Olimjon Eshkobilov, G. Manno, G. Moreno, Katja Sagerschnig
Abstract In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n + 1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a Ph.D course given by two of the authors (G. M. and G. M.). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.
{"title":"Contact manifolds, Lagrangian Grassmannians and PDEs","authors":"Olimjon Eshkobilov, G. Manno, G. Moreno, Katja Sagerschnig","doi":"10.1515/coma-2018-0003","DOIUrl":"https://doi.org/10.1515/coma-2018-0003","url":null,"abstract":"Abstract In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n + 1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a Ph.D course given by two of the authors (G. M. and G. M.). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"5 1","pages":"26 - 88"},"PeriodicalIF":0.5,"publicationDate":"2017-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2018-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42477458","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 present a set of necessary and sufficient conditions that provides a Schwarz lemma for the tetrablock E. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc G2. In either case, our results generalize all previous results in this direction for E and G2.
{"title":"A generalized Schwarz lemma for two domains related to μ-synthesis","authors":"S. Pal, Samriddho Roy","doi":"10.1515/coma-2018-0001","DOIUrl":"https://doi.org/10.1515/coma-2018-0001","url":null,"abstract":"Abstract We present a set of necessary and sufficient conditions that provides a Schwarz lemma for the tetrablock E. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc G2. In either case, our results generalize all previous results in this direction for E and G2.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"5 1","pages":"1 - 8"},"PeriodicalIF":0.5,"publicationDate":"2017-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2018-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43138858","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 paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].
{"title":"Transverse Hilbert schemes and completely integrable systems","authors":"Niccolò Lora Lamia Donin","doi":"10.1515/coma-2017-0015","DOIUrl":"https://doi.org/10.1515/coma-2017-0015","url":null,"abstract":"Abstract In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"263 - 272"},"PeriodicalIF":0.5,"publicationDate":"2017-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45706015","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 prove the existence of a closed regularization of the integration current associated to an effective analytic cycle, with a bounded negative part. By means of the King formula, we are reduced to regularize a closed differential form with L1loc coefficients, which by extension has a test value on any positive current with the same support as the cycle. As a consequence, the restriction of a closed positive current to a closed analytic submanifold is well defined as a closed positive current. Lastly, given a closed smooth differential (qʹ, qʹ)-form on a closed analytic submanifold, we prove the existence of a closed (qʹ, qʹ)-current having a restriction equal to that differential form. After blowing up we deal with the case of a hypersurface and then the extension current is obtained as a solution of a linear differential equation of order 1.
{"title":"Regularization of closed positive currents and intersection theory","authors":"M. Meo","doi":"10.1515/coma-2017-0008","DOIUrl":"https://doi.org/10.1515/coma-2017-0008","url":null,"abstract":"Abstract We prove the existence of a closed regularization of the integration current associated to an effective analytic cycle, with a bounded negative part. By means of the King formula, we are reduced to regularize a closed differential form with L1loc coefficients, which by extension has a test value on any positive current with the same support as the cycle. As a consequence, the restriction of a closed positive current to a closed analytic submanifold is well defined as a closed positive current. Lastly, given a closed smooth differential (qʹ, qʹ)-form on a closed analytic submanifold, we prove the existence of a closed (qʹ, qʹ)-current having a restriction equal to that differential form. After blowing up we deal with the case of a hypersurface and then the extension current is obtained as a solution of a linear differential equation of order 1.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"120 - 136"},"PeriodicalIF":0.5,"publicationDate":"2017-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42949529","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 gives a full characterization of the reducing subspaces for the multiplication operator Mϕ on the Dirichlet space with symbol of finite Blaschke product ϕ of order 5I 6I 7. The reducing subspaces of Mϕ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surfaces to study the reducing subspaces of Mϕ on the Bergman space. By this means, we determine the reducing subspaces of Mϕ on the Dirichlet space and answer some questions of Douglas-Putinar-Wang in [6].
