We show that the Looijenga--Lunts--Verbitsky Lie algebra acting on the cohomology of a hyperkahler variety is a derived invariant. We use this to give upper bounds for the image of the group of derived auto-equivalences on the cohomology of a hyperkahler variety. For certain Hilbert squares of K3 surfaces, we show that the obtained upper bound is close to being sharp.
{"title":"Derived equivalences of hyperkähler varieties","authors":"Lenny Taelman","doi":"10.2140/gt.2023.27.2649","DOIUrl":"https://doi.org/10.2140/gt.2023.27.2649","url":null,"abstract":"We show that the Looijenga--Lunts--Verbitsky Lie algebra acting on the cohomology of a hyperkahler variety is a derived invariant. We use this to give upper bounds for the image of the group of derived auto-equivalences on the cohomology of a hyperkahler variety. For certain Hilbert squares of K3 surfaces, we show that the obtained upper bound is close to being sharp.","PeriodicalId":49200,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135011237","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}
We define a collection of cohomology classes $Theta_{g,n}in H^{4g-4+2n}(overline{cal M}_{g,n})$ for $2g-2+n>0$ that restrict naturally to boundary divisors. We prove that a generating function for the intersection numbers $int_{overline{cal M}_{g,n}}Theta_{g,n}prod_{i=1}^npsi_i^{m_i}$ is a tau function of the KdV hierarchy. This is analogous to the theorem conjectured by Witten and proven by Kontsevich that a generating function for the intersection numbers $int_{overline{cal M}_{g,n}}prod_{i=1}^npsi_i^{m_i}$ is a tau function of the KdV hierarchy.
{"title":"A new cohomology class on the moduli space of curves","authors":"Paul Norbury","doi":"10.2140/gt.2023.27.2695","DOIUrl":"https://doi.org/10.2140/gt.2023.27.2695","url":null,"abstract":"We define a collection of cohomology classes $Theta_{g,n}in H^{4g-4+2n}(overline{cal M}_{g,n})$ for $2g-2+n>0$ that restrict naturally to boundary divisors. We prove that a generating function for the intersection numbers $int_{overline{cal M}_{g,n}}Theta_{g,n}prod_{i=1}^npsi_i^{m_i}$ is a tau function of the KdV hierarchy. This is analogous to the theorem conjectured by Witten and proven by Kontsevich that a generating function for the intersection numbers $int_{overline{cal M}_{g,n}}prod_{i=1}^npsi_i^{m_i}$ is a tau function of the KdV hierarchy.","PeriodicalId":49200,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135010181","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}
In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first higher-dimensional Hamiltonian no-torsion theorems beyond the symplectically aspherical case. We start by showing that closed symplectic Calabi-Yau and negative monotone symplectic manifolds do not admit Hamiltonian torsion. Going still beyond topological constraints, we prove that every closed positive monotone symplectic manifold $(M,omega)$ admitting Hamiltonian torsion is geometrically uniruled by holomorphic spheres for every $omega$-compatible almost complex structure, partially answering a question of McDuff-Salamon. This provides many additional no-torsion results, and as a corollary yields the geometric uniruledness of monotone Hamiltonian $S^1$-manifolds, a fact closely related to a celebrated result of McDuff from 2009. Moreover, the non-existence of Hamiltonian torsion implies the triviality of Hamiltonian actions of lattices like $SL(k,mathbb{Z})$ for $k geq 2,$ as well as those of compact Lie groups. Finally, for monotone symplectic manifolds admitting Hamiltonian torsion, we prove an analogue of Newman's theorem on finite transformation groups for several natural norms on the Hamiltonian group: such subgroups cannot be contained in arbitrarily small neighborhoods of the identity. Our arguments rely on generalized Morse-Bott methods, as well as on quantum Steenrod powers and Smith theory in filtered Floer homology.
