Pub Date : 2023-04-26DOI: 10.59277/rrmpa.2023.61.82
Dan Coman, G. Marinescu, Hua Wang
We obtain asymptotic estimates of the dimension of cohomology on possibly non-compact complex manifolds for line bundles endowed with Hermitian metrics with algebraic singularities. We give a unified approach to establishing singular holomorphic Morse inequalities for hyperconcave manifolds, pseudoconvex domains, q-convex manifolds and q-concave manifolds, and we generalize related estimates of Berndtsson. We also consider the case of metrics with more general than algebraic singularities.
{"title":"SINGULAR HOLOMORPHIC MORSE INEQUALITIES ON NON-COMPACT MANIFOLDS","authors":"Dan Coman, G. Marinescu, Hua Wang","doi":"10.59277/rrmpa.2023.61.82","DOIUrl":"https://doi.org/10.59277/rrmpa.2023.61.82","url":null,"abstract":"We obtain asymptotic estimates of the dimension of cohomology on possibly non-compact complex manifolds for line bundles endowed with Hermitian metrics with algebraic singularities. We give a unified approach to establishing singular holomorphic Morse inequalities for hyperconcave manifolds, pseudoconvex domains, q-convex manifolds and q-concave manifolds, and we generalize related estimates of Berndtsson. We also consider the case of metrics with more general than algebraic singularities.","PeriodicalId":45738,"journal":{"name":"REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES","volume":"160 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76079959","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}
Pub Date : 2023-01-01DOI: 10.59277/rrmpa.2023.141.148
N. Manolache
One gives a characterization of the Gorenstein nilpotent scheme structures on a smooth algebraic variety as support, in terms of a duality property of the graded objects associated to two canonical filtrations.
{"title":"DUALITY FOR GORENSTEIN MULTIPLE STRUCTURES ON SMOOTH ALGEBRAIC VARIETIES","authors":"N. Manolache","doi":"10.59277/rrmpa.2023.141.148","DOIUrl":"https://doi.org/10.59277/rrmpa.2023.141.148","url":null,"abstract":"One gives a characterization of the Gorenstein nilpotent scheme structures on a smooth algebraic variety as support, in terms of a duality property of the graded objects associated to two canonical filtrations.","PeriodicalId":45738,"journal":{"name":"REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES","volume":"462 2-3 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78403840","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}
Pub Date : 2023-01-01DOI: 10.59277/rrmpa.2023.149.167
Eugen Mihailescu
In this survey, we present some methods in the dynamics and dimension theory for invariant measures of hyperbolic endomorphisms (smooth non-invertible maps), and for conformal iterated function systems with overlaps. For endomorphisms, we recall the notion of asymptotic degree of an equilibrium measure, which is shown to be related to the folding entropy; this degree is then applied to dimension estimates. For finite iterated function systems, we present the notion of overlap number of a measure, which is related to the folding entropy of a lift transformation, and also give some examples when it can be computed or estimated. We apply overlap numbers to prove the exact dimensionality of invariant measures, and to obtain a geometric formula for their dimension. Then, for countable conformal iterated function systems with overlaps, the projections of ergodic measures are shown to be exact dimensional, and we give a dimension formula. Relations with ergodic number theory, continued fractions, and random dynamical systems are also presented.
{"title":"DIMENSIONS OF MEASURES, DEGREES, AND FOLDING ENTROPY IN DYNAMICS","authors":"Eugen Mihailescu","doi":"10.59277/rrmpa.2023.149.167","DOIUrl":"https://doi.org/10.59277/rrmpa.2023.149.167","url":null,"abstract":"In this survey, we present some methods in the dynamics and dimension theory for invariant measures of hyperbolic endomorphisms (smooth non-invertible maps), and for conformal iterated function systems with overlaps. For endomorphisms, we recall the notion of asymptotic degree of an equilibrium measure, which is shown to be related to the folding entropy; this degree is then applied to dimension estimates. For finite iterated function systems, we present the notion of overlap number of a measure, which is related to the folding entropy of a lift transformation, and also give some examples when it can be computed or estimated. We apply overlap numbers to prove the exact dimensionality of invariant measures, and to obtain a geometric formula for their dimension. Then, for countable conformal iterated function systems with overlaps, the projections of ergodic measures are shown to be exact dimensional, and we give a dimension formula. Relations with ergodic number theory, continued fractions, and random dynamical systems are also presented.","PeriodicalId":45738,"journal":{"name":"REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES","volume":"29 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90016441","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}
Pub Date : 2023-01-01DOI: 10.59277/rrmpa.2023.191.198
Ovidiu Preda
In this short survey, we present the results obtained so far in the geometry of lcK singular spaces.
