number one.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47813537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-05DOI: 10.4310/ajm.2021.v25.n4.a7
Yanwen Luo
In this paper, we define a new metric structure on the shape space of a high genus surface. We introduce a rigorous definition of a shape of a surface and construct a metric based on two energies measuring the area distortion and the angle distortion of a quasiconformal homeomorphism. We show that the energy minimizer in a fixed homotopy class is achieved by a quasiconformal homeomorphism by the lower semicontinuity property of these two energies. Finally, we explore the properties of several energies related to the Dirichlet energy.
{"title":"Comparing shapes of high genus surfaces","authors":"Yanwen Luo","doi":"10.4310/ajm.2021.v25.n4.a7","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n4.a7","url":null,"abstract":"In this paper, we define a new metric structure on the shape space of a high genus surface. We introduce a rigorous definition of a shape of a surface and construct a metric based on two energies measuring the area distortion and the angle distortion of a quasiconformal homeomorphism. We show that the energy minimizer in a fixed homotopy class is achieved by a quasiconformal homeomorphism by the lower semicontinuity property of these two energies. Finally, we explore the properties of several energies related to the Dirichlet energy.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46345189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-19DOI: 10.4310/ajm.2021.v25.n2.a3
K. Hare, K. Hare, Wanchun Shen
It is known that the heuristic principle, referred to as the multifractal formalism, need not hold for self-similar measures with overlap, such as the $3$-fold convolution of the Cantor measure and certain Bernoulli convolutions. In this paper we study an important function in the multifractal theory, the $L^{q}$-spectrum, $tau (q)$, for measures of finite type, a class of self-similar measures that includes these examples. Corresponding to each measure, we introduce finitely many variants on the $% L^{q}$-spectrum which arise naturally from the finite type structure and are often easier to understand than $tau $. We show that $tau$ is always bounded by the minimum of these variants and is equal to the minimum variant for $qgeq 0$. This particular variant coincides with the $L^{q}$-spectrum of the measure $mu$ restricted to appropriate subsets of its support. If the IFS satisfies particular structural properties, which do hold for the above examples, then $tau$ is shown to be the minimum of these variants for all $q$. Under certain assumptions on the local dimensions of $mu$, we prove that the minimum variant for $q ll 0$ coincides with the straight line having slope equal to the maximum local dimension of $mu $. Again, this is the case with the examples above. More generally, bounds are given for $tau$ and its variants in terms of notions closely related to the local dimensions of $mu $.
{"title":"The $L^q$-spectrum for a class of self-similar measures with overlap","authors":"K. Hare, K. Hare, Wanchun Shen","doi":"10.4310/ajm.2021.v25.n2.a3","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n2.a3","url":null,"abstract":"It is known that the heuristic principle, referred to as the multifractal formalism, need not hold for self-similar measures with overlap, such as the $3$-fold convolution of the Cantor measure and certain Bernoulli convolutions. In this paper we study an important function in the multifractal theory, the $L^{q}$-spectrum, $tau (q)$, for measures of finite type, a class of self-similar measures that includes these examples. Corresponding to each measure, we introduce finitely many variants on the $% L^{q}$-spectrum which arise naturally from the finite type structure and are often easier to understand than $tau $. We show that $tau$ is always bounded by the minimum of these variants and is equal to the minimum variant for $qgeq 0$. This particular variant coincides with the $L^{q}$-spectrum of the measure $mu$ restricted to appropriate subsets of its support. If the IFS satisfies particular structural properties, which do hold for the above examples, then $tau$ is shown to be the minimum of these variants for all $q$. Under certain assumptions on the local dimensions of $mu$, we prove that the minimum variant for $q ll 0$ coincides with the straight line having slope equal to the maximum local dimension of $mu $. Again, this is the case with the examples above. More generally, bounds are given for $tau$ and its variants in terms of notions closely related to the local dimensions of $mu $.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44549061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-11DOI: 10.4310/ajm.2019.v23.n6.a3
Oussama Hijazi, S. Montiel, S. Raulot
In this paper, we generalize a theorem {a} la Alexandrov of Wang, Wang and Zhang [WWZ] for closed codimension-two spacelike submanifolds in the Minkowski spacetime for an adapted CMC condition .
