Abstract In this article, the cosymplectic analogue of the symplectic flux homomorphism of a compact connected cosymplectic manifold ( M , η , ω ) left(M,eta ,omega ) with ∂ M = ∅ partial M=varnothing is studied. This is a continuous map with respect to the C 0 {C}^{0} -metric, whose kernel is connected by smooth arcs and coincides with the subgroup of all weakly Hamiltonian diffeomorphisms. We discuss the cosymplectic analogue of the Weinstein’s chart, and derive that the group G η , ω ( M ) {G}_{eta ,omega }left(M) of all cosymplectic diffeomorphisms isotopic to the identity map is locally contractible. A study of an analogue of Polterovich’s regularization process for co-Hamiltonian isotopies follows. Finally, we study Moser’s stability theorems for locally conformal cosymplectic manifolds.
{"title":"Towards the cosymplectic topology","authors":"S. Tchuiaga","doi":"10.1515/coma-2022-0151","DOIUrl":"https://doi.org/10.1515/coma-2022-0151","url":null,"abstract":"Abstract In this article, the cosymplectic analogue of the symplectic flux homomorphism of a compact connected cosymplectic manifold ( M , η , ω ) left(M,eta ,omega ) with ∂ M = ∅ partial M=varnothing is studied. This is a continuous map with respect to the C 0 {C}^{0} -metric, whose kernel is connected by smooth arcs and coincides with the subgroup of all weakly Hamiltonian diffeomorphisms. We discuss the cosymplectic analogue of the Weinstein’s chart, and derive that the group G η , ω ( M ) {G}_{eta ,omega }left(M) of all cosymplectic diffeomorphisms isotopic to the identity map is locally contractible. A study of an analogue of Polterovich’s regularization process for co-Hamiltonian isotopies follows. Finally, we study Moser’s stability theorems for locally conformal cosymplectic manifolds.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"70 14","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41285390","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 define a dual of the Chow transformation of currents on any complex projective manifold. This integral transformation is a factor of a left inverse of the Chow transformation and its composition with the Chow transformation is a right inverse of a linear differential operator, which does not commute with ∂ partial or ∂ ¯ overline{partial } . We obtain a complete intrinsic resolution of the problem of the algebraicity of the cohomology classes. On another hand, in the case of the complex projective space, we give the translation in terms of real-analytic D {mathcal{D}} -modules of the properties of the Chow transformation. Then, the proofs can be simplified by using the conormal currents, which exist for all currents of bidimension ( p , p ) left(p,p) on the complex projective space, even not closed. This is a consequence of the existence of dual currents, defined on the dual complex projective space. In particular, we obtain a linear differential system of order lower than that of the Gelfand-Gindikin-Graev differential system, characterizing the images by the Chow transformation of smooth differential forms on the complex projective space.
{"title":"Chow transformation of coherent sheaves","authors":"M. Meo","doi":"10.1515/coma-2022-0147","DOIUrl":"https://doi.org/10.1515/coma-2022-0147","url":null,"abstract":"Abstract We define a dual of the Chow transformation of currents on any complex projective manifold. This integral transformation is a factor of a left inverse of the Chow transformation and its composition with the Chow transformation is a right inverse of a linear differential operator, which does not commute with ∂ partial or ∂ ¯ overline{partial } . We obtain a complete intrinsic resolution of the problem of the algebraicity of the cohomology classes. On another hand, in the case of the complex projective space, we give the translation in terms of real-analytic D {mathcal{D}} -modules of the properties of the Chow transformation. Then, the proofs can be simplified by using the conormal currents, which exist for all currents of bidimension ( p , p ) left(p,p) on the complex projective space, even not closed. This is a consequence of the existence of dual currents, defined on the dual complex projective space. In particular, we obtain a linear differential system of order lower than that of the Gelfand-Gindikin-Graev differential system, characterizing the images by the Chow transformation of smooth differential forms on the complex projective space.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46762966","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}
I. Biswas, U. Dubey, Manish Kumar, A. J. Parameswaran
Abstract We consider several related examples of Fourier-Mukai transformations involving the quot scheme. A method of showing conservativity of these Fourier-Mukai transformations is described.
{"title":"Quot schemes and Fourier-Mukai transformation","authors":"I. Biswas, U. Dubey, Manish Kumar, A. J. Parameswaran","doi":"10.1515/coma-2023-0152","DOIUrl":"https://doi.org/10.1515/coma-2023-0152","url":null,"abstract":"Abstract We consider several related examples of Fourier-Mukai transformations involving the quot scheme. A method of showing conservativity of these Fourier-Mukai transformations is described.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45006998","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 note, we determine the structure of the associative algebra generated by the differential operators μ ¯ , ∂ ¯ , ∂ overline{mu },overline{partial },partial , and μ mu that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential [ d , − ] left[d,-] , as well as its cohomology with respect to all its inner differentials.
