Abstract We give an exposition of Sullivan’s theorem on realizing rational homotopy types by closed smooth manifolds, including a discussion of the necessary rational homotopy and surgery theory, adapted to the realization problem for almost complex manifolds: namely, we give a characterization of the possible simply connected rational homotopy types, along with a choice of rational Chern classes and fundamental class, realized by simply connected closed almost complex manifolds in real dimensions six and greater. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a simply connected closed almost complex manifold depends only on its cohomology ring. We conclude with some computations and examples.
{"title":"On the characterization of rational homotopy types and Chern classes of closed almost complex manifolds","authors":"A. Milivojević","doi":"10.1515/coma-2021-0133","DOIUrl":"https://doi.org/10.1515/coma-2021-0133","url":null,"abstract":"Abstract We give an exposition of Sullivan’s theorem on realizing rational homotopy types by closed smooth manifolds, including a discussion of the necessary rational homotopy and surgery theory, adapted to the realization problem for almost complex manifolds: namely, we give a characterization of the possible simply connected rational homotopy types, along with a choice of rational Chern classes and fundamental class, realized by simply connected closed almost complex manifolds in real dimensions six and greater. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a simply connected closed almost complex manifold depends only on its cohomology ring. We conclude with some computations and examples.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"138 - 169"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48223089","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}
Abstract We propose a new method to construct degenerate 3-(α, δ)-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperkähler manifolds. Subsequently, we study homogeneous degenerate 3-(α, δ)-Sasakian manifolds and prove that no non-trivial compact examples exist aswell as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.
{"title":"On Degenerate 3-(α, δ)-Sasakian Manifolds","authors":"Oliver Goertsches, Leon Roschig, Leander Stecker","doi":"10.1515/coma-2021-0142","DOIUrl":"https://doi.org/10.1515/coma-2021-0142","url":null,"abstract":"Abstract We propose a new method to construct degenerate 3-(α, δ)-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperkähler manifolds. Subsequently, we study homogeneous degenerate 3-(α, δ)-Sasakian manifolds and prove that no non-trivial compact examples exist aswell as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"337 - 344"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45789782","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}
Abstract The purpose in this paper is to determine a locally conformal Kähler solvmanifold such that the nilradical of the solvable Lie group is constructed by a Heisenberg Lie group.
{"title":"On LCK solvmanifolds with a property of Vaisman solvmanifolds","authors":"H. Sawai","doi":"10.1515/coma-2021-0135","DOIUrl":"https://doi.org/10.1515/coma-2021-0135","url":null,"abstract":"Abstract The purpose in this paper is to determine a locally conformal Kähler solvmanifold such that the nilradical of the solvable Lie group is constructed by a Heisenberg Lie group.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"196 - 205"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47671582","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}
Abstract Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kähler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable in the sense of geometric invariant theory with respect to the linear action of the automorphism group of the bundle on its space of in˝nitesimal deformations.
{"title":"Polystable bundles and representations of their automorphisms","authors":"N. Buchdahl, G. Schumacher","doi":"10.1515/coma-2021-0131","DOIUrl":"https://doi.org/10.1515/coma-2021-0131","url":null,"abstract":"Abstract Using a quasi-linear version of Hodge theory, holomorphic vector bundles in a neighbourhood of a given polystable bundle on a compact Kähler manifold are shown to be (poly)stable if and only if their corresponding classes are (poly)stable in the sense of geometric invariant theory with respect to the linear action of the automorphism group of the bundle on its space of in˝nitesimal deformations.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"78 - 113"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49137594","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}
Abstract Recently, Dinew and Popovici introduced and studied an energy functional F acting on the metrics in the Aeppli cohomology class of a Hermitian-symplectic metric and showed that in dimension 3 its critical points (if any) are Kähler. In this article we further investigate the critical points of this functional in higher dimensions and under holomorphic deformations. We first prove that being a critical point for F is a closed property under holomorphic deformations. We then show that the existence of a Kähler metric ω in the Aeppli cohomology class is an open property under holomorphic deformations. Furthermore, we consider the case when the (2, 0)-torsion form ρω 2, 0 of ω is ∂-exact and prove that this property is closed under holomorphic deformations. Finally, we give an explicit formula for the differential of F when the (2, 0)-torsion form ρω2, 0 is ∂-exact.
{"title":"Properties of Critical Points of the Dinew-Popovici Energy Functional","authors":"Erfan Soheil","doi":"10.1515/coma-2021-0144","DOIUrl":"https://doi.org/10.1515/coma-2021-0144","url":null,"abstract":"Abstract Recently, Dinew and Popovici introduced and studied an energy functional F acting on the metrics in the Aeppli cohomology class of a Hermitian-symplectic metric and showed that in dimension 3 its critical points (if any) are Kähler. In this article we further investigate the critical points of this functional in higher dimensions and under holomorphic deformations. We first prove that being a critical point for F is a closed property under holomorphic deformations. We then show that the existence of a Kähler metric ω in the Aeppli cohomology class is an open property under holomorphic deformations. Furthermore, we consider the case when the (2, 0)-torsion form ρω 2, 0 of ω is ∂-exact and prove that this property is closed under holomorphic deformations. Finally, we give an explicit formula for the differential of F when the (2, 0)-torsion form ρω2, 0 is ∂-exact.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"355 - 369"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43884510","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}
Abstract We study symmetric minimal surfaces in the three-dimensional Heisenberg group Nil3 using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will present a general scheme for how to construct minimal surfaces in Nil3 with non-trivial geometry. Special emphasis will be put on equivariant minimal surfaces. Moreover, we will classify equivariant minimal surfaces given by one-parameter subgroups of the isometry group Iso◦(Nil3) of Nil3.
