Pub Date : 2022-04-05DOI: 10.2422/2036-2145.202205_014
Vladimiro Benedetti, L. Manivel
Tevelev has given a remarkable explicit formula for the discriminant of a complex simple Lie algebra, which can be defined as the equation of the dual hypersurface of the minimal nilpotent orbit, or of the so-called adjoint variety. In this paper we extend this formula to the setting of graded Lie algebras, and express the equation of the corresponding dual hypersurfaces in terms of the reflections in the little Weyl groups, the associated complex reflection groups. This explains for example why the codegree of the Grassmannian $G(4, 8)$ is equal to the number of roots of $mathfrak{e}_7$ .
{"title":"Discriminants of theta-representations","authors":"Vladimiro Benedetti, L. Manivel","doi":"10.2422/2036-2145.202205_014","DOIUrl":"https://doi.org/10.2422/2036-2145.202205_014","url":null,"abstract":"Tevelev has given a remarkable explicit formula for the discriminant of a complex simple Lie algebra, which can be defined as the equation of the dual hypersurface of the minimal nilpotent orbit, or of the so-called adjoint variety. In this paper we extend this formula to the setting of graded Lie algebras, and express the equation of the corresponding dual hypersurfaces in terms of the reflections in the little Weyl groups, the associated complex reflection groups. This explains for example why the codegree of the Grassmannian $G(4, 8)$ is equal to the number of roots of $mathfrak{e}_7$ .","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86901723","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 : 2022-03-30DOI: 10.2422/2036-2145.202203_026
F. Cacciafesta, A. Suzzoni, Long Meng
In this paper we prove Strichartz estimates for the Dirac equation on asymptotically flat manifolds. The proof combines the weak dispersive estimates proved by the first two authors with the Strichartz and smoothing estimates for the wave and Klein-Gordon flows, exploiting some recent results in the same geometrical setting.
{"title":"Strichartz estimates for the Dirac equation on asymptotically flat manifolds","authors":"F. Cacciafesta, A. Suzzoni, Long Meng","doi":"10.2422/2036-2145.202203_026","DOIUrl":"https://doi.org/10.2422/2036-2145.202203_026","url":null,"abstract":"In this paper we prove Strichartz estimates for the Dirac equation on asymptotically flat manifolds. The proof combines the weak dispersive estimates proved by the first two authors with the Strichartz and smoothing estimates for the wave and Klein-Gordon flows, exploiting some recent results in the same geometrical setting.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81657032","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 : 2022-03-15DOI: 10.2422/2036-2145.202203_022
Valeria Bertini, Franco Giovenzana
We investigate the Hodge structure of the singular O'Grady's six and ten dimensional examples of irreducible symplectic varieties. In particular, we compute some of their Betti numbers and their Euler characteristic. As consequence, we deduce that these varieties do not have finite quotient singularities answering a question of Bakker and Lehn.
{"title":"Hodge structure of O'Grady's singular moduli spaces","authors":"Valeria Bertini, Franco Giovenzana","doi":"10.2422/2036-2145.202203_022","DOIUrl":"https://doi.org/10.2422/2036-2145.202203_022","url":null,"abstract":"We investigate the Hodge structure of the singular O'Grady's six and ten dimensional examples of irreducible symplectic varieties. In particular, we compute some of their Betti numbers and their Euler characteristic. As consequence, we deduce that these varieties do not have finite quotient singularities answering a question of Bakker and Lehn.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84307244","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 : 2022-02-21DOI: 10.2422/2036-2145.202202_018
C. Miebach
We show that the quotient of any bounded homogeneous domain by a unipotent discrete group of automorphisms is holomorphically separable. Then we give a necessary condition for the quotient to be Stein and prove that in some cases this condition is also sufficient.
