scholarly journals Betti numbers and pseudoeffective cones in 2-Fano varieties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Giosuè Emanuele Muratore
Keyword(s):  
Large Class ◽  
Betti Numbers ◽  
Fano Varieties ◽  
Higher Dimensional ◽  
Special Case ◽  
Answering Questions

Abstract The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher-dimensional analogous properties of Fano varieties. We consider (weak) k-Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties, in analogy with the case k = 1. Then we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index at least n − 2, and we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in [2].

scholarly journals Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

2020 ◽  
pp. 1-17
Author(s):  
THOMAS BARTHELMÉ ◽  
SERGIO R. FENLEY ◽  
STEVEN FRANKEL ◽  
RAFAEL POTRIE
Keyword(s):  
Large Class ◽  
Universal Cover ◽  
Isotopy Class ◽  
Seifert Manifold ◽  
Hyperbolic Diffeomorphisms ◽  
Surface Homeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

scholarly journals Constructing Carrollian CFTs

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Nishant Gupta ◽  
Nemani V. Suryanarayana
Keyword(s):  
Scalar Fields ◽  
Higher Dimensions ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Relativistic Limit ◽  
Conformal Field ◽  
Local Symmetries ◽  
Derivatives Of ◽  
Special Case

Abstract We construct classical theories for scalar fields in arbitrary Carroll spacetimes that are invariant under Carrollian diffeomorphisms and Weyl transformations. When the local symmetries are gauge fixed these theories become Carrollian conformal field theories. We show that generically there are at least two types of such theories: one in which only time derivatives of the fields appear and the other in which both space and time derivatives appear. A classification of such scalar field theories in three (and higher) dimensions up to two derivative order is provided. We show that only a special case of our theories arises in the ultra-relativistic limit of a covariant parent theory.

scholarly journals Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1077
Author(s):  
Yarema A. Prykarpatskyy
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Structure ◽  
Necessary Condition ◽  
Type Representation ◽  
Infinite Hierarchy ◽  
Polynomial Coefficients ◽  
Kdv Type Equations ◽  
Special Case

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

Asymptotic stability of heteroclinic cycles in systems with symmetry. II

2004 ◽  
Vol 134 (6) ◽  
pp. 1177-1197 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
Systematic Investigation ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Heteroclinic Cycles ◽  
Higher Dimensional ◽  
Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

Homotopy invariants and continuous mappings

1950 ◽  
Vol 202 (1069) ◽  
pp. 253-263 ◽  
Keyword(s):  
Homotopy Type ◽  
Betti Numbers ◽  
Continuous Mappings ◽  
Numerical Invariants ◽  
Homotopy Invariants

This paper contributes new numerical invariants to the topology of a certain class of polyhedra. These invariants, together with the Betti numbers and coefficients of torsion, characterize the homotopy type of one of these polyhedra. They are also applied to the classification of continuous mappings of an ( n + 2)-dimensional polyhedron into an ( n + 1)-sphere ( n > 2).

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

scholarly journals The Lattice of Definable Equivalence Relations in Homogeneous $n$-Dimensional Permutation Structures

10.37236/5980 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Samuel Braunfeld
Keyword(s):  
Distributive Lattice ◽  
Equivalence Relations ◽  
Electronic Journal ◽  
Finite Distributive Lattice ◽  
Linear Orders ◽  
Finite Dimensional ◽  
Homogeneous Structures ◽  
Special Case

In Homogeneous permutations, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous $n$-dimensional permutation structures (homogeneous structures with $n$ linear orders) for all finite $n$. We prove here that the lattice of $\emptyset$-definable equivalence relations in such a structure can be any finite distributive lattice, providing many new imprimitive examples of homogeneous finite dimensional permutation structures. We conjecture that the distributivity of the lattice of $\emptyset$-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of $\emptyset$-definable equivalence relations is homogeneous. Finally, we conjecture a classification of the primitive examples, and confirm this in the special case where all minimal forbidden structures have order 2. 

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