Advances in Mathematics最新文献

{"title":"Tame maximal weights, relative types and valuations","authors":"Shijie Bao ,&nbsp;Qi'an Guan ,&nbsp;Zhitong Mi ,&nbsp;Zheng Yuan","doi":"10.1016/j.aim.2025.110364","DOIUrl":"10.1016/j.aim.2025.110364","url":null,"abstract":"<div><div>In this article, we obtain a class of tame maximal weights (Zhou weights). Using Tian functions (the function of jumping numbers with respect to the exponents of a holomorphic function or the multiples of a plurisubharmonic function) as a main tool, we establish an expression of relative types (Zhou numbers) to these tame maximal weights in integral form, which shows that the relative types satisfy tropical multiplicativity and tropical additivity. Thus, the relative types to Zhou weights are valuations (Zhou valuations) on the ring of germs of holomorphic functions. We use Tian functions and Zhou numbers to measure the singularities of plurisubharmonic functions, involving jumping numbers and multiplier ideal sheaves. Especially, the relative types to Zhou weights characterize the division relations of the ring of germs of holomorphic functions. Finally, we consider a global version of Zhou weights on domains in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, which is a generalization of the pluricomplex Green functions, and we obtain some properties of them, including continuity and some approximation results.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"477 ","pages":"Article 110364"},"PeriodicalIF":1.5,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134706","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}

{"title":"Characteristic forms of complex Cartan geometries II","authors":"Benjamin McKay","doi":"10.1016/j.aim.2025.110360","DOIUrl":"10.1016/j.aim.2025.110360","url":null,"abstract":"<div><div>Characteristic class relations in Dolbeault cohomology follow from the existence of a holomorphic Cartan geometry (for example, a holomorphic conformal structure or a holomorphic projective connection). These relations can be calculated directly from the representation theory of the structure group, without selecting any metric or connection or having any knowledge of the Dolbeault cohomology groups of the manifold. This paper improves on its predecessor <span><span>[35]</span></span> by allowing noncompact and non-Kähler manifolds and by deriving invariants in cohomology of vector bundles, not just in scalar Dolbeault cohomology, and computing relations involving Chern–Simons invariants in Dolbeault cohomology. For the geometric structures previously considered in its predecessor, this paper gives stronger results and simplifies the computations. It gives the first results on Chern–Simons invariants of Cartan geometries.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"477 ","pages":"Article 110360"},"PeriodicalIF":1.5,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134705","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}

{"title":"Vojta's abc conjecture for algebraic tori and applications over function fields","authors":"Ji Guo ,&nbsp;Khoa D. Nguyen ,&nbsp;Chia-Liang Sun ,&nbsp;Julie Tzu-Yueh Wang","doi":"10.1016/j.aim.2025.110358","DOIUrl":"10.1016/j.aim.2025.110358","url":null,"abstract":"<div><div>We prove Vojta's generalized abc conjecture for algebraic tori over function fields with exceptional sets that can be determined effectively. Additionally, we establish a version of the conjecture for toric varieties. As an application, we investigate the Lang-Vojta Conjecture for varieties of log general type that are ramified covers of <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> over function fields. In particular, we consider the case of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>∖</mo><mi>D</mi></math></span>, where <em>D</em> is a hypersurface over a function field in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> irreducible components and <span><math><mi>deg</mi><mo>⁡</mo><mi>D</mi><mo>≥</mo><mi>n</mi><mo>+</mo><mn>2</mn></math></span>. Our methods also apply to the complex situation, enabling us to find explicit exceptional sets for the corresponding case of Vojta's general abc conjecture (complex version) and the Green-Griffiths-Lang conjecture.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110358"},"PeriodicalIF":1.5,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115047","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}

{"title":"On global smooth solutions to the 2D isentropic and irrotational Chaplygin gases with short pulse data","authors":"Bingbing Ding ,&nbsp;Zhouping Xin ,&nbsp;Huicheng Yin","doi":"10.1016/j.aim.2025.110362","DOIUrl":"10.1016/j.aim.2025.110362","url":null,"abstract":"<div><div>This paper establishes the global existence of smooth solutions to the 2D isentropic and irrotational Euler equations for Chaplygin gases with a general class of short pulse initial data, which, in particular, resolves in this special case, the Majda's conjecture on the non-formation of shock waves of solutions from smooth initial data for multi-dimensional nonlinear symmetric systems which are totally linearly degenerate. Comparing to the 4D case, the major difficulties in this paper are caused by the slower time decay and the largeness of the solutions to the 2D quasilinear wave equation, some new auxiliary energies and multipliers are introduced to overcome these difficulties.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110362"},"PeriodicalIF":1.5,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115050","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}

{"title":"Isotropic constants and regular polytopes","authors":"Christian Kipp","doi":"10.1016/j.aim.2025.110361","DOIUrl":"10.1016/j.aim.2025.110361","url":null,"abstract":"<div><div>We discuss first-order optimality conditions for the isotropic constant and combine them with RS-movements to obtain structural information about polytopal maximizers. Strengthening a result by Rademacher, it is shown that a polytopal local maximizer with a simplicial vertex must be a simplex. A similar statement is shown for a centrally symmetric local maximizer with a simplicial vertex: it has to be a cross-polytope. Moreover, we show that a zonotope that maximizes the isotropic constant and that has a cubical zone must be a cube. Finally, we consider the class of zonotopes with at most <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span> generators and determine the extremals in this class.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110361"},"PeriodicalIF":1.5,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115051","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}

