Abstract In this work we show that the Weil-Petersson volume (which coincides with the CM degree) in the case of weighted points in the projective line is continuous when approaching the Calabi-Yau geometry from the Fano geometry. More specifically, the CM volume computed via localization converges to the geometric volume, computed by McMullen with different techniques, when the sum of the weights approaches the Calabi-Yau geometry.
{"title":"On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line","authors":"Salvatore Tambasco","doi":"10.1515/coma-2021-0137","DOIUrl":"https://doi.org/10.1515/coma-2021-0137","url":null,"abstract":"Abstract In this work we show that the Weil-Petersson volume (which coincides with the CM degree) in the case of weighted points in the projective line is continuous when approaching the Calabi-Yau geometry from the Fano geometry. More specifically, the CM volume computed via localization converges to the geometric volume, computed by McMullen with different techniques, when the sum of the weights approaches the Calabi-Yau geometry.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"206 - 222"},"PeriodicalIF":0.5,"publicationDate":"2021-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66884627","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 (M, ∇, 〈, 〉) be a manifold endowed with a flat torsionless connection r and a Riemannian metric 〈, 〉 and (TkM)k≥1 the sequence of tangent bundles given by TkM = T(Tk−1M) and T1M = TM. We show that, for any k ≥ 1, TkM carries a Hermitian structure (Jk, gk) and a flat torsionless connection ∇k and when M is a Lie group and (∇, 〈, 〉) are left invariant there is a Lie group structure on each TkM such that (Jk, gk, ∇k) are left invariant. It is well-known that (TM, J1, g1) is Kähler if and only if 〈, 〉 is Hessian, i.e, in each system of affine coordinates (x1, . . ., xn), 〈 ∂xi,∂xj 〉=∂2φ∂xi∂xj leftlangle {{partial _x}_{_i},{partial _{{x_j}}}} rightrangle = {{{partial ^2}phi } over {{partial _x}_{_i}{partial _x}_j}} . Having in mind many generalizations of the Kähler condition introduced recently, we give the conditions on (∇, 〈, 〉) so that (TM, J1, g1) is balanced, locally conformally balanced, locally conformally Kähler, pluriclosed, Gauduchon, Vaisman or Calabi-Yau with torsion. Moreover, we can control at the level of (∇, 〈, 〉) the conditions insuring that some (TkM, Jk, gk) or all of them satisfy a generalized Kähler condition. For instance, we show that there are some classes of (M, ∇, 〈, 〉) such that, for any k ≥ 1, (TkM, Jk, gk) is balanced non-Kähler and Calabi-Yau with torsion. By carefully studying the geometry of (M, ∇, 〈, 〉), we develop a powerful machinery to build a large classes of generalized Kähler manifolds.
{"title":"On the Hermitian structures of the sequence of tangent bundles of an affine manifold endowed with a Riemannian metric","authors":"M. Boucetta","doi":"10.1515/coma-2021-0128","DOIUrl":"https://doi.org/10.1515/coma-2021-0128","url":null,"abstract":"Abstract Let (M, ∇, 〈, 〉) be a manifold endowed with a flat torsionless connection r and a Riemannian metric 〈, 〉 and (TkM)k≥1 the sequence of tangent bundles given by TkM = T(Tk−1M) and T1M = TM. We show that, for any k ≥ 1, TkM carries a Hermitian structure (Jk, gk) and a flat torsionless connection ∇k and when M is a Lie group and (∇, 〈, 〉) are left invariant there is a Lie group structure on each TkM such that (Jk, gk, ∇k) are left invariant. It is well-known that (TM, J1, g1) is Kähler if and only if 〈, 〉 is Hessian, i.e, in each system of affine coordinates (x1, . . ., xn), 〈 ∂xi,∂xj 〉=∂2φ∂xi∂xj leftlangle {{partial _x}_{_i},{partial _{{x_j}}}} rightrangle = {{{partial ^2}phi } over {{partial _x}_{_i}{partial _x}_j}} . Having in mind many generalizations of the Kähler condition introduced recently, we give the conditions on (∇, 〈, 〉) so that (TM, J1, g1) is balanced, locally conformally balanced, locally conformally Kähler, pluriclosed, Gauduchon, Vaisman or Calabi-Yau with torsion. Moreover, we can control at the level of (∇, 〈, 〉) the conditions insuring that some (TkM, Jk, gk) or all of them satisfy a generalized Kähler condition. For instance, we show that there are some classes of (M, ∇, 〈, 〉) such that, for any k ≥ 1, (TkM, Jk, gk) is balanced non-Kähler and Calabi-Yau with torsion. By carefully studying the geometry of (M, ∇, 〈, 〉), we develop a powerful machinery to build a large classes of generalized Kähler manifolds.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"18 - 51"},"PeriodicalIF":0.5,"publicationDate":"2021-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43631550","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 give a complete classification of left invariant para-Kähler structures on four-dimensional simply connected Lie groups up to an automorphism. As an application we discuss some curvatures properties of the canonical connection associated to these structures as flat, Ricci flat and existence of Ricci solitons.
