{"title":"On the total curvature and Betti numbers of complex projective manifolds","authors":"J. Hoisington","doi":"10.2140/gt.2022.26.1","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1","url":null,"abstract":"We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"153 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114641645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Topological dualities in the Ising model","authors":"D. Freed, C. Teleman","doi":"10.2140/gt.2022.26.1907","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1907","url":null,"abstract":"Author(s): Freed, Daniel S; Teleman, Constantin | Abstract: We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124444790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Alexandrov spaces with maximal radius","authors":"K. Grove, P. Petersen","doi":"10.2140/gt.2022.26.1635","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1635","url":null,"abstract":"Abstract. In this paper we prove several rigidity theorems related to and including Lytchak's problem. The focus is on Alexandrov spaces with curvgeq1, nonempty boundary, and maximal radius frac{pi}{2}. We exhibit many such spaces that indicate that this class is remarkably flexible. Nevertheless, we also show that when the boundary is either geometrically or topologically spherical, then it is possible to obtain strong rigidity results. In contrast to this one can show that with general lower curvature bounds and strictly convex boundary only cones can have maximal radius. We also mention some connections between our problems and the positive mass conjectures. This paper is an expanded version and replacement of the two previous versions","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"230 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128176804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Towards logarithmic GLSM : the r–spin\u0000case","authors":"Qile Chen, F. Janda, Y. Ruan, Adrien Sauvaget","doi":"10.2140/gt.2022.26.2855","DOIUrl":"https://doi.org/10.2140/gt.2022.26.2855","url":null,"abstract":"In this article, we establish the logarithmic foundation for compactifying the moduli stacks of the gauged linear sigma model using stable log maps of Abramovich-Chen-Gross-Siebert. We then illustrate our method via the key example of Witten's $r$-spin class to construct a proper moduli stack with a reduced perfect obstruction theory whose virtual cycle recovers the $r$-spin virtual cycle of Chang-Li-Li. Indeed, our construction of the reduced virtual cycle is built upon the work of Chang-Li-Li by appropriately extending and modifying the Kiem-Li cosection along certain logarithmic boundary. In the subsequent article, we push the technique to a general situation. One motivation of our construction is to fit the gauged linear sigma model in the broader setting of Gromov-Witten theory so that powerful tools such as virtual localization can be applied. A project along this line is currently in progress leading to applications including computing loci of holomorphic differentials, and calculating higher genus Gromov-Witten invariants of quintic threefolds.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"109 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122887595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Positivity and the Kodaira embedding theorem","authors":"Lei Ni, F. Zheng","doi":"10.2140/gt.2022.26.2491","DOIUrl":"https://doi.org/10.2140/gt.2022.26.2491","url":null,"abstract":"In his recent work arXiv:1708.06713, X. Yang proved a conjecture raised by Yau in 1982, which states that any compact K\"ahler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity. In this note, we prove a generalization to this statement by showing that any compact K\"ahler manifold with positive 2nd scalar curvature (which is the average of holomorphic sectional curvature over $2$-dimensional subspaces of the tangent space) must be projective. In view of generic 2-tori being non-Abelian, this condition is sharp in some sense. Vanishing theorems are also proved for the Hodge numbers when the condition is replaced by the positivity of the weaker interpolating $k$-scalar curvature.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129004648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Chromatic splitting for the K(2)–local sphere at\u0000p = 2","authors":"A. Beaudry, P. Goerss, H. Henn","doi":"10.2140/gt.2022.26.377","DOIUrl":"https://doi.org/10.2140/gt.2022.26.377","url":null,"abstract":"We calculate the homotopy type of $L_1L_{K(2)}S^0$ and $L_{K(1)}L_{K(2)}S^0$ at the prime 2, where $L_{K(n)}$ is localization with respect to Morava $K$-theory and $L_1$ localization with respect to $2$-local $K$ theory. In $L_1L_{K(2)}S^0$ we find all the summands predicted by the Chromatic Splitting Conjecture, but we find some extra summands as well. An essential ingredient in our approach is the analysis of the continuous group cohomology $H^ast(mathbb{G}_2,E_0)$ where $mathbb{G}_2$ is the Morava stabilizer group and $E_0 = mathbb{W}[[u_1]]$ is the ring of functions on the height $2$ Lubin-Tate space. We show that the inclusion of the constants $mathbb{W} to E_0$ induces an isomorphism on group cohomology, a radical simplification.