{"title":"On the algebra generated by μ ¯ , ∂ ¯ , ∂ , μ overline{mu },overline{partial },partial ,mu","authors":"S. Auyeung, Jin-Cheng Guu, Jiahao Hu","doi":"10.1515/coma-2022-0149","DOIUrl":"https://doi.org/10.1515/coma-2022-0149","url":null,"abstract":"Abstract In this note, we determine the structure of the associative algebra generated by the differential operators μ ¯ , ∂ ¯ , ∂ overline{mu },overline{partial },partial , and μ mu that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential [ d , − ] left[d,-] , as well as its cohomology with respect to all its inner differentials.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49105508","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 four-dimensional second Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis, the Riemannian dual of the Lee form is a Killing vector field. We use that observation to describe four-dimensional compact second Chern-Einstein locally conformally symplectic manifolds, and we give some examples of such manifolds. Finally, we study the second Chern-Einstein problem on unimodular almost-abelian Lie algebras, classifying those that admit a left-invariant second Chern-Einstein metric with a parallel non-zero Lee form.
{"title":"Second Chern-Einstein metrics on four-dimensional almost-Hermitian manifolds","authors":"G. Barbaro, Mehdi Lejmi","doi":"10.1515/coma-2022-0150","DOIUrl":"https://doi.org/10.1515/coma-2022-0150","url":null,"abstract":"Abstract We study four-dimensional second Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis, the Riemannian dual of the Lee form is a Killing vector field. We use that observation to describe four-dimensional compact second Chern-Einstein locally conformally symplectic manifolds, and we give some examples of such manifolds. Finally, we study the second Chern-Einstein problem on unimodular almost-abelian Lie algebras, classifying those that admit a left-invariant second Chern-Einstein metric with a parallel non-zero Lee form.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47786483","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 aim of this article is to study the geometry of Bott-Chern hypercohomology from the bimeromorphic point of view. We construct some new bimeromorphic invariants involving the cohomology for the sheaf of germs of pluriharmonic functions, the truncated holomorphic de Rham cohomology, and the de Rham cohomology. To define these invariants, by using a sheaf-theoretic approach, we establish a blow-up formula together with a canonical morphism for the Bott-Chern hypercohomology. In particular, we compute the invariants of some compact complex threefolds, such as Iwasawa manifolds and quintic threefolds.
{"title":"Bott-Chern hypercohomology and bimeromorphic invariants","authors":"Song Yang, Xiangdong Yang","doi":"10.1515/coma-2022-0148","DOIUrl":"https://doi.org/10.1515/coma-2022-0148","url":null,"abstract":"Abstract The aim of this article is to study the geometry of Bott-Chern hypercohomology from the bimeromorphic point of view. We construct some new bimeromorphic invariants involving the cohomology for the sheaf of germs of pluriharmonic functions, the truncated holomorphic de Rham cohomology, and the de Rham cohomology. To define these invariants, by using a sheaf-theoretic approach, we establish a blow-up formula together with a canonical morphism for the Bott-Chern hypercohomology. In particular, we compute the invariants of some compact complex threefolds, such as Iwasawa manifolds and quintic threefolds.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"10 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43496581","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 Let X X be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if X X is d d -semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus of X X , and the first author’s existence result of holomorphic volume forms on global smoothings of X X . In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples of d d -semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces, and K 3 K3 surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable to K 3 K3 surfaces.
摘要设X X是一个具有平凡正则丛的简单正交（SNC）紧致复曲面，它包含三个交。我们证明了如果X是d-半稳定的，那么在微分几何意义上存在一个光滑族。这可以被解释为代数几何中Friedman、Kawamata Namikawa、Felten Filip Ruddat、Chan Leung Ma等人的光滑性结果的微分几何模拟。该证明基于X X奇异轨迹上局部光滑的显式构造，以及第一作者关于X X全局光滑上全纯体形式的存在性结果。特别地，这些体积形式被给出为非线性椭圆偏微分方程的解。作为一个应用，我们提供了几个具有平凡正则丛的d-半稳定SNC复曲面的例子，这些平凡正则丛包括双曲，它们可以光滑到复环面、主Kodaira曲面和K3曲面。我们还提供了包括三点的这种复杂曲面的几个例子，这些三点可以平滑到K3曲面。
{"title":"Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle","authors":"Mamoru Doi, N. Yotsutani","doi":"10.1515/coma-2022-0143","DOIUrl":"https://doi.org/10.1515/coma-2022-0143","url":null,"abstract":"Abstract Let X X be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if X X is d d -semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus of X X , and the first author’s existence result of holomorphic volume forms on global smoothings of X X . In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples of d d -semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces, and K 3 K3 surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable to K 3 K3 surfaces.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44968431","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 Fu-Yau equation on compact almost astheno-Kähler manifolds and show an a priori C0-estiamte for a smooth solution of the equation.