{"title":"On quotients of bounded homogeneous domains by unipotent discrete groups","authors":"C. Miebach","doi":"10.2422/2036-2145.202202_018","DOIUrl":"https://doi.org/10.2422/2036-2145.202202_018","url":null,"abstract":"We show that the quotient of any bounded homogeneous domain by a unipotent discrete group of automorphisms is holomorphically separable. Then we give a necessary condition for the quotient to be Stein and prove that in some cases this condition is also sufficient.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84887481","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 : 2022-02-20DOI: 10.2422/2036-2145.202203_018
G. Colombo, Marco Mariani, M. Rigoli
We prove that a compact Riemannian manifold of dimension m ≥ 3 with harmonic curvature and ⌊ 2 ⌋-positive curvature operator has constant sectional curvature, extending the classical Tachibana theorem for manifolds with positive curvature operator. The condition of ⌊ 2 ⌋-positivity originates from recent work of Petersen and Wink, who proved a similar Tachibana-type theorem under the stronger condition that the manifold be Einstein. We show that the same rigidity property holds for complete manifolds assuming either parabolicity, an integral bound on the Weyl tensor or a stronger pointwise positive lower bound on the average of the first ⌊ 2 ⌋ eigenvalues of the curvature operator. For 3-manifolds, we show that positivity of the curvature operator can be relaxed to positivity of the Ricci tensor. MSC 2020 Primary: 53B20, 53C20, 53C21; Secondary: 31C12.
{"title":"Tachibana-type theorems on complete manifolds","authors":"G. Colombo, Marco Mariani, M. Rigoli","doi":"10.2422/2036-2145.202203_018","DOIUrl":"https://doi.org/10.2422/2036-2145.202203_018","url":null,"abstract":"We prove that a compact Riemannian manifold of dimension m ≥ 3 with harmonic curvature and ⌊ 2 ⌋-positive curvature operator has constant sectional curvature, extending the classical Tachibana theorem for manifolds with positive curvature operator. The condition of ⌊ 2 ⌋-positivity originates from recent work of Petersen and Wink, who proved a similar Tachibana-type theorem under the stronger condition that the manifold be Einstein. We show that the same rigidity property holds for complete manifolds assuming either parabolicity, an integral bound on the Weyl tensor or a stronger pointwise positive lower bound on the average of the first ⌊ 2 ⌋ eigenvalues of the curvature operator. For 3-manifolds, we show that positivity of the curvature operator can be relaxed to positivity of the Ricci tensor. MSC 2020 Primary: 53B20, 53C20, 53C21; Secondary: 31C12.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83719112","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 : 2022-02-14DOI: 10.2422/2036-2145.202106_009
H. Frankowska, Thomas Lorenz
{"title":"Filippov’s Theorem for mutational inclusions in a metric space","authors":"H. Frankowska, Thomas Lorenz","doi":"10.2422/2036-2145.202106_009","DOIUrl":"https://doi.org/10.2422/2036-2145.202106_009","url":null,"abstract":"","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79010069","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}
{"title":"Regularity of primes associated with polynomial parametrisations","authors":"Francesca Cioffi, A. Conca","doi":"10.2422/2036-2145.202107_020","DOIUrl":"https://doi.org/10.2422/2036-2145.202107_020","url":null,"abstract":"We prove a doubly exponential bound for the Castelnuovo-Mumford regularity of prime ideals defining varieties with polynomial parametrisation.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82512180","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 : 2022-02-11DOI: 10.2422/2036-2145.202205_016
A. Boscaggin, W. Dambrosio, D. Papini
We study relativistic Kepler problems in the plane. At first, using non-smooth critical point theory, we show that under a general time-periodic external force of gradient type there are two infinite families of T -periodic solutions, parameterized by their winding number around the singularity: the first family is a sequence of local minima, while the second one comes from a mountain pass-type geometry of the action functional. Secondly, we investigate the minimality of the circular and noncircular periodic solutions of the unforced problem, via Morse index theory and level estimates of the action functional.