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121509725","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hilbert schemes and y–ification of\u0000Khovanov–Rozansky homology","authors":"E. Gorsky, Matthew Hogancamp","doi":"10.2140/gt.2022.26.587","DOIUrl":"https://doi.org/10.2140/gt.2022.26.587","url":null,"abstract":"Author(s): Gorsky, Eugene; Hogancamp, Matthew | Abstract: We define a deformation of the triply graded Khovanov-Rozansky homology of a link $L$ depending on a choice of parameters $y_c$ for each component of $L$, which satisfies link-splitting properties similar to the Batson-Seed invariant. Keeping the $y_c$ as formal variables yields a link homology valued in triply graded modules over $mathbb{Q}[x_c,y_c]_{cin pi_0(L)}$. We conjecture that this invariant restores the missing $Qleftrightarrow TQ^{-1}$ symmetry of the triply graded Khovanov-Rozansky homology, and in addition satisfies a number of predictions coming from a conjectural connection with Hilbert schemes of points in the plane. We compute this invariant for all positive powers of the full twist and match it to the family of ideals appearing in Haiman's description of the isospectral Hilbert scheme.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134270705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
E. Elmanto, M. Levine, Markus Spitzweck, P. Ostvaer
{"title":"Algebraic cobordism and étale\u0000cohomology","authors":"E. Elmanto, M. Levine, Markus Spitzweck, P. Ostvaer","doi":"10.2140/gt.2022.26.477","DOIUrl":"https://doi.org/10.2140/gt.2022.26.477","url":null,"abstract":"Thomason's '{e}tale descent theorem for Bott periodic algebraic $K$-theory cite{aktec} is generalized to any $MGL$ module over a regular Noetherian scheme of finite dimension. Over arbitrary Noetherian schemes of finite dimension, this generalizes the analog of Thomason's theorem for Weibel's homotopy $K$-theory. This is achieved by amplifying the effects from the case of motivic cohomology, using the slice spectral sequence in the case of the universal example of algebraic cobordism. We also obtain integral versions of these statements: Bousfield localization at 'etale motivic cohomology is the universal way to impose 'etale descent for these theories. As applications, we describe the 'etale local objects in modules over these spectra and show that they satisfy the full six functor formalism, construct an 'etale descent spectral sequence converging to Bott-inverted motivic Landweber exact theories, and prove cellularity and effectivity of the '{e}tale versions of these motivic spectra.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131535759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Pseudoholomorphic curves relative to a normal crossings symplectic divisor: compactification","authors":"Mohammad Farajzadeh-Tehrani","doi":"10.2140/gt.2022.26.989","DOIUrl":"https://doi.org/10.2140/gt.2022.26.989","url":null,"abstract":"Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli space is characterized by a second homology class, genus, and contact data. For certain almost complex structures, we show that the moduli space of stable log J-holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than the relative compactification and there is a projection map from the latter onto the former. The latter is constructed via expanded degenerations of the target. Our construction does not need any modification of (or any extra structure on) the target. Unlike the classical moduli spaces of stable maps, these log moduli spaces are often virtually singular. We describe an explicit toric model for the normal cone (i.e. the space of gluing parameters) to each stratum in terms of the defining combinatorial data of that stratum. In [FT2], we introduce a natural set up for studying the deformation theory of log (and relative) curves and obtain a logarithmic analog of the space of Ruan-Tian perturbations for these moduli spaces. In a forthcoming paper, we will prove a gluing theorem for smoothing log curves in the normal direction to each stratum. With some modifications to the theory of Kuranishi spaces, the latter will allow us to construct a virtual fundamental class for every such log moduli space and define relative GW invariants without any restriction.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130398085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Surface group representations in SL2(ℂ) with\u0000finite mapping class orbits","authors":"I. Biswas, Subhojoy Gupta, Mahan Mj, J. Whang","doi":"10.2140/gt.2022.26.679","DOIUrl":"https://doi.org/10.2140/gt.2022.26.679","url":null,"abstract":". Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123654382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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