Geometric Consequences Research Articles

Symmetries of Fano varieties

Abstract Prokhorov and Shramov proved that the BAB conjecture, which Birkar later proved, implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of finite semi-simple groups (i.e., those with no nontrivial normal abelian subgroups) acting faithfully on 𝑛-dimensional complex Fano varieties, and this bound only depends on 𝑛. We investigate the geometric consequences of an action by a certain semi-simple group: the symmetric group. We give an effective upper bound for the maximal symmetric group action on an 𝑛-dimensional Fano variety. For certain classes of varieties – toric varieties and Fano weighted complete intersections – we obtain optimal upper bounds. Finally, we draw a connection between large symmetric actions and boundedness of varieties, by showing that the maximally symmetric Fano fourfolds form a bounded family. Along the way, we also show analogues of some of our results for Calabi–Yau varieties and log terminal singularities.

Spherical objects in dimensions two and three

This paper classifies spherical objects in various geometric settings in dimensions two and three, including both minimal and partial crepant resolutions of Kleinian singularities, as well as arbitrary flopping 3 -fold contractions with only Gorenstein terminal singularities. The main result is much more general: in each such setting, we prove that all objects in the associated null category \mathcal{C} with no negative Ext groups are the image, under the action of an appropriate braid or pure braid group, of some object in the heart of a bounded t-structure. The corollary is that all objects x which admit no negative Exts, and for which \mathrm{Hom}_{\mathcal{C}}(x,x)=\mathbb{C} , are the images of the simples. A variation on this argument goes further and classifies all bounded t-structures on \mathcal{C} . There are multiple geometric, topological and algebraic consequences, primarily to autoequivalences and stability conditions. Our main new technique also extends into representation theory, and we establish that in the derived category of a finite-dimensional algebra which is silting discrete, every object with no negative Ext groups lies in the heart of a bounded t-structure. As a consequence, every semibrick complex can be completed to a simple-minded collection.

Gopakumar-Vafa invariants and the Emergent String Conjecture

The Emergent String Conjecture of Lee, Lerche, and Weigand holds that every infinite-distance limit in the moduli space of a quantum gravity represents either a decompactification limit or an emergent string limit in some duality frame. Within the context of 5d supergravities coming from M-theory compactifications on Calabi-Yau threefolds, we find evidence for this conjecture by studying (a) the gauge couplings and (b) the BPS spectrum, which is encoded in the Gopakumar-Vafa invariants of the threefold. In the process, we disuss a testable geometric consequence of the Emergent String Conjecture, and we verify that it is satisfied in all complete intersection Calabi-Yau threefolds in products of projective spaces (CICYs).

Effects of polydispersity on the plastic behaviors of dense two-dimensional granular systems under shear.

We study particle-scale motion in sheared highly polydisperse amorphous materials, in which the largest particles are as much as ten times the size of the smallest. We find strikingly different behavior from the more commonly studied amorphous systems with low polydispersity. In particular, an analysis of the nonaffine motion of particles reveals qualitative differences between large and small particles: The smaller particles have dramatically more nonaffine motion, which is induced by the presence of the large particles. We characterize how the nonaffine motion changes from the low- to high-polydispersity regimes. We further demonstrate a quantitative way to distinguish between "large" and "small" particles in systems with broad distributions of particle sizes. A macroscopic consequence of the nonaffine motion is a decrease in the energy dissipation rate for highly polydisperse samples, which is due both to a geometric consequence of the changing jamming conditions for higher polydispersity and to the changing character of nonaffine motion.

On the rigidity of cosmological space-times

In this paper, we analyse a family of geometrically well-behaved cosmological space-times (V^{n+1},g), which are foliated by intrinsically isotropic space-like hypersurfaces {M_t}_{tin mathbb {R}}, which are orthogonal to a family of co-moving observers defined by a global time-like vector field U. In particular, this implies such space-times satisfy several of the well-known criteria for isotropic cosmological space-times, although, in the family in question, the simultaneity spaces (M_t,g_t) associated with U can have as sectional curvature a sign-changing function k(t). Being this clearly impossible in the FLRW family of standard cosmological space-times, it motivates us to revisit the geometric rigidity consequences of different definitions of isotropy available in the literature. In this analysis, we divide such definition according to whether the isometries involved are taken to be (local) space-time (STI space-times) or (local) space isometries (SI space-times) of (M_t,g_t) for each t. This subtlety will be shown to be critical, proving that only when space-time isometries are considered one obtains the well-known rigidity properties associated with isotropic cosmological space-times. In particular, SI space-times will be shown to be a strictly larger class than the STI ones, allowing a family of basic cosmological curvature change models which are not even locally isometric to any FLRW space-time.

Ricci and scalar curvature inequalities for semi-slant Riemannian submersions in complex space forms

This paper is devoted to the obtain Chen inequalities containing the Ricci and scalar inequalities on the vertical and the horizontal distributions for semi-slant Riemannian submersions from complex space forms onto Riemannian manifolds. The equality case of the obtained inequalities is discussed. At the end, some geometric consequences are obtained. Moreover, many examples are constructed.

