Pub Date : 2025-02-05DOI: 10.1016/j.aim.2025.110131
A.J. Homburg , J.S.W. Lamb , D.V. Turaev
We consider reversible vector fields in such that the set of fixed points of the involutory reversing symmetry is n-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that the topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection.
{"title":"Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy","authors":"A.J. Homburg , J.S.W. Lamb , D.V. Turaev","doi":"10.1016/j.aim.2025.110131","DOIUrl":"10.1016/j.aim.2025.110131","url":null,"abstract":"<div><div>We consider reversible vector fields in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span> such that the set of fixed points of the involutory reversing symmetry is <em>n</em>-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that the topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110131"},"PeriodicalIF":1.5,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-04DOI: 10.1016/j.aim.2025.110130
Luya Wang
Given two closed contact three-manifolds, one can form their contact connected sum via the Weinstein one-handle attachment. We study how pseudo-holomorphic curves in the symplectization behave under this operation. As a result, we give a connected sum formula for embedded contact homology.
{"title":"A connected sum formula for embedded contact homology","authors":"Luya Wang","doi":"10.1016/j.aim.2025.110130","DOIUrl":"10.1016/j.aim.2025.110130","url":null,"abstract":"<div><div>Given two closed contact three-manifolds, one can form their contact connected sum via the Weinstein one-handle attachment. We study how pseudo-holomorphic curves in the symplectization behave under this operation. As a result, we give a connected sum formula for embedded contact homology.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"464 ","pages":"Article 110130"},"PeriodicalIF":1.5,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143165801","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}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110088
Abbas Fakhari , Meysam Nassiri , Hesam Rajabzadeh
In this paper, we study stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. Such actions are known to exist only on one-dimensional manifolds. The aim of this paper is to introduce a geometric method to overcome this restriction and to construct higher dimensional examples. In particular, we show that every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class . We also prove the stable ergodicity of certain algebraic actions, including the natural action of a generic pair of matrices near the identity on a sphere of arbitrary dimension. These are consequences of the quasi-conformal blender, a local and stable mechanism/phenomenon introduced in this paper, which encapsulates our method for proving stable local ergodicity by providing quasi-conformal orbits with fine controlled geometry. The quasi-conformal blender is developed in the context of pseudo-semigroup actions of locally defined smooth diffeomorphisms, which allows for applications in diverse settings.
{"title":"Stable local dynamics: Expansion, quasi-conformality and ergodicity","authors":"Abbas Fakhari , Meysam Nassiri , Hesam Rajabzadeh","doi":"10.1016/j.aim.2024.110088","DOIUrl":"10.1016/j.aim.2024.110088","url":null,"abstract":"<div><div>In this paper, we study stable ergodicity of the action of groups of diffeomorphisms on smooth manifolds. Such actions are known to exist only on one-dimensional manifolds. The aim of this paper is to introduce a geometric method to overcome this restriction and to construct higher dimensional examples. In particular, we show that every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>α</mi></mrow></msup></math></span>. We also prove the stable ergodicity of certain algebraic actions, including the natural action of a generic pair of matrices near the identity on a sphere of arbitrary dimension. These are consequences of the <em>quasi-conformal blender</em>, a local and stable mechanism/phenomenon introduced in this paper, which encapsulates our method for proving stable local ergodicity by providing quasi-conformal orbits with fine controlled geometry. The quasi-conformal blender is developed in the context of pseudo-semigroup actions of locally defined smooth diffeomorphisms, which allows for applications in diverse settings.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110088"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149612","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}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110089
Tom Bridgeland
Joyce structures are a class of geometric structures which first arose in relation to holomorphic generating functions for Donaldson-Thomas invariants. They can be thought of as non-linear analogues of Frobenius structures, or as special classes of complex hyperkähler manifolds. We give a detailed introduction to Joyce structures, with particular focus on the geometry of the associated twistor space. We also prove several new results.
{"title":"Joyce structures and their twistor spaces","authors":"Tom Bridgeland","doi":"10.1016/j.aim.2024.110089","DOIUrl":"10.1016/j.aim.2024.110089","url":null,"abstract":"<div><div>Joyce structures are a class of geometric structures which first arose in relation to holomorphic generating functions for Donaldson-Thomas invariants. They can be thought of as non-linear analogues of Frobenius structures, or as special classes of complex hyperkähler manifolds. We give a detailed introduction to Joyce structures, with particular focus on the geometry of the associated twistor space. We also prove several new results.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110089"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110065
J.P. May, Ruoqi Zhang, Foling Zou
We give a description of unital operads in a symmetric monoidal category as monoids in a monoidal category of unital -sequences. This is a new variant of Kelly's old description of operads as monoids in the monoidal category of symmetric sequences. The monads associated to unital operads are the ones of interest in iterated loop space theory and factorization homology, among many other applications. Our new description of unital operads allows an illuminating comparison between the two-sided monadic bar constructions used in such applications and “classical” monoidal two-sided bar constructions. It also allows a more conceptual understanding of the scanning map central to non-abelian Poincaré duality in factorization homology.