{"title":"Minimal surfaces with non-trivial geometry in the three-dimensional Heisenberg group","authors":"J. Dorfmeister, J. Inoguchi, Shimpei Kobayashi","doi":"10.1515/coma-2021-0141","DOIUrl":"https://doi.org/10.1515/coma-2021-0141","url":null,"abstract":"Abstract We study symmetric minimal surfaces in the three-dimensional Heisenberg group Nil3 using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will present a general scheme for how to construct minimal surfaces in Nil3 with non-trivial geometry. Special emphasis will be put on equivariant minimal surfaces. Moreover, we will classify equivariant minimal surfaces given by one-parameter subgroups of the isometry group Iso◦(Nil3) of Nil3.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"285 - 336"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42580958","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}
Abstract We describe examples showing the sharpness of Fujita’s conjecture on adjoint bundles also in the general type case, and use these examples to formulate related bold conjectures on pluricanonical maps of varieties of general type.
{"title":"Pluricanonical Maps and the Fujita Conjecture","authors":"F. Catanese","doi":"10.1515/coma-2021-0136","DOIUrl":"https://doi.org/10.1515/coma-2021-0136","url":null,"abstract":"Abstract We describe examples showing the sharpness of Fujita’s conjecture on adjoint bundles also in the general type case, and use these examples to formulate related bold conjectures on pluricanonical maps of varieties of general type.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"192 - 195"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49619701","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}
Abstract In this continuation of [4], we investigate the deformations of holomorphic Cartan geometries where the underlying complex manifold is allowed to move. The space of infinitesimal deformations of a flat holomorphic Cartan geometry is computed. We show that the natural forgetful map, from the infinitesimal deformations of a flat holomorphic Cartan geometry to the infinitesimal deformations of the underlying flat principal bundle on the topological manifold, is an isomorphism.
{"title":"Deformation theory of holomorphic Cartan geometries, II","authors":"I. Biswas, Sorin Dumitrescu, G. Schumacher","doi":"10.1515/coma-2021-0129","DOIUrl":"https://doi.org/10.1515/coma-2021-0129","url":null,"abstract":"Abstract In this continuation of [4], we investigate the deformations of holomorphic Cartan geometries where the underlying complex manifold is allowed to move. The space of infinitesimal deformations of a flat holomorphic Cartan geometry is computed. We show that the natural forgetful map, from the infinitesimal deformations of a flat holomorphic Cartan geometry to the infinitesimal deformations of the underlying flat principal bundle on the topological manifold, is an isomorphism.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"52 - 64"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47767695","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}
Abstract We investigate the pseudo-hyperkähler geometry of higher degree rational curves in the twistor space of a hyperkähler 4-manifold.
摘要我们研究了超kähler 4-流形的扭曲空间中的高阶有理曲线的拟超káhler几何。
{"title":"Hyperkähler geometry of rational curves in twistor spaces","authors":"R. Bielawski, Naizhen Zhang","doi":"10.1515/coma-2021-0145","DOIUrl":"https://doi.org/10.1515/coma-2021-0145","url":null,"abstract":"Abstract We investigate the pseudo-hyperkähler geometry of higher degree rational curves in the twistor space of a hyperkähler 4-manifold.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"345 - 354"},"PeriodicalIF":0.5,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49532341","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}
Abstract We compute almost-complex invariants h∂¯p,o h_{bar partial }^{p,o} , hDolp,o h_{Dol}^{p,o} and almost-Hermitian invariants hδ¯p,o h_{bar delta }^{p,o} on families of almost-Kähler and almost-Hermitian 6-dimensional solvmanifolds. Finally, as a consequence of almost-Kähler identities we provide an obstruction to the existence of a compatible symplectic structure on a given compact almost-complex manifold. Notice that, when (X, J, g, ω) is a compact almost Hermitian manifold of real dimension greater than four, not much is known concerning the numbers h∂¯p,q h_{bar partial }^{p,q} .
{"title":"Almost-complex invariants of families of six-dimensional solvmanifolds","authors":"Nicoletta Tardini, A. Tomassini","doi":"10.1515/coma-2021-0139","DOIUrl":"https://doi.org/10.1515/coma-2021-0139","url":null,"abstract":"Abstract We compute almost-complex invariants h∂¯p,o h_{bar partial }^{p,o} , hDolp,o h_{Dol}^{p,o} and almost-Hermitian invariants hδ¯p,o h_{bar delta }^{p,o} on families of almost-Kähler and almost-Hermitian 6-dimensional solvmanifolds. Finally, as a consequence of almost-Kähler identities we provide an obstruction to the existence of a compatible symplectic structure on a given compact almost-complex manifold. Notice that, when (X, J, g, ω) is a compact almost Hermitian manifold of real dimension greater than four, not much is known concerning the numbers h∂¯p,q h_{bar partial }^{p,q} .","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"238 - 260"},"PeriodicalIF":0.5,"publicationDate":"2021-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45841663","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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