本文推广了Wang, Wang和Zhang [WWZ]关于Minkowski时空中闭上维-二类空间子流形的一个定理{a} la Alexandrov [a]。
{"title":"An Alexandrov theorem in Minkowski spacetime","authors":"Oussama Hijazi, S. Montiel, S. Raulot","doi":"10.4310/ajm.2019.v23.n6.a3","DOIUrl":"https://doi.org/10.4310/ajm.2019.v23.n6.a3","url":null,"abstract":"In this paper, we generalize a theorem {a} la Alexandrov of Wang, Wang and Zhang [WWZ] for closed codimension-two spacelike submanifolds in the Minkowski spacetime for an adapted CMC condition .","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47604896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-07-02DOI: 10.4310/ajm.2021.v25.n4.a3
L. Borisov, Chengxi Wang
We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $mathbb{P}_{mathbf{Sigma}}$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of $mathbb{P}_{mathbf{Sigma}}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $mathbb{P}_{mathbf{Sigma}}$.
{"title":"On strong exceptional collections of line bundles of maximal length on fano toric Deligne–Mumford stacks","authors":"L. Borisov, Chengxi Wang","doi":"10.4310/ajm.2021.v25.n4.a3","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n4.a3","url":null,"abstract":"We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $mathbb{P}_{mathbf{Sigma}}$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of $mathbb{P}_{mathbf{Sigma}}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $mathbb{P}_{mathbf{Sigma}}$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48683406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-29DOI: 10.4310/ajm.2021.v25.n4.a1
I. Biswas, J. Hurtubise
We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the ``sheaf of holomorphic connections'' (the sheaf of splittings of the one-jet sequence) for the determinant (Quillen) line bundle over the moduli space of vector bundles on a compact connected Riemann surface. This equivalence is shown to be holomorphically symplectic. The equivalences, both holomorphic and symplectic, seem to be quite general, in that they extend to other general families of holomorphic bundles and holomorphic connections, in particular those arising from ``Tyurin families" of stable bundles over the surface. These families generalize the Tyurin parametrization of stable vector bundles $E$ over a compact connected Riemann surface, and one can build above them spaces of (equivalence classes of) connections, which are again symplectic. These spaces are also symplectically biholomorphically equivalent to the sheaf of connections for the determinant bundle over the Tyurin family. The last portion of the paper shows how this extends to moduli of framed bundles.
{"title":"Meromorphic connections, determinant line bundles and the Tyurin parametrization","authors":"I. Biswas, J. Hurtubise","doi":"10.4310/ajm.2021.v25.n4.a1","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n4.a1","url":null,"abstract":"We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the ``sheaf of holomorphic connections'' (the sheaf of splittings of the one-jet sequence) for the determinant (Quillen) line bundle over the moduli space of vector bundles on a compact connected Riemann surface. This equivalence is shown to be holomorphically symplectic. The equivalences, both holomorphic and symplectic, seem to be quite general, in that they extend to other general families of holomorphic bundles and holomorphic connections, in particular those arising from ``Tyurin families\" of stable bundles over the surface. These families generalize the Tyurin parametrization of stable vector bundles $E$ over a compact connected Riemann surface, and one can build above them spaces of (equivalence classes of) connections, which are again symplectic. These spaces are also symplectically biholomorphically equivalent to the sheaf of connections for the determinant bundle over the Tyurin family. The last portion of the paper shows how this extends to moduli of framed bundles.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45994715","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-25DOI: 10.4310/ajm.2021.v25.n5.a4
Y. Hu, J. Niu
We construct a smooth moduli stack of genus one weighted curves with twisted fields. We show the moduli is isomorphic to Hu-Li's blowup stack of the moduli of genus one weighted curves. This leads to a modular interpretation of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. Our results should extend to resolutions of the moduli of stable maps of higher genera.