{"title":"Residues of quadratic Weyl group multiple Dirichlet series","authors":"Adrian Diaconu ,&nbsp;Bogdan Ion ,&nbsp;Vicenţiu Paşol ,&nbsp;Alexandru A. Popa","doi":"10.1016/j.aim.2025.110359","DOIUrl":"10.1016/j.aim.2025.110359","url":null,"abstract":"<div><div>We give explicit formulas for the residue of the Chinta-Gunnells average attached to a finite irreducible root system, at the polar divisor corresponding to a simple short root. The formula describes the residue in terms of the average attached to the root subsystem orthogonal to the relevant simple root. As a consequence, we obtain similar formulas for the residues of quadratic Weyl group multiple Dirichlet series over the rational function field and over the Gaussian field. The residue formula also allows us to obtain a new expression for the Chinta-Gunnells average of a finite irreducible root system, as an average over a maximal parabolic subgroup of a rational function that has an explicit description reflecting the combinatorics of the root system.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110359"},"PeriodicalIF":1.5,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115052","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}

{"title":"Cohomology of (φ,τ)-modules","authors":"Hui Gao ,&nbsp;Luming Zhao","doi":"10.1016/j.aim.2025.110363","DOIUrl":"10.1016/j.aim.2025.110363","url":null,"abstract":"<div><div>We construct cohomology theories for <span><math><mo>(</mo><mi>φ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>-modules, and study their relation with cohomology of <span><math><mo>(</mo><mi>φ</mi><mo>,</mo><mi>Γ</mi><mo>)</mo></math></span>-modules, as well as Galois cohomology. The method is axiomatic, and can treat the étale case, the overconvergent case, and the rigid-overconvergent case simultaneously. We use recent advances in locally analytic cohomology as a key ingredient.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110363"},"PeriodicalIF":1.5,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115053","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}

{"title":"Towards a generic absoluteness theorem for Chang models","authors":"Sandra Müller ,&nbsp;Grigor Sargsyan","doi":"10.1016/j.aim.2025.110357","DOIUrl":"10.1016/j.aim.2025.110357","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; be the set of all universally Baire sets of reals. Inspired by the work done in &lt;span&gt;&lt;span&gt;[22]&lt;/span&gt;&lt;/span&gt; and &lt;span&gt;&lt;span&gt;[21]&lt;/span&gt;&lt;/span&gt;, we introduce a new technique for establishing generic absoluteness results for models containing &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;Our main technical tool is an iteration that realizes &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; as the sets of reals in a derived model of some iterate of &lt;em&gt;V&lt;/em&gt;. We show, from a supercompact cardinal &lt;em&gt;κ&lt;/em&gt; and a proper class of Woodin cardinals, that whenever &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;Col&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is &lt;em&gt;V&lt;/em&gt;-generic and &lt;em&gt;h&lt;/em&gt; is &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-generic for some poset &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, there is an elementary embedding &lt;span&gt;&lt;math&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; as computed in &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a derived model of &lt;em&gt;M&lt;/em&gt; at &lt;span&gt;&lt;math&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Here &lt;em&gt;j&lt;/em&gt; is obtained by iteratively taking ultrapowers of &lt;em&gt;V&lt;/em&gt; by extenders with critical point &lt;em&gt;κ&lt;/em&gt; and its images.&lt;/div&gt;&lt;div&gt;As a corollary we obtain that &lt;span&gt;&lt;math&gt;&lt;mi&gt;Sealing&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; holds in &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, which was previously demonstrated by Woodin using the stationary tower forcing. Also, using a theorem of Woodin, we conclude that the derived model of &lt;em&gt;V&lt;/em&gt; at &lt;em&gt;κ&lt;/em&gt; satisfies &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;AD&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mtext&gt;“&lt;/mtext&gt;&lt;mi&gt;Θ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a regular cardinal”.&lt;/div&gt;&lt;div&gt;Inspired by core model induction, we introduce the definable powerset &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and use our derived model representation mentioned above to show that the theory of &lt;span&gt;&lt;math&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; cannot be changed by forcing (see &lt;span&gt;&lt;span&gt;Theorem 1","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110357"},"PeriodicalIF":1.5,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144106163","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}

{"title":"Five-variable p-adic L-functions for U(3)×U(2)","authors":"Ming-Lun Hsieh ,&nbsp;Shunsuke Yamana","doi":"10.1016/j.aim.2025.110355","DOIUrl":"10.1016/j.aim.2025.110355","url":null,"abstract":"<div><div>We construct a five-variable <em>p</em>-adic <em>L</em>-function attached to Hida families on the definite unitary groups <span><math><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span> and <span><math><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></math></span> by using the Ichino-Ikeda formula. The interpolation formula fits into the conjectural shape of <em>p</em>-adic <em>L</em>-functions predicted by Coates and Perrin-Riou.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110355"},"PeriodicalIF":1.5,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144099153","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}

{"title":"The structure of algebraic families of birational transformations","authors":"Andriy Regeta ,&nbsp;Christian Urech ,&nbsp;Immanuel van Santen","doi":"10.1016/j.aim.2025.110354","DOIUrl":"10.1016/j.aim.2025.110354","url":null,"abstract":"<div><div>We give a description of the algebraic families of birational transformations of an algebraic variety <em>X</em>. As an application, we show that the morphisms to <span><math><mi>Bir</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> given by algebraic families satisfy a Chevalley type result and a certain fibre-dimension formula. Moreover, we show that the algebraic subgroups of <span><math><mi>Bir</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> are exactly the closed finite-dimensional subgroups with finitely many components. We also study algebraic families of birational transformations preserving a fibration. This builds on previous work of Blanc-Furter <span><span>[2]</span></span>, Hanamura <span><span>[9]</span></span>, and Ramanujam <span><span>[20]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"476 ","pages":"Article 110354"},"PeriodicalIF":1.5,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144099154","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}