16 results
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2. Maximally algebraic potentially irrational cubic fourfolds.
- Author
-
Laza, Radu
- Subjects
LOGICAL prediction ,IRRATIONAL numbers ,MATHEMATICS ,GEOMETRY - Abstract
A well known conjecture due to Hassett asserts that a cubic fourfold X whose transcendental cohomology T
X cannot be realized as the transcendental cohomology of a K3 surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic 2-cycles on them, it is natural to ask for the most algebraic cubic fourfolds X to which this conjecture is still applicable. In this paper, we show that for an appropriate "algebraicity index" κX ∈ Q+ , there exists a unique class of cubics maximizing κX , not having an associated K3 surface; namely, the cubic fourfolds with an Eckardt point (previously investigated in by Laza, Pearlstein, and Zhang [Adv. Math. 340 (2018), pp. 684-722]). Arguably, they are the most algebraic conjecturally irrational cubic fourfolds, and thus a good testing ground for Hassett's irrationality conjecture for cubic fourfolds. [ABSTRACT FROM AUTHOR]- Published
- 2021
- Full Text
- View/download PDF
3. Hessenberg varieties, intersections of quadrics, and the Springer correspondence.
- Author
-
Chen, Tsao-Hsien, Vilonen, Kari, and Xue, Ting
- Subjects
SYMMETRIC spaces ,QUADRICS ,FOURIER transforms ,LETTERS ,GEOMETRY ,MATHEMATICS - Abstract
In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [Compos. Math. 154 (2018), pp. 2403-2425]. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
4. On the geometry of the second fundamental form of the Torelli map.
- Author
-
Frediani, Paola and Pirola, Gian Pietro
- Subjects
GEOMETRY ,GEODESICS ,MATHEMATICS ,CURVES ,HYPERELLIPTIC integrals - Abstract
In this paper we give a geometric interpretation of the second fundamental form of the period map of curves and we use it to improve the upper bounds on the dimension of a totally geodesic subvariety Y of A
g generically contained in the Torelli locus obtained by Elisabetta Colombo, Paola Frediani, and Alessandro Ghigi [Internat. J. Math. 26 (2015), no. 1, 1550005] and A. Ghigi, P. Pirola, and S. Torelli (to appear on Communications in Contemporary Mathematics, https:// doi.org/10.1142/S0219199720500200). We get dim Y ≤ 2g − 1 if g is even, dim Y ≤ 2g if g is odd. We also study totally geodesic subvarieties Z of Ag generically contained in the hyperelliptic Torelli locus and we show that dim Z ≤ g + 1. [ABSTRACT FROM AUTHOR]- Published
- 2021
- Full Text
- View/download PDF
5. Enumerative geometry of del Pezzo surfaces.
- Author
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Lin, Yu-Shen
- Subjects
GEOMETRY ,TORUS ,MATHEMATICS - Abstract
We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin [Duke Math. J. 170 (2021), pp. 1291–1375]. We also include some explicit calculations for the projective plane, which confirm some folklore conjectures in this case. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
6. ON ENDOMORPHISMS OF ARRANGEMENT COMPLEMENTS.
- Author
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KURUL, ŞEVDA and WERNER, ANNETTE
- Subjects
ANALYTIC geometry ,FINITE fields ,PROJECTIVE spaces ,AUTOMORPHISMS ,ENDOMORPHISMS ,MATHEMATICS ,GEOMETRY - Abstract
Let Ω be the complement of a connected, essential hyperplane arrangement. We prove that every dominant endomorphism of Ω extends to an endomorphism of the tropical compactification X of Ω associated to the Bergman fan structure on the tropical variety trop(Ω). This generalizes a result in [Compos. Math. 149 (2013), pp. 1211–1224], which states that every automorphism of Drinfeld’s half-space over a finite field F
q extends to an automorphism of the successive blow-up of projective space at all Fq -rational linear subspaces. This successive blow-up is in fact the minimal wonderful compactification by de Concini and Procesi, which coincides with X by results of Feichtner and Sturmfels. Whereas the proof in [Compos. Math. 149 (2013), pp. 1211–1224] is based on Berkovich analytic geometry over the trivially valued finite ground field, the generalization proved in the present paper relies on matroids and tropical geometry. [ABSTRACT FROM AUTHOR]- Published