{"title":"Moduli of curves of genus one with twisted fields","authors":"Y. Hu, J. Niu","doi":"10.4310/ajm.2021.v25.n5.a4","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n5.a4","url":null,"abstract":"We construct a smooth moduli stack of genus one weighted curves with twisted fields. We show the moduli is isomorphic to Hu-Li's blowup stack of the moduli of genus one weighted curves. This leads to a modular interpretation of Vakil-Zinger's desingularization of the moduli of genus one stable maps to projective spaces. Our results should extend to resolutions of the moduli of stable maps of higher genera.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43791447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-29DOI: 10.4310/AJM.2020.V24.N5.A1
Helena Cobo, M. Soto, J. Tornero
The seminal concept of characteristic polygon of an embedded algebroid surface, first developed by Hironaka, seems well suited for combinatorially (perhaps even effectively) tracking of a resolution process. However, the way this object evolves through the resolution of singularities was not really well understood, as some references had pointed out. The aim of this paper is to explain, in a clear way, how this object changes as the surface gets resolved. In order to get a precise description of the phenomena involved, we need to use different techniques and ideas. Eventually, some effective results regarding the number of blow-ups needed to decrease the multiplicity are obtained as a side product.
{"title":"Combinatorics and their evolution in resolution of embedded algebroid surfaces","authors":"Helena Cobo, M. Soto, J. Tornero","doi":"10.4310/AJM.2020.V24.N5.A1","DOIUrl":"https://doi.org/10.4310/AJM.2020.V24.N5.A1","url":null,"abstract":"The seminal concept of characteristic polygon of an embedded algebroid surface, first developed by Hironaka, seems well suited for combinatorially (perhaps even effectively) tracking of a resolution process. However, the way this object evolves through the resolution of singularities was not really well understood, as some references had pointed out. The aim of this paper is to explain, in a clear way, how this object changes as the surface gets resolved. In order to get a precise description of the phenomena involved, we need to use different techniques and ideas. Eventually, some effective results regarding the number of blow-ups needed to decrease the multiplicity are obtained as a side product.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43407256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-23DOI: 10.4310/ajm.2021.v25.n4.a6
P. Laurain, Mircea Petrache
Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in $mathbb R^d$ has remained open, except for $d=1$ and for compactly supported measures in $d=2$, and for codimension $1$. In this paper we study $1$-dimensional measures in $mathbb R^d$ for all $d$ and classify uniform measures with connected $1$-dimensional support, which turn out to be homogeneous measures. We provide as well a partial classification of general uniform measures of dimension $1$ in the absence of the connected support hypothesis.
{"title":"Classification of uniformly distributed measures of dimension $1$ in general codimension","authors":"P. Laurain, Mircea Petrache","doi":"10.4310/ajm.2021.v25.n4.a6","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n4.a6","url":null,"abstract":"Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in $mathbb R^d$ has remained open, except for $d=1$ and for compactly supported measures in $d=2$, and for codimension $1$. In this paper we study $1$-dimensional measures in $mathbb R^d$ for all $d$ and classify uniform measures with connected $1$-dimensional support, which turn out to be homogeneous measures. We provide as well a partial classification of general uniform measures of dimension $1$ in the absence of the connected support hypothesis.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44556482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-13DOI: 10.4310/ajm.2021.v25.n6.a4
M. Hirose, H. Murahara, Shingo Saito
Ohno's relation states that a certain sum, which we call an Ohno type sum, of multiple zeta values remains unchanged if we replace the base index by its dual index. In view of Oyama's theorem concerning Ohno type sums of finite and symmetric multiple zeta values, Kaneko looked at Ohno type sums of finite and symmetric multiple zeta-star values and made a conjecture on the generating function for a specific index of depth three. In this paper, we confirm this conjecture and further give a formula for arbitrary indices of depth three.
{"title":"Generating functions for Ohno type sums of finite and symmetric multiple zeta-star values","authors":"M. Hirose, H. Murahara, Shingo Saito","doi":"10.4310/ajm.2021.v25.n6.a4","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n6.a4","url":null,"abstract":"Ohno's relation states that a certain sum, which we call an Ohno type sum, of multiple zeta values remains unchanged if we replace the base index by its dual index. In view of Oyama's theorem concerning Ohno type sums of finite and symmetric multiple zeta values, Kaneko looked at Ohno type sums of finite and symmetric multiple zeta-star values and made a conjecture on the generating function for a specific index of depth three. In this paper, we confirm this conjecture and further give a formula for arbitrary indices of depth three.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49034128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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