{"title":"The classification of left-invariant para-Kähler structures on simply connected four-dimensional Lie groups","authors":"M. W. Mansouri, A. Oufkou","doi":"10.1515/coma-2021-0127","DOIUrl":"https://doi.org/10.1515/coma-2021-0127","url":null,"abstract":"Abstract We give a complete classification of left invariant para-Kähler structures on four-dimensional simply connected Lie groups up to an automorphism. As an application we discuss some curvatures properties of the canonical connection associated to these structures as flat, Ricci flat and existence of Ricci solitons.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"88 4","pages":"1 - 17"},"PeriodicalIF":0.5,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41275108","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 obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ2, which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ3 with one cusp (for δ ≤ 2).
摘要本文给出了一个具有顶点的有理次d曲线的个数公式，该曲线的象位于一个经过r条线和s个点的(其中r + 2s = 3d + 1)。这个问题可以看作是先前由Z. Ran([13])、r . Pandharipande([12])和a . Zinger([16])研究的经典的关于计算有理次d曲线的问题的族版本。我们通过计算相关束的欧拉类，然后找出相应的简并对欧拉类的贡献来得到这个数。我们使用的方法与A. Zinger([1])和I. Biswas、S. D 'Mello、R. Mukherjee和V. Pingali([1])所采用的方法密切相关。我们还验证了有理尖形平面三次和四分之一的特征数的答案与N. Das和第一作者([2])得到的答案是一致的，他们计算了一个尖形(δ≤2)的δ-节点平面曲线的特征数。
{"title":"Rational cuspidal curves in a moving family of ℙ2","authors":"R. Mukherjee, R. Singh","doi":"10.1515/coma-2020-0110","DOIUrl":"https://doi.org/10.1515/coma-2020-0110","url":null,"abstract":"Abstract In this paper we obtain a formula for the number of rational degree d curves in ℙ3 having a cusp, whose image lies in a ℙ2 and that passes through r lines and s points (where r + 2s = 3d + 1). This problem can be viewed as a family version of the classical question of counting rational cuspidal curves in ℙ2, which has been studied earlier by Z. Ran ([13]), R. Pandharipande ([12]) and A. Zinger ([16]). We obtain this number by computing the Euler class of a relevant bundle and then finding out the corresponding degenerate contribution to the Euler class. The method we use is closely based on the method followed by A. Zinger ([16]) and I. Biswas, S. D’Mello, R. Mukherjee and V. Pingali ([1]). We also verify that our answer for the characteristic numbers of rational cuspidal planar cubics and quartics is consistent with the answer obtained by N. Das and the first author ([2]), where they compute the characteristic number of δ-nodal planar curves in ℙ3 with one cusp (for δ ≤ 2).","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"125 - 137"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0110","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45545967","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 short paper, we will review the proposal of a correspondence between the doubled geometry of Double Field Theory and the higher geometry of bundle gerbes. Double Field Theory is T-duality covariant formulation of the supergravity limit of String Theory, which generalises Kaluza-Klein theory by unifying metric and Kalb-Ramond field on a doubled-dimensional space. In light of the proposed correspondence, this doubled geometry is interpreted as an atlas description of the higher geometry of bundle gerbes. In this sense, Double Field Theory can be interpreted as a field theory living on the total space of the bundle gerbe, just like Kaluza-Klein theory is set on the total space of a principal bundle. This correspondence provides a higher geometric interpretation for para-Hermitian geometry which opens the door to its generalisation to Exceptional Field Theory. This review is based on, but not limited to, my talk at the workshop Generalized Geometry and Applications at Universität Hamburg on 3rd of March 2020.