{"title":"Hamiltonian no-torsion","authors":"Marcelo S Atallah, Egor Shelukhin","doi":"10.2140/gt.2023.27.2833","DOIUrl":"https://doi.org/10.2140/gt.2023.27.2833","url":null,"abstract":"In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first higher-dimensional Hamiltonian no-torsion theorems beyond the symplectically aspherical case. We start by showing that closed symplectic Calabi-Yau and negative monotone symplectic manifolds do not admit Hamiltonian torsion. Going still beyond topological constraints, we prove that every closed positive monotone symplectic manifold $(M,omega)$ admitting Hamiltonian torsion is geometrically uniruled by holomorphic spheres for every $omega$-compatible almost complex structure, partially answering a question of McDuff-Salamon. This provides many additional no-torsion results, and as a corollary yields the geometric uniruledness of monotone Hamiltonian $S^1$-manifolds, a fact closely related to a celebrated result of McDuff from 2009. Moreover, the non-existence of Hamiltonian torsion implies the triviality of Hamiltonian actions of lattices like $SL(k,mathbb{Z})$ for $k geq 2,$ as well as those of compact Lie groups. Finally, for monotone symplectic manifolds admitting Hamiltonian torsion, we prove an analogue of Newman's theorem on finite transformation groups for several natural norms on the Hamiltonian group: such subgroups cannot be contained in arbitrarily small neighborhoods of the identity. Our arguments rely on generalized Morse-Bott methods, as well as on quantum Steenrod powers and Smith theory in filtered Floer homology.","PeriodicalId":49200,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135108537","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}
In recent works, [20],[21], descendent integrals on the moduli space of Riemann surfaces with boundary were defined. It was conjectured in [20] that the generating function of these integrals satisfies the open KdV equations. In this paper we develop the notions of symmetric Strebel-Jenkins differentials and of Kasteleyn orientations for graphs embedded in open surfaces. In addition we write an explicit expression for the angular form of the sum of line bundles. Using these tools we prove a formula for the descendent integrals in terms of sums over weighted graphs. Based on this formula, the conjecture of [20] was proved in [5].
{"title":"The combinatorial formula for open gravitational descendents","authors":"Ran J. Tessler","doi":"10.2140/gt.2023.27.2497","DOIUrl":"https://doi.org/10.2140/gt.2023.27.2497","url":null,"abstract":"In recent works, [20],[21], descendent integrals on the moduli space of Riemann surfaces with boundary were defined. It was conjectured in [20] that the generating function of these integrals satisfies the open KdV equations. In this paper we develop the notions of symmetric Strebel-Jenkins differentials and of Kasteleyn orientations for graphs embedded in open surfaces. In addition we write an explicit expression for the angular form of the sum of line bundles. Using these tools we prove a formula for the descendent integrals in terms of sums over weighted graphs. Based on this formula, the conjecture of [20] was proved in [5].","PeriodicalId":49200,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135010182","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":"A higher-rank rigidity theorem for convex real projective manifolds","authors":"Andrew Zimmer","doi":"10.2140/gt.2023.27.2899","DOIUrl":"https://doi.org/10.2140/gt.2023.27.2899","url":null,"abstract":"For convex real projective manifolds we prove an analogue of the higher rank rigidity theorem of Ballmann and Burns-Spatzier.","PeriodicalId":49200,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135011235","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}
We determine the image of the 2-primary tmf-Hurewicz homomorphism, where tmf is the spectrum of topological modular forms. We do this by lifting elements of tmf_* to the homotopy groups of the generalized Moore spectrum M(8,v_1^8) using a modified form of the Adams spectral sequence and the tmf-resolution, and then proving the existence of a v_2^32-self map on M(8,v_1^8) to generate 192-periodic families in the stable homotopy groups of spheres.
{"title":"The 2–primary Hurewicz image of tmf","authors":"Mark Behrens, Mark Mahowald, J D Quigley","doi":"10.2140/gt.2023.27.2763","DOIUrl":"https://doi.org/10.2140/gt.2023.27.2763","url":null,"abstract":"We determine the image of the 2-primary tmf-Hurewicz homomorphism, where tmf is the spectrum of topological modular forms. We do this by lifting elements of tmf_* to the homotopy groups of the generalized Moore spectrum M(8,v_1^8) using a modified form of the Adams spectral sequence and the tmf-resolution, and then proving the existence of a v_2^32-self map on M(8,v_1^8) to generate 192-periodic families in the stable homotopy groups of spheres.","PeriodicalId":49200,"journal":{"name":"Geometry & Topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135108534","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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