{"title":"An a priori C0-estimate for the Fu-Yau equation on compact almost astheno-Kähler manifolds","authors":"Masaya Kawamura","doi":"10.1515/coma-2021-0138","DOIUrl":"https://doi.org/10.1515/coma-2021-0138","url":null,"abstract":"Abstract We investigate the Fu-Yau equation on compact almost astheno-Kähler manifolds and show an a priori C0-estiamte for a smooth solution of the equation.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"223 - 237"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41740965","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 characterize the k-th Gauduchon condition and by applying its characterization, we reprove that a compact k-th Gauduchon, semi-Kähler manifold becomes quasi-Kähler, which tells us that in particular, a compact almost pluriclosed, semi-Kähler manifold is quasi-Kähler.
{"title":"On a k-th Gauduchon almost Hermitian manifold","authors":"Masaya Kawamura","doi":"10.1515/coma-2021-0130","DOIUrl":"https://doi.org/10.1515/coma-2021-0130","url":null,"abstract":"Abstract We characterize the k-th Gauduchon condition and by applying its characterization, we reprove that a compact k-th Gauduchon, semi-Kähler manifold becomes quasi-Kähler, which tells us that in particular, a compact almost pluriclosed, semi-Kähler manifold is quasi-Kähler.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"65 - 77"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45926604","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 This paper shows that given 0 < p < 3 and a complex Borel measure µ on the unit disk 𝔻 the inhomogeneous Cauchy-Riemann ̄∂-equation ∂z¯u(z)=dμ(z)(2πi)-1dz¯∧dz {partial _{bar z}}uleft( z right) = {{dmu left( z right)} over {{{left( {2pi i} right)}^{ - 1}}dbar z wedge dz}} − a complex Gauss curvature of the weighted disk (𝔻, µ) ᗄ z ∈ 𝔻, has a distributional solution (initially defined on ̄𝔻 = 𝔻 ∪ 𝕋) u ∈ ℒ2,p(𝕋) (formed of: (i) Morrey’s space M2,0
摘要本文证明了给定0
{"title":"Cauchy-Riemann ̄∂-equations with some applications","authors":"J. Xiao, Cheng Yuan","doi":"10.1515/coma-2021-0134","DOIUrl":"https://doi.org/10.1515/coma-2021-0134","url":null,"abstract":"Abstract This paper shows that given 0 < p < 3 and a complex Borel measure µ on the unit disk 𝔻 the inhomogeneous Cauchy-Riemann ̄∂-equation ∂z¯u(z)=dμ(z)(2πi)-1dz¯∧dz {partial _{bar z}}uleft( z right) = {{dmu left( z right)} over {{{left( {2pi i} right)}^{ - 1}}dbar z wedge dz}} − a complex Gauss curvature of the weighted disk (𝔻, µ) ᗄ z ∈ 𝔻, has a distributional solution (initially defined on ̄𝔻 = 𝔻 ∪ 𝕋) u ∈ ℒ2,p(𝕋) (formed of: (i) Morrey’s space M2,0<p<1(𝕋); (ii) John-Nirenberg’s space BMO(𝕋) = 2,1(𝕋); (iii) Hölder-Lipschitz’s space C C0<p-12<1 {C^{0 < {{p - 1} over 2} < 1}} (𝕋)), if and only if 𝔻¯∋z↦∫𝔻(1-zw¯)-1dμ¯(w) mathbb{D} z mapsto intlimits_mathbb{D} {{{left( {1 - zbar w} right)}^{ - 1}}dbar mu } left( w right) belongs to the analytic Campanato space ϱ𝒜p(𝔻), thereby not only extending Carleson’s corona & Wolff’s ideal theorems to the algebra M ϱ𝒜p(𝔻) of all analytic pointwise multiplications of ϱ𝒜p(𝔻), but quadratically generalizing Brownawell’s result on Hilbert’s Nullstellensatz for the analytic polynomial class 𝒫(ℂ).","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"170 - 191"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45303352","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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