{"title":"Towards an extended/higher correspondence","authors":"L. Alfonsi","doi":"10.1515/coma-2020-0121","DOIUrl":"https://doi.org/10.1515/coma-2020-0121","url":null,"abstract":"Abstract In this short paper, we will review the proposal of a correspondence between the doubled geometry of Double Field Theory and the higher geometry of bundle gerbes. Double Field Theory is T-duality covariant formulation of the supergravity limit of String Theory, which generalises Kaluza-Klein theory by unifying metric and Kalb-Ramond field on a doubled-dimensional space. In light of the proposed correspondence, this doubled geometry is interpreted as an atlas description of the higher geometry of bundle gerbes. In this sense, Double Field Theory can be interpreted as a field theory living on the total space of the bundle gerbe, just like Kaluza-Klein theory is set on the total space of a principal bundle. This correspondence provides a higher geometric interpretation for para-Hermitian geometry which opens the door to its generalisation to Exceptional Field Theory. This review is based on, but not limited to, my talk at the workshop Generalized Geometry and Applications at Universität Hamburg on 3rd of March 2020.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"302 - 328"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48111426","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 locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.
{"title":"Locally conformally balanced metrics on almost abelian Lie algebras","authors":"Fabio Paradiso","doi":"10.1515/coma-2020-0111","DOIUrl":"https://doi.org/10.1515/coma-2020-0111","url":null,"abstract":"Abstract We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"196 - 207"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0111","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43182066","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 some estimates for a real-valued smooth function φ on almost Hermitian manifolds. In the present paper, we show that ∂∂∂̄ φ and ∂̄∂∂̄ φ can be estimated by the gradient of the function φ.
{"title":"Estimates for a function on almost Hermitian manifolds","authors":"Masaya Kawamura","doi":"10.1515/coma-2020-0118","DOIUrl":"https://doi.org/10.1515/coma-2020-0118","url":null,"abstract":"Abstract We study some estimates for a real-valued smooth function φ on almost Hermitian manifolds. In the present paper, we show that ∂∂∂̄ φ and ∂̄∂∂̄ φ can be estimated by the gradient of the function φ.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"267 - 273"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41366376","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 is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.
{"title":"Gerbes in Geometry, Field Theory, and Quantisation","authors":"Severin Bunk","doi":"10.1515/coma-2020-0112","DOIUrl":"https://doi.org/10.1515/coma-2020-0112","url":null,"abstract":"Abstract This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"150 - 182"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/coma-2020-0112","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42058437","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 After a review on the development of deformation theory of abelian complex structures from both the classical and generalized sense, we propose the concept of semi-abelian generalized complex structure. We present some observations on such structure and illustrate this new concept with a variety of examples.
{"title":"Abelian Complex Structures and Generalizations","authors":"Y. Poon","doi":"10.1515/coma-2020-0117","DOIUrl":"https://doi.org/10.1515/coma-2020-0117","url":null,"abstract":"Abstract After a review on the development of deformation theory of abelian complex structures from both the classical and generalized sense, we propose the concept of semi-abelian generalized complex structure. We present some observations on such structure and illustrate this new concept with a variety of examples.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"247 - 266"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42108128","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}
C. Boyer, Hongnian Huang, Christina W. Tønnesen-Friedman
Abstract We study the transverse Kähler holonomy groups on Sasaki manifolds (M, S) and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number b1(M) and the basic Hodge number h0,2B(S) vanish, then S is stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure is S-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is S-stable when dim M ≥ 7. Finally, we prove that the standard Sasaki join operation (transverse holonomy U(n1) × U(n2)) as well as the fiber join operation preserve S-stability.
{"title":"Transverse Kähler holonomy in Sasaki Geometry and S-Stability","authors":"C. Boyer, Hongnian Huang, Christina W. Tønnesen-Friedman","doi":"10.1515/coma-2020-0123","DOIUrl":"https://doi.org/10.1515/coma-2020-0123","url":null,"abstract":"Abstract We study the transverse Kähler holonomy groups on Sasaki manifolds (M, S) and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number b1(M) and the basic Hodge number h0,2B(S) vanish, then S is stable under deformations of the transverse Kähler flow. In addition we show that an irreducible transverse hyperkähler Sasakian structure is S-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is S-stable when dim M ≥ 7. Finally, we prove that the standard Sasaki join operation (transverse holonomy U(n1) × U(n2)) as well as the fiber join operation preserve S-stability.","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"8 1","pages":"336 - 353"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43490207","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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