{"title":"Periodic solutions to relativistic Kepler problems: a variational approach","authors":"A. Boscaggin, W. Dambrosio, D. Papini","doi":"10.2422/2036-2145.202205_016","DOIUrl":"https://doi.org/10.2422/2036-2145.202205_016","url":null,"abstract":"We study relativistic Kepler problems in the plane. At first, using non-smooth critical point theory, we show that under a general time-periodic external force of gradient type there are two infinite families of T -periodic solutions, parameterized by their winding number around the singularity: the first family is a sequence of local minima, while the second one comes from a mountain pass-type geometry of the action functional. Secondly, we investigate the minimality of the circular and noncircular periodic solutions of the unforced problem, via Morse index theory and level estimates of the action functional.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"120 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91468721","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 : 2022-01-26DOI: 10.2422/2036-2145.202204_004
Serge Lvovski
. Using an adjunction-theoretic result due to A. J. Sommese together with a proposition from SGA7, we obtain a complete list of smooth threefolds for which the monodromy group acting on H 2 of its smooth hyperplane section is Z / 2 Z . The possibility of such a classification was announced by F. L. Zak in 1991.
． 利用a . J. Sommese的一个合论结果和SGA7的一个命题，我们得到了一个光滑三折的完整列表，其中作用在其光滑超平面截面h2上的单群为Z / 2z。这种分类的可能性是由f·l·扎克在1991年宣布的。
{"title":"On threefolds with the smallest nontrivial monodromy group","authors":"Serge Lvovski","doi":"10.2422/2036-2145.202204_004","DOIUrl":"https://doi.org/10.2422/2036-2145.202204_004","url":null,"abstract":". Using an adjunction-theoretic result due to A. J. Sommese together with a proposition from SGA7, we obtain a complete list of smooth threefolds for which the monodromy group acting on H 2 of its smooth hyperplane section is Z / 2 Z . The possibility of such a classification was announced by F. L. Zak in 1991.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"139 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78895679","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 : 2021-12-28DOI: 10.2422/2036-2145.202201_009
Joachim Jelisiejew, H. Keneshlou
A continuous map $mathbb{C}^nto Gr(tau, N)$ is $k$-regular if the $tau$-dimensional subspaces corresponding to images of any $k$ distinct points span a $tau k$-dimensional space. For $tau = 1$ this essentially recovers the classical notion of a $k$-regular map $mathbb{C}^nto mathbb{C}^N$. We provide new examples of $k$-regular maps, both in the classical setting $tau = 1$ and for $taugeq 2$, where these are the first examples known. Our methods come from algebraic geometry, following and generalizing Buczy'{n}ski-Januszkiewicz-Jelisiejew-Micha{l}ek. The key and highly nontrivial part of the argument is proving that certain loci of the Hilbert scheme of points have expected dimension. As an important side result, we prove irreducibility of the punctual Hilbert scheme of $k$ points on a threefold, for $kleq 11$.
{"title":"On construction of k-regular maps to Grassmannians via algebras of socle dimension two","authors":"Joachim Jelisiejew, H. Keneshlou","doi":"10.2422/2036-2145.202201_009","DOIUrl":"https://doi.org/10.2422/2036-2145.202201_009","url":null,"abstract":"A continuous map $mathbb{C}^nto Gr(tau, N)$ is $k$-regular if the $tau$-dimensional subspaces corresponding to images of any $k$ distinct points span a $tau k$-dimensional space. For $tau = 1$ this essentially recovers the classical notion of a $k$-regular map $mathbb{C}^nto mathbb{C}^N$. We provide new examples of $k$-regular maps, both in the classical setting $tau = 1$ and for $taugeq 2$, where these are the first examples known. Our methods come from algebraic geometry, following and generalizing Buczy'{n}ski-Januszkiewicz-Jelisiejew-Micha{l}ek. The key and highly nontrivial part of the argument is proving that certain loci of the Hilbert scheme of points have expected dimension. As an important side result, we prove irreducibility of the punctual Hilbert scheme of $k$ points on a threefold, for $kleq 11$.","PeriodicalId":8132,"journal":{"name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78174566","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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