{"title":"Unital operads, monoids, monads, and bar constructions","authors":"J.P. May, Ruoqi Zhang, Foling Zou","doi":"10.1016/j.aim.2024.110065","DOIUrl":"10.1016/j.aim.2024.110065","url":null,"abstract":"<div><div>We give a description of unital operads in a symmetric monoidal category as monoids in a monoidal category of unital <figure><img></figure>-sequences. This is a new variant of Kelly's old description of operads as monoids in the monoidal category of symmetric sequences. The monads associated to unital operads are the ones of interest in iterated loop space theory and factorization homology, among many other applications. Our new description of unital operads allows an illuminating comparison between the two-sided monadic bar constructions used in such applications and “classical” monoidal two-sided bar constructions. It also allows a more conceptual understanding of the scanning map central to non-abelian Poincaré duality in factorization homology.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110065"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164287","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}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110075
Jean-Baptiste Campesato , Goulwen Fichou , Adam Parusiński
We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with only normal crossing singularities by introducing their common extension: the complete Milnor fibration. We give two equivalent constructions: the first one extending the classical Kato–Nakayama log-space, and the second one, more geometric, based on the real oriented multigraph construction, a version of the real oriented deformation to the normal cone. As an application, we recover A'Campo's model of the topological Milnor fibration, by quotienting the motivic Milnor fibration with suitable powers of , and show that it determines the classical motivic Milnor fibre.
We also give precise formulae expressing how the introduced objects change under blowings-up. As an application, we show that the motivic Milnor fibre is well-defined as an element of a suitable Grothendieck ring without requiring that the Lefschetz motive be invertible.
{"title":"Motivic, logarithmic, and topological Milnor fibrations","authors":"Jean-Baptiste Campesato , Goulwen Fichou , Adam Parusiński","doi":"10.1016/j.aim.2024.110075","DOIUrl":"10.1016/j.aim.2024.110075","url":null,"abstract":"<div><div>We compare the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with only normal crossing singularities by introducing their common extension: the complete Milnor fibration. We give two equivalent constructions: the first one extending the classical Kato–Nakayama log-space, and the second one, more geometric, based on the real oriented multigraph construction, a version of the real oriented deformation to the normal cone. As an application, we recover A'Campo's model of the topological Milnor fibration, by quotienting the motivic Milnor fibration with suitable powers of <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>></mo><mn>0</mn></mrow></msub></math></span>, and show that it determines the classical motivic Milnor fibre.</div><div>We also give precise formulae expressing how the introduced objects change under blowings-up. As an application, we show that the motivic Milnor fibre is well-defined as an element of a suitable Grothendieck ring without requiring that the Lefschetz motive be invertible.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110075"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164291","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}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110067
Marco Abbadini , Vincenzo Marra , Luca Spada
We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points carry an arithmetic character to be preserved by continuous maps. The arithmetic character of a point is (the complete isomorphism invariant of) a metrically complete additive subgroup of the real numbers containing 1—namely, either for an integer , or the whole of . The main result needed to establish the extended duality theorem is a substantial generalisation of Urysohn's Lemma to such “arithmetic” compact Hausdorff spaces. The original duality is obtained by considering the full subcategory of spaces every point of which is assigned the entire group of real numbers. In the Introduction we indicate motivations from and connections with the theory of dimension groups.
{"title":"Stone-Gelfand duality for metrically complete lattice-ordered groups","authors":"Marco Abbadini , Vincenzo Marra , Luca Spada","doi":"10.1016/j.aim.2024.110067","DOIUrl":"10.1016/j.aim.2024.110067","url":null,"abstract":"<div><div>We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points carry an arithmetic character to be preserved by continuous maps. The arithmetic character of a point is (the complete isomorphism invariant of) a metrically complete additive subgroup of the real numbers containing 1—namely, either <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><mi>Z</mi></math></span> for an integer <span><math><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></math></span>, or the whole of <span><math><mi>R</mi></math></span>. The main result needed to establish the extended duality theorem is a substantial generalisation of Urysohn's Lemma to such “arithmetic” compact Hausdorff spaces. The original duality is obtained by considering the full subcategory of spaces every point of which is assigned the entire group of real numbers. In the Introduction we indicate motivations from and connections with the theory of dimension groups.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110067"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110053
Joav Orovitz, Raz Slutsky, Itamar Vigdorovich
We show that the space of traces of free products of the form , where and are compact metrizable spaces without isolated points, is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. In particular, the space of traces of the free group on generators is a Poulsen simplex, and we demonstrate that this is no longer true for many virtually free groups. Using a similar strategy, we show that the space of traces of the free product of matrix algebras is a Poulsen simplex as well, answering a question of Musat and Rørdam for . Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.
