Abstract Let${mathbb M}$ be an affine variety equipped with a foliation, both defined over a number field ${mathbb K}$. For an algebraic $Vsubset {mathbb M}$ over ${mathbb K}$, write $delta _{V}$ for the maximum of the degree and log-height of V. Write $Sigma _{V}$ for the points where the leaves intersect V improperly. Fix a compact subset ${mathcal B}$ of a leaf ${mathcal L}$. We prove effective bounds on the geometry of the intersection ${mathcal B}cap V$. In particular, when $operatorname {codim} V=dim {mathcal L}$ we prove that $#({mathcal B}cap V)$ is bounded by a polynomial in $delta _{V}$ and $log operatorname {dist}^{-1}({mathcal B},Sigma _{V})$. Using these bounds we prove a result on the interpolation of algebraic points in images of ${mathcal B}cap V$ by an algebraic map $Phi $. For instance, under suitable conditions we show that $Phi ({mathcal B}cap V)$ contains at most $operatorname {poly}(g,h)$ algebraic points of log-height h and degree g. We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections $P,Q$ of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever $P,Q$ are simultaneously torsion their order of torsion is bounded effectively by a polynomial in $delta _{P},delta _{Q}$; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given $Vsubset {mathbb C}^{n}$, there is an (ineffective) upper bound, polynomial in $delta _{V}$, for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galois-orbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.
{"title":"Point counting for foliations over number fields","authors":"Gal Binyamini","doi":"10.1017/fmp.2021.20","DOIUrl":"https://doi.org/10.1017/fmp.2021.20","url":null,"abstract":"Abstract Let${mathbb M}$ be an affine variety equipped with a foliation, both defined over a number field ${mathbb K}$. For an algebraic $Vsubset {mathbb M}$ over ${mathbb K}$, write $delta _{V}$ for the maximum of the degree and log-height of V. Write $Sigma _{V}$ for the points where the leaves intersect V improperly. Fix a compact subset ${mathcal B}$ of a leaf ${mathcal L}$. We prove effective bounds on the geometry of the intersection ${mathcal B}cap V$. In particular, when $operatorname {codim} V=dim {mathcal L}$ we prove that $#({mathcal B}cap V)$ is bounded by a polynomial in $delta _{V}$ and $log operatorname {dist}^{-1}({mathcal B},Sigma _{V})$. Using these bounds we prove a result on the interpolation of algebraic points in images of ${mathcal B}cap V$ by an algebraic map $Phi $. For instance, under suitable conditions we show that $Phi ({mathcal B}cap V)$ contains at most $operatorname {poly}(g,h)$ algebraic points of log-height h and degree g. We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections $P,Q$ of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever $P,Q$ are simultaneously torsion their order of torsion is bounded effectively by a polynomial in $delta _{P},delta _{Q}$; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given $Vsubset {mathbb C}^{n}$, there is an (ineffective) upper bound, polynomial in $delta _{V}$, for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galois-orbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $mathrm {SL}_n$- and $mathrm {PGL}_n$-Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $mathrm {SL}_n$-Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p-adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler.
{"title":"Endoscopic decompositions and the Hausel–Thaddeus conjecture","authors":"D. Maulik, Junliang Shen","doi":"10.1017/fmp.2021.7","DOIUrl":"https://doi.org/10.1017/fmp.2021.7","url":null,"abstract":"Abstract We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $mathrm {SL}_n$- and $mathrm {PGL}_n$-Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $mathrm {SL}_n$-Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p-adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48720065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $mathfrak {gl}_2(mathbb {Q}_p)$ . As applications, we prove a classicality result for overconvergent eigenforms of weight 1 and give a new proof of the Fontaine–Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight k, we show its corresponding Galois representation has Hodge–Tate–Sen weights $0,k-1$ and prove a converse result.
{"title":"On locally analytic vectors of the completed cohomology of modular curves","authors":"Lue Pan","doi":"10.1017/fmp.2022.1","DOIUrl":"https://doi.org/10.1017/fmp.2022.1","url":null,"abstract":"Abstract We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of \u0000$mathfrak {gl}_2(mathbb {Q}_p)$\u0000 . As applications, we prove a classicality result for overconvergent eigenforms of weight 1 and give a new proof of the Fontaine–Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight k, we show its corresponding Galois representation has Hodge–Tate–Sen weights \u0000$0,k-1$\u0000 and prove a converse result.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45502370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions. More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions. These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the $phi ^4$ problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.