在这篇简短的综述中，我们给出了迄今为止在lcK奇异空间几何中得到的一些结果。
{"title":"LCK SPACES. A SHORT SURVEY","authors":"Ovidiu Preda","doi":"10.59277/rrmpa.2023.191.198","DOIUrl":"https://doi.org/10.59277/rrmpa.2023.191.198","url":null,"abstract":"In this short survey, we present the results obtained so far in the geometry of lcK singular spaces.","PeriodicalId":45738,"journal":{"name":"REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES","volume":"11 4 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78339884","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}
Pub Date : 2023-01-01DOI: 10.59277/rrmpa.2023.169.189
T. Ohsawa
After recalling basic results on the L2 ¯∂-cohomology groups and known existence criteria for bounded plurisubharmonic exhaustion functions on locally pseudoconvex bounded domains, results on the Bergman kernel on hyperconvex domains will be reviewed. Then, on locally pseudoconvex domains with certain regularity constraints on the boundary, a result on the asymptotics of the Bergman kernel is proved without assuming the existence of plurisubharmonic exhaustion functions, as an application of the finite-dimensionality of L2 ¯∂-cohomology groups.
{"title":"ON HYPERCONVEXITY AND TOWARDS BUNDLE-VALUED KERNEL ASYMPTOTICS ON LOCALLY PSEUDOCONVEX DOMAINS","authors":"T. Ohsawa","doi":"10.59277/rrmpa.2023.169.189","DOIUrl":"https://doi.org/10.59277/rrmpa.2023.169.189","url":null,"abstract":"After recalling basic results on the L2 ¯∂-cohomology groups and known existence criteria for bounded plurisubharmonic exhaustion functions on locally pseudoconvex bounded domains, results on the Bergman kernel on hyperconvex domains will be reviewed. Then, on locally pseudoconvex domains with certain regularity constraints on the boundary, a result on the asymptotics of the Bergman kernel is proved without assuming the existence of plurisubharmonic exhaustion functions, as an application of the finite-dimensionality of L2 ¯∂-cohomology groups.","PeriodicalId":45738,"journal":{"name":"REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES","volume":"44 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76974661","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}
Pub Date : 2023-01-01DOI: 10.59277/rrmpa.2023.95.114
A. Iordan
Let L be a hypothetical smooth Levi flat hypersurface in CP2 and r the signed distance to L by means of the Fubini-Study metric g. Denote Lru = cru the second order elliptic equation for the infinitesimal Levi-flat deformations of L, where cr = dbJbr + br ∧ Jbr, br = ιXrdγr, Xr = gradgr/ ∥gradgr∥2 g, γr is the restriction of dcr to L and db is the differentiation along the leafs of the Levi foliation. Then −cr ≥ H as leaf-wise (1, 1)-forms, where H is the holomorphic bisectional curvature of CP2. We give also an example of a Levi-flat manifold L of dimension 3 verifying that there exists a (1, 0)-form α on L such that ∂α is a K¨ahler form on every leaf of the Levi foliation, but L is not embeddable in CP2.
{"title":"ON THE EMBEDDING OF LEVI-FLAT HYPERSURFACES IN THE COMPLEX PROJECTIVE PLANE (AND AN APPENDIX WITH L´ASZL´O LEMPERT)","authors":"A. Iordan","doi":"10.59277/rrmpa.2023.95.114","DOIUrl":"https://doi.org/10.59277/rrmpa.2023.95.114","url":null,"abstract":"Let L be a hypothetical smooth Levi flat hypersurface in CP2 and r the signed distance to L by means of the Fubini-Study metric g. Denote Lru = cru the second order elliptic equation for the infinitesimal Levi-flat deformations of L, where cr = dbJbr + br ∧ Jbr, br = ιXrdγr, Xr = gradgr/ ∥gradgr∥2 g, γr is the restriction of dcr to L and db is the differentiation along the leafs of the Levi foliation. Then −cr ≥ H as leaf-wise (1, 1)-forms, where H is the holomorphic bisectional curvature of CP2. We give also an example of a Levi-flat manifold L of dimension 3 verifying that there exists a (1, 0)-form α on L such that ∂α is a K¨ahler form on every leaf of the Levi foliation, but L is not embeddable in CP2.","PeriodicalId":45738,"journal":{"name":"REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES","volume":"44 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87890390","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}
Pub Date : 2023-01-01DOI: 10.59277/rrmpa.2023.33.66
I. Coandă
We clarify the undecided case c2 = 3 of a result of Ein, Hartshorne and Vogelaar [8] about the restriction of a stable rank 3 vector bundle with c1 = 0 on the projective 3-space to a general plane. It turns out that there are more exceptions to the stable restriction property than those conjectured by the three authors. One of them is a Schwarzenberger bundle (twisted by −1); it has c3 = 6. There are also some exceptions with c3 = 2 (plus, of course, their duals).