- 2019
- Full Text
- View/download PDF
7. EIGENVALUES AND EIGENFUNCTIONS OF DOUBLE LAYER POTENTIALS.
- Author
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YOSHIHISA MIYANISHI and TAKASHI SUZUKI
- Subjects
EIGENVALUES ,EIGENFUNCTIONS ,GEOMETRY ,EIGENANALYSIS ,MATHEMATICS - Abstract
Eigenvalues and eigenfunctions of two- and three-dimensional double layer potentials are considered. Let Ω be a C
2 bounded region in Rn (n = 2, 3). The double layer potential K : L2 (∂Ω) → L2 (∂Ω) is defined by (Kψ)(x) ≡ ∫∂ Ω ψ(y)·vy E(x, y) dsy , where E(x, y) = ∫1/2π log1/∣x-y∣ , if n = 2,1/π log1/∣x-y∣ , if n = 3, dsy is the line or surface element and vy is the outer normal derivative on ∂Ω. It is known that K is a compact operator on L2 (∂Ω) and consists of at most a countable number of eigenvalues, with 0 as the only possible limit point. This paper aims to establish some relationships among the eigenvalues, the eigenfunctions, and the geometry of ∂Ω. [ABSTRACT FROM AUTHOR]- Published
- 2017
- Full Text
- View/download PDF
8. Z-GRADED SIMPLE RINGS.
- Author
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BELL, J. and ROGALSKI, D.
- Subjects
WEYL groups ,WEYL space ,GEOMETRY ,INTEGERS ,MATHEMATICS - Abstract
The Weyl algebra over a field k of characteristic 0 is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all Z-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study Z-graded simple rings A of any dimension which have a graded quotient ring of the form K[t, t
-1 ; σ] for a field K. Under some further hypotheses, we classify all such A in terms of a new construction of simple rings which we introduce in this paper. In the important special case that GKdimA = tr. deg(K/k) + 1, we show that K and σ must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry. [ABSTRACT FROM AUTHOR]- Published
- 2016
- Full Text
- View/download PDF
9. Poisson approximation and Weibull asymptotics in the geometry of numbers.
- Author
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Björklund, Michael and Gorodnik, Alexander
- Subjects
EUCLIDEAN domains ,GEOMETRY ,MATHEMATICS ,LOGARITHMS - Abstract
Minkowski's First Theorem and Dirichlet's Approximation Theorem provide upper bounds on certain minima taken over lattice points contained in domains of Euclidean spaces. We study the distribution of such minima and show, under some technical conditions, that they exhibit Weibull asymptotics with respect to different natural measures on the space of unimodular lattices in \mathbb {R}^d. This follows from very general Poisson approximation results for shrinking targets which should be of independent interest. Furthermore, we show in the appendix that the logarithm laws of Kleinbock-Margulis [Invent. Math. 138 (1999), pp. 451–494], Khinchin and Gallagher [J. London Math. Soc. 37 (1962), pp. 387–390] can be deduced from our distributional results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
10. Complex nilmanifolds with constant holomorphic sectional curvature.
- Author
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Li, Yulu and Zheng, Fangyang
- Subjects
CURVATURE ,LOGICAL prediction ,MATHEMATICS ,GEOMETRY - Abstract
A well known conjecture in complex geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler if the constant is non-zero and must be Chern flat if the constant is zero. The conjecture is confirmed in complex dimension 2, by the work of Balas-Gauduchon [Math. Z. 189 (1985), pp. 193–210]. (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov [Trans. Amer. Math. Soc. 348 (1996), pp. 3051–3063] (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to the class of complex nilmanifolds and confirm the conjecture in that case. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