{"title":"Quadratic Klein-Gordon equations with a potential in one dimension","authors":"P. Germain, F. Pusateri","doi":"10.1017/fmp.2022.9","DOIUrl":"https://doi.org/10.1017/fmp.2022.9","url":null,"abstract":"Abstract This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions. More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions. These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the \u0000$phi ^4$\u0000 problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46588163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Matteo Costantini, Martin Möller, Jonathan Zachhuber
Abstract For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification. The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of multi-scale differentials and computational tools in the Chow ring, such as a description of normal bundles to boundary divisors.
{"title":"The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials","authors":"Matteo Costantini, Martin Möller, Jonathan Zachhuber","doi":"10.1017/fmp.2022.10","DOIUrl":"https://doi.org/10.1017/fmp.2022.10","url":null,"abstract":"Abstract For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification. The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of multi-scale differentials and computational tools in the Chow ring, such as a description of normal bundles to boundary divisors.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42174161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension $n+1geqslant 3$ and with partial data. We consider the case when the set $Omega _{mathrm{in}}$ , where the sources are supported, and the set $Omega _{mathrm{out}}$ , where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point $p_{in}in Omega _{mathrm{in}}$ and the past of the point $p_{out}in Omega _{mathrm{out}}$ . In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in $Omega _{mathrm{in}}$ and observations in $Omega _{mathrm{out}}$ , determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.
{"title":"Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers","authors":"A. Feizmohammadi, M. Lassas, L. Oksanen","doi":"10.1017/fmp.2021.11","DOIUrl":"https://doi.org/10.1017/fmp.2021.11","url":null,"abstract":"Abstract The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension \u0000$n+1geqslant 3$\u0000 and with partial data. We consider the case when the set \u0000$Omega _{mathrm{in}}$\u0000 , where the sources are supported, and the set \u0000$Omega _{mathrm{out}}$\u0000 , where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point \u0000$p_{in}in Omega _{mathrm{in}}$\u0000 and the past of the point \u0000$p_{out}in Omega _{mathrm{out}}$\u0000 . In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in \u0000$Omega _{mathrm{in}}$\u0000 and observations in \u0000$Omega _{mathrm{out}}$\u0000 , determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45589284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider the $mathbb {T}^{4}$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the $H^{1}$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $H^{1}$ uniqueness for the $ mathbb {R}^{3}/mathbb {R}^{4}/mathbb {T}^{3}/mathbb {T}^{4}$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.
{"title":"Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on $mathbb {T}^{4}$","authors":"Xuwen Chen, J. Holmer","doi":"10.1017/fmp.2021.16","DOIUrl":"https://doi.org/10.1017/fmp.2021.16","url":null,"abstract":"Abstract We consider the $mathbb {T}^{4}$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the $H^{1}$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $H^{1}$ uniqueness for the $ mathbb {R}^{3}/mathbb {R}^{4}/mathbb {T}^{3}/mathbb {T}^{4}$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42851234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.
{"title":"Resonance-based schemes for dispersive equations via decorated trees","authors":"Y. Bruned, Katharina Schratz","doi":"10.1017/fmp.2021.13","DOIUrl":"https://doi.org/10.1017/fmp.2021.13","url":null,"abstract":"Abstract We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48874055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfaces at generalised Eckardt points and for cubic surfaces at all points, and (c) provide a new algebraic proof of Tian’s criterion for K-stability, amongst other applications.