{"title":"The space of traces of the free group and free products of matrix algebras","authors":"Joav Orovitz, Raz Slutsky, Itamar Vigdorovich","doi":"10.1016/j.aim.2024.110053","DOIUrl":"10.1016/j.aim.2024.110053","url":null,"abstract":"<div><div>We show that the space of traces of free products of the form <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>⁎</mo><mi>C</mi><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are compact metrizable spaces without isolated points, is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. In particular, the space of traces of the free group <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> on <span><math><mn>2</mn><mo>≤</mo><mi>d</mi><mo>≤</mo><mo>∞</mo></math></span> generators is a Poulsen simplex, and we demonstrate that this is no longer true for many virtually free groups. Using a similar strategy, we show that the space of traces of the free product of matrix algebras <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo><mo>⁎</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> is a Poulsen simplex as well, answering a question of Musat and Rørdam for <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110053"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110091
Parker Evans
<div><div>We give an explicit geometric structures interpretation of the <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>-Hitchin component <span><math><mrow><mi>Hit</mi></mrow><mo>(</mo><mi>S</mi><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>)</mo><mo>⊂</mo><mi>χ</mi><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>S</mi><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>)</mo></math></span> of a closed oriented surface <em>S</em> of genus <span><math><mi>g</mi><mo>≥</mo><mn>2</mn></math></span>. In particular, we prove <span><math><mrow><mi>Hit</mi></mrow><mo>(</mo><mi>S</mi><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>)</mo></math></span> is naturally homeomorphic to a moduli space <span><math><mi>M</mi></math></span> of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span>-structures for <span><math><mi>G</mi><mo>=</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> and <span><math><mi>X</mi><mo>=</mo><msup><mrow><mi>Ein</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msup></math></span> on a fiber bundle <span><math><mi>C</mi></math></span> over <em>S</em> via the descended holonomy map. Explicitly, <span><math><mi>C</mi></math></span> is the direct sum of fiber bundles <figure><img></figure> with fiber <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>S</mi><mo>×</mo><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>S</mi><mo>×</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>, where <em>UTS</em> denotes the unit tangent bundle.</div><div>The geometric structure associated to a <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>-Hitchin representation <em>ρ</em> is explicitly constructed from the unique associated <em>ρ</em>-equivariant alternating almost-complex curve <span><math><mover><mrow><mi>ν</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>:</mo><mover><mrow><mi>S</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>→</mo><msup><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mn>2</mn><mo>,</mo><mn>4</mn></mrow></msup></math></span>; we critically use recent work of Collier-Toulisse on the moduli space of such curves. Our explicit geometric structures are examined in the <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>-Fuchsian case and shown to be unrelated to the <span><math><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><msup><mrow><mi>
{"title":"Geometric structures for the G2′-Hitchin component","authors":"Parker Evans","doi":"10.1016/j.aim.2024.110091","DOIUrl":"10.1016/j.aim.2024.110091","url":null,"abstract":"<div><div>We give an explicit geometric structures interpretation of the <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>-Hitchin component <span><math><mrow><mi>Hit</mi></mrow><mo>(</mo><mi>S</mi><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>)</mo><mo>⊂</mo><mi>χ</mi><mo>(</mo><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>S</mi><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>)</mo></math></span> of a closed oriented surface <em>S</em> of genus <span><math><mi>g</mi><mo>≥</mo><mn>2</mn></math></span>. In particular, we prove <span><math><mrow><mi>Hit</mi></mrow><mo>(</mo><mi>S</mi><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>)</mo></math></span> is naturally homeomorphic to a moduli space <span><math><mi>M</mi></math></span> of <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span>-structures for <span><math><mi>G</mi><mo>=</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> and <span><math><mi>X</mi><mo>=</mo><msup><mrow><mi>Ein</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>3</mn></mrow></msup></math></span> on a fiber bundle <span><math><mi>C</mi></math></span> over <em>S</em> via the descended holonomy map. Explicitly, <span><math><mi>C</mi></math></span> is the direct sum of fiber bundles <figure><img></figure> with fiber <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>S</mi><mo>×</mo><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>S</mi><mo>×</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>, where <em>UTS</em> denotes the unit tangent bundle.</div><div>The geometric structure associated to a <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>-Hitchin representation <em>ρ</em> is explicitly constructed from the unique associated <em>ρ</em>-equivariant alternating almost-complex curve <span><math><mover><mrow><mi>ν</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>:</mo><mover><mrow><mi>S</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>→</mo><msup><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mn>2</mn><mo>,</mo><mn>4</mn></mrow></msup></math></span>; we critically use recent work of Collier-Toulisse on the moduli space of such curves. Our explicit geometric structures are examined in the <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>-Fuchsian case and shown to be unrelated to the <span><math><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><msup><mrow><mi>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"462 ","pages":"Article 110091"},"PeriodicalIF":1.5,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143149613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-01DOI: 10.1016/j.aim.2024.110066
Nicholas Rungi , Andrea Tamburelli
The aim of this paper is to show the existence and give an explicit description of a semi-pseudo-Riemannian metric and a symplectic form on the -Hitchin component, both compatible with Labourie and Loftin's complex structure. In particular, they are non-degenerate on a neighborhood of the Fuchsian locus, where they give rise to a mapping class group invariant pseudo-Kähler structure that restricts to a multiple of the Weil-Petersson metric on Teichmüller space. By comparing our symplectic form with Goldman's , we prove that the pair cannot define a Kähler structure on the Hitchin component.
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