{"title":"STABLE RANK 3 VECTOR BUNDLES ON P3 WITH c1 = 0, c2 = 3","authors":"I. Coandă","doi":"10.59277/rrmpa.2023.33.66","DOIUrl":"https://doi.org/10.59277/rrmpa.2023.33.66","url":null,"abstract":"We clarify the undecided case c2 = 3 of a result of Ein, Hartshorne and Vogelaar [8] about the restriction of a stable rank 3 vector bundle with c1 = 0 on the projective 3-space to a general plane. It turns out that there are more exceptions to the stable restriction property than those conjectured by the three authors. One of them is a Schwarzenberger bundle (twisted by −1); it has c3 = 6. There are also some exceptions with c3 = 2 (plus, of course, their duals).","PeriodicalId":45738,"journal":{"name":"REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES","volume":"50 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74440016","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}
Pub Date : 2022-12-12DOI: 10.59277/RRMPA.2023.9.17
Ying Chen, Mihai Tibùar
We discuss the most general condition under which a singular local tube fibration exists. We give an application to composition of map germs.
我们讨论了奇异局部管颤振存在的最一般条件。给出了在地图细菌合成中的一个应用。
{"title":"ON SINGULAR MAPS WITH LOCAL FIBRATION","authors":"Ying Chen, Mihai Tibùar","doi":"10.59277/RRMPA.2023.9.17","DOIUrl":"https://doi.org/10.59277/RRMPA.2023.9.17","url":null,"abstract":"We discuss the most general condition under which a singular local tube fibration exists. We give an application to composition of map germs.","PeriodicalId":45738,"journal":{"name":"REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES","volume":"76 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85726838","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}
Pub Date : 2022-04-14DOI: 10.59277/rrmpa.2023.83.94
F. Forstnerič
We show that every bordered Riemann surface, M, with smooth boundary bM admits a proper holomorphic map M → Ω into any bounded strongly pseudoconvex domain Ω in Cn, n > 1, extending to a smooth map f : M → Ω which can be chosen an immersion if n ≥ 3 and an embedding if n ≥ 4. Furthermore, f can be chosen to approximate a given holomorphic map M → Ω on compacts in M and interpolate it at finitely many given points in M.
我们证明了在Cn, n > 1中，具有光滑边界bM的每一个有边界的Riemann曲面M允许一个适当的全纯映射M→Ω进入任何有界的强伪凸域Ω，并扩展到一个光滑映射f: M→Ω，当n≥3时可以选择浸入，当n≥4时可以选择嵌入。此外，可以选择f来近似M中的紧簇上的给定全纯映射M→Ω，并将其插值到M中的有限多个给定点。
{"title":"EMBEDDING BORDERED RIEMANN SURFACES IN STRONGLY PSEUDOCONVEX DOMAINS","authors":"F. Forstnerič","doi":"10.59277/rrmpa.2023.83.94","DOIUrl":"https://doi.org/10.59277/rrmpa.2023.83.94","url":null,"abstract":"We show that every bordered Riemann surface, M, with smooth boundary bM admits a proper holomorphic map M → Ω into any bounded strongly pseudoconvex domain Ω in Cn, n > 1, extending to a smooth map f : M → Ω which can be chosen an immersion if n ≥ 3 and an embedding if n ≥ 4. Furthermore, f can be chosen to approximate a given holomorphic map M → Ω on compacts in M and interpolate it at finitely many given points in M.","PeriodicalId":45738,"journal":{"name":"REVUE ROUMAINE DE MATHEMATIQUES PURES ET APPLIQUEES","volume":"10 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79510430","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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