{"title":"K-stability of Fano varieties via admissible flags","authors":"Hamid Abban, Ziquan Zhuang","doi":"10.1017/fmp.2022.11","DOIUrl":"https://doi.org/10.1017/fmp.2022.11","url":null,"abstract":"Abstract We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfaces at generalised Eckardt points and for cubic surfaces at all points, and (c) provide a new algebraic proof of Tian’s criterion for K-stability, amongst other applications.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44573644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let $Gamma _{g}$ be the fundamental group of a closed connected orientable surface of genus $ggeq 2$ . We develop a new method for integrating over the representation space $mathbb {X}_{g,n}=mathrm {Hom}(Gamma _{g},S_{n})$ , where $S_{n}$ is the symmetric group of permutations of ${1,ldots ,n}$ . Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g. Given $phi in mathbb {X}_{g,n}$ and $gamma in Gamma _{g}$ , we let $mathsf {fix}_{gamma }(phi )$ be the number of fixed points of the permutation $phi (gamma )$ . The function $mathsf {fix}_{gamma }$ is a special case of a natural family of functions on $mathbb {X}_{g,n}$ called Wilson loops. Our new methodology leads to an asymptotic formula, as $nto infty $ , for the expectation of $mathsf {fix}_{gamma }$ with respect to the uniform probability measure on $mathbb {X}_{g,n}$ , which is denoted by $mathbb {E}_{g,n}[mathsf {fix}_{gamma }]$ . We prove that if $gamma in Gamma _{g}$ is not the identity and q is maximal such that $gamma $ is a q th power in $Gamma _{g}$ , then $$begin{align*}mathbb{E}_{g,n}left[mathsf{fix}_{gamma}right]=d(q)+O(n^{-1}) end{align*}$$ as $nto infty $ , where $dleft (qright )$ is the number of divisors of q. Even the weaker corollary that $mathbb {E}_{g,n}[mathsf {fix}_{gamma }]=o(n)$ as $nto infty $ is a new result of this paper. We also prove that $mathbb {E}_{g,n}[mathsf {fix}_{gamma }]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$ .
{"title":"The Asymptotic Statistics of Random Covering Surfaces","authors":"Michael Magee, Doron Puder","doi":"10.1017/fmp.2023.13","DOIUrl":"https://doi.org/10.1017/fmp.2023.13","url":null,"abstract":"Abstract Let \u0000$Gamma _{g}$\u0000 be the fundamental group of a closed connected orientable surface of genus \u0000$ggeq 2$\u0000 . We develop a new method for integrating over the representation space \u0000$mathbb {X}_{g,n}=mathrm {Hom}(Gamma _{g},S_{n})$\u0000 , where \u0000$S_{n}$\u0000 is the symmetric group of permutations of \u0000${1,ldots ,n}$\u0000 . Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g. Given \u0000$phi in mathbb {X}_{g,n}$\u0000 and \u0000$gamma in Gamma _{g}$\u0000 , we let \u0000$mathsf {fix}_{gamma }(phi )$\u0000 be the number of fixed points of the permutation \u0000$phi (gamma )$\u0000 . The function \u0000$mathsf {fix}_{gamma }$\u0000 is a special case of a natural family of functions on \u0000$mathbb {X}_{g,n}$\u0000 called Wilson loops. Our new methodology leads to an asymptotic formula, as \u0000$nto infty $\u0000 , for the expectation of \u0000$mathsf {fix}_{gamma }$\u0000 with respect to the uniform probability measure on \u0000$mathbb {X}_{g,n}$\u0000 , which is denoted by \u0000$mathbb {E}_{g,n}[mathsf {fix}_{gamma }]$\u0000 . We prove that if \u0000$gamma in Gamma _{g}$\u0000 is not the identity and q is maximal such that \u0000$gamma $\u0000 is a q th power in \u0000$Gamma _{g}$\u0000 , then \u0000$$begin{align*}mathbb{E}_{g,n}left[mathsf{fix}_{gamma}right]=d(q)+O(n^{-1}) end{align*}$$\u0000 as \u0000$nto infty $\u0000 , where \u0000$dleft (qright )$\u0000 is the number of divisors of q. Even the weaker corollary that \u0000$mathbb {E}_{g,n}[mathsf {fix}_{gamma }]=o(n)$\u0000 as \u0000$nto infty $\u0000 is a new result of this paper. We also prove that \u0000$mathbb {E}_{g,n}[mathsf {fix}_{gamma }]$\u0000 can be approximated to any order \u0000$O(n^{-M})$\u0000 by a polynomial in \u0000$n^{-1}$\u0000 .","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":null,"pages":null},"PeriodicalIF":2.3,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44854396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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