{"title":"Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surfaces","authors":"Caixing Gu, S. Luo, J. Xiao","doi":"10.1515/coma-2017-0007","DOIUrl":"https://doi.org/10.1515/coma-2017-0007","url":null,"abstract":"Abstract This paper gives a full characterization of the reducing subspaces for the multiplication operator Mϕ on the Dirichlet space with symbol of finite Blaschke product ϕ of order 5I 6I 7. The reducing subspaces of Mϕ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surfaces to study the reducing subspaces of Mϕ on the Bergman space. By this means, we determine the reducing subspaces of Mϕ on the Dirichlet space and answer some questions of Douglas-Putinar-Wang in [6].","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"119 - 84"},"PeriodicalIF":0.5,"publicationDate":"2017-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49219770","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 of this paper is to prove that there exists a lattice on a certain solvable Lie group and construct a six-dimensional locally conformal Kähler solvmanifold with non-parallel Lee form.
{"title":"Example of a six-dimensional LCK solvmanifold","authors":"H. Sawai","doi":"10.1515/coma-2017-0004","DOIUrl":"https://doi.org/10.1515/coma-2017-0004","url":null,"abstract":"Abstract The purpose of this paper is to prove that there exists a lattice on a certain solvable Lie group and construct a six-dimensional locally conformal Kähler solvmanifold with non-parallel Lee form.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"37 - 42"},"PeriodicalIF":0.5,"publicationDate":"2017-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42438932","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 A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical მ̄-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex manifolds are nilmanifolds with an abelian complex structure. For a particular class of holomorphic Poisson structures, this result leads to a Hodge-type decomposition of the holomorphic Poisson cohomology. We provide examples when the nilmanifolds are 2-step.
{"title":"A Hodge-type decomposition of holomorphic Poisson cohomology on nilmanifolds","authors":"Y. Poon, John Simanyi","doi":"10.1515/coma-2017-0009","DOIUrl":"https://doi.org/10.1515/coma-2017-0009","url":null,"abstract":"Abstract A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bicomplex where one of the two operators is the classical მ̄-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex manifolds are nilmanifolds with an abelian complex structure. For a particular class of holomorphic Poisson structures, this result leads to a Hodge-type decomposition of the holomorphic Poisson cohomology. We provide examples when the nilmanifolds are 2-step.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"137 - 154"},"PeriodicalIF":0.5,"publicationDate":"2017-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45697773","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 N be a simply connected real nilpotent Lie group, n its Lie algebra, and € a lattice in N. If a left-invariant complex structure on N is Γ-rational, then HƏ̄s,t(Γ/N) ≃ HƏ̄s,t(nC) for each s; t. We can construct different left-invariant complex structures on one nilpotent Lie group by using the complexification and the scalar restriction. We investigate relationships to Hodge numbers of associated compact complex nilmanifolds.
{"title":"Some relations between Hodge numbers and invariant complex structures on compact nilmanifolds","authors":"Takumi Yamada","doi":"10.1515/coma-2017-0006","DOIUrl":"https://doi.org/10.1515/coma-2017-0006","url":null,"abstract":"Abstract Let N be a simply connected real nilpotent Lie group, n its Lie algebra, and € a lattice in N. If a left-invariant complex structure on N is Γ-rational, then HƏ̄s,t(Γ/N) ≃ HƏ̄s,t(nC) for each s; t. We can construct different left-invariant complex structures on one nilpotent Lie group by using the complexification and the scalar restriction. We investigate relationships to Hodge numbers of associated compact complex nilmanifolds.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"73 - 83"},"PeriodicalIF":0.5,"publicationDate":"2017-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46610532","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 survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.
{"title":"Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves","authors":"R. X. Dong","doi":"10.1515/coma-2017-0002","DOIUrl":"https://doi.org/10.1515/coma-2017-0002","url":null,"abstract":"Abstract We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"4 1","pages":"15 - 7"},"PeriodicalIF":0.5,"publicationDate":"2017-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2017-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47427335","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 work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow.We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.
{"title":"On the stability of harmonic maps under the homogeneous Ricci flow","authors":"Rafaela F. do Prado, L. Grama","doi":"10.1515/coma-2018-0007","DOIUrl":"https://doi.org/10.1515/coma-2018-0007","url":null,"abstract":"Abstract In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow.We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"5 1","pages":"122 - 132"},"PeriodicalIF":0.5,"publicationDate":"2017-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2018-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44384418","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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