11. ON PROJECTIVIZED VECTOR BUNDLES AND POSITIVE HOLOMORPHIC SECTIONAL CURVATURE.
- Author
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ALVAREZ, ANGELYNN, HEIER, GORDON, and FANGYANG ZHENG
- Subjects
HOLOMORPHIC functions ,VECTOR bundles ,MANIFOLDS (Mathematics) ,GEOMETRY ,MATHEMATICS - Abstract
We generalize a construction of Hitchin to prove that, given any compact Kähler manifold M with positive holomorphic sectional curvature and any holomorphic vector bundle E over M, the projectivized vector bundle ℙ(E) admits a Kähler metric with positive holomorphic sectional curvature. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
12. ON COMPUTATIONS WITH DESSINS D'ENFANTS.
- Author
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KAMALINEJAD, ALI and SHAHSHAHANI, MEHRDAD
- Subjects
DIFFERENTIAL equations ,CALCULUS ,JORDAN curves ,ALGORITHMS ,GEOMETRY ,MATHEMATICS - Abstract
The geometric theory of dessins d'enfants is used to make explicit calculations on curves. In particular, an algorithmic procedure for the construction of ramified covering of curves over number fields with prescribed ramifications and for the explicit construction of Jenkins-Strebel differentials are developed. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
13. MODULI SPACES AND MACROMOLECULES.
- Author
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PENNER, R. C.
- Subjects
MODULI theory ,MATHEMATICS ,MACROMOLECULES ,PHYSICS ,GEOMETRY ,POLYSACCHARIDES ,MATHEMATICIANS - Abstract
The article surveys the progress in applying techniques from moduli spaces in mathematics and physics to studying the geometry and topology of three families of macromolecules of interest in biology, including RNAs, proteins and polysaccharides. It is intended for audience such as mathematicians whose expertise does not necessarily extend to either moduli spaces or macromolecules and who may be interested in the nascent application of geometry to problems in biology.
- Published
- 2016
- Full Text
- View/download PDF
14. THE IDEAL OF THE TRIFOCAL VARIETY.
- Author
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AHOLT, CHRIS and OEDING, LUKE
- Subjects
ALGEBRAIC geometry ,MATHEMATICAL analysis ,GEOMETRY ,MATHEMATICS ,ALGEBRA - Abstract
Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor is a trifocal tensor is also given. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
15. A characterization of virtually embedded subsurfaces in 3-manifolds
- Author
-
Yi Liu
- Subjects
Applied Mathematics ,General Mathematics ,57M05 (Primary) ,010102 general mathematics ,Taut foliation ,Fibered knot ,Geometry ,Geometric Topology (math.GT) ,16. Peace & justice ,01 natural sciences ,Mathematics::Geometric Topology ,Physics::Geophysics ,Mathematics - Geometric Topology ,0103 physical sciences ,Suspension flow ,FOS: Mathematics ,010307 mathematical physics ,Mathematics::Differential Geometry ,0101 mathematics ,Invariant (mathematics) ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this case, the subsurface is virtually a leaf of a taut foliation. Besides other consequences, examples are exhibited that non-geometric 3-manifolds with no Seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces., Comment: 28 pages. Errors of previous Proposition 3.1 and Formula 7.2 corrected
- Published
- 2017
16. Real and Complex Singularities
- Author
-
Farid Tari and Graham Mark Reeve
- Subjects
Plane curve ,Medial axis ,Euclidean geometry ,Minkowski space ,Mathematics::Metric Geometry ,Geometry ,Gravitational singularity ,Mathematics - Abstract
In this paper a Minkowski analogue of the Euclidean medial axis of a closed and smooth plane curve is introduced. Its generic local configurations are studied and the types of shocks that occur on these are also determined.
- Published
- 2016
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