Automorphisms of algebraic varieties and infinite transitivity

This is a survey of recent results on multiple transitivity for automorphism groups of affine algebraic varieties. The property of infinite transitivity of the special automorphism group is treated, which is equivalent to the flexibility of the corresponding affine variety. These properties have important algebraic and geometric consequences. At the same time they are fulfilled for wide classes of varieties. Also, the situations are studied where infinite transitivity occurs for automorphism groups generated by finitely many one-parameter subgroups. In the appendices to the paper, the results on infinitely transitive actions in complex analysis and in combinatorial group theory are discussed.

Geometric and dosimetric consequences of intra-fractional movement in single isocenter non-coplanar stereotactic radiosurgery

PurposeTo investigate the geometric and dosimetric impacts of intra-fractional movement for patients with single or multiple brain metastasis treated using Varian Hyperarc™ mono-isocentric radiosurgery.MethodsA total of 50 single or hypo-fractionated Hyperarc™ treatment courses (118 lesions) were included in the analysis. Intra-fractional translational and rotational movements were quantified according to the post-treatment cone-beam CT (CBCT). Geometric displacements of all targets were calculated individually based on the assessed head movement in each treatment fraction and their relationships with treatment time and target-to-isocenter distances were studied. For dosimetric analysis, only single-fraction treatments (56 lesions) were included. Re-planning was performed with 0, 1, and 2 mm planning target volume (PTV) margins. Doses were then re-calculated on rotated CT images with isocenter shifted which emulate the change in patient treatment position. Target coverage, target and normal brain doses before and after intra-fractional movement were compared.ResultsThe mean 3D target displacements was 0.6 ± 0.3 (SD) mm. Target shifts for patients treated within 10 min were significantly smaller than those treated in longer sessions. No correlation was found between target shift and target-to-isocenter distance as the origin of head rotation was not located at the isocenter. Loss of target coverage and minimum Gross Tumor Volume (GTV) dose due to intra-fractional movement were apparent only when no margin was used, leading to an extra 23% of the targets violating the dose acceptance criteria, in contrast, the effects on normal brain V12Gy were negligible regardless of the margin used. The use of 1 mm PTV margin can compensate clinically significant geographical miss caused by intra-fractional movements while limiting V12Gy to within dose criteria for 88% of the cases. The plan acceptance rate (fulfillment of both target and normal brain dose criteria) after intra-fractional movement was also the highest with the 1 mm margin.ConclusionAlthough intra-fractional movements during Hyperarc™ treatments were small, there were substantial dosimetric effects due to the sharp dose fall-off near target boundaries. These effects could be mitigated by using a 1 mm PTV margin and maintaining the effective treatment time to within 10 min.

Thrust Faults Bound an Elevated Mantle Plug Beneath Several Lunar Basins

AbstractThe lunar maria are large expanses of basalt that infill antecedent impact basins and show evidence for postemplacement deformation. Landforms within many of these basins suggest a period of compressive tectonics, although their formation mechanism has yet to be established. Previous work for Mare Crisium demonstrated that basin‐circumferential wrinkle ridges, which typically demarcate the inner edge of an annulus of elevated terrain, are the result of deep‐seated thrust faults that preferentially form along the boundary of an elevated superisostatic portion of mantle and a thick, subisostatic collar of crustal material. Here, we show that a similar fault architecture exists for several other mascon‐bearing basins, including Maria Serenitatis, Nectaris, Moscoviense, and, to a lesser degree, Humorum and Imbrium. These deeply penetrating basin‐circumferential thrust faults, as for Mare Crisium, form a (partial) outward‐dipping ring‐fault system that bounds the elevated mantle plug beneath each basin as a geometric consequence of mascon evolution. If this geometric arrangement is unique to the Moon, then some characteristic(s) of lunar mascon evolution enables the formation of such mascon‐bounding faults. Despite the ubiquitous nature of mascon‐bound thrust ring faults at several lunar basins, whether such structures exist at mascon basins on other terrestrial worlds remains an open question.

Families of faces and the normal cycle of a convex semi-algebraic set

We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a patch—a term recently coined by Ciripoi, Kaihnsa, Löhne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences.

Families of convex tilings

We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile ``orientations'' determine the tiling. We associate to a tiling an underlying bipartite planar graph $G$ and its corresponding Kasteleyn matrix $K$. If $G$ has quadrilateral faces, we show that $K$ is the differential of the map from edge intercepts to tile areas, and extract some geometric and probabilistic consequences.

On the geometry of Cayley automatic groups

In contrast to being automatic, being Cayley automatic a priori has no geometric consequences. Specifically, Cayley graphs of automatic groups enjoy a fellow traveler property. Here, we study a distance function introduced by the first author and Trakuldit which aims to measure how far a Cayley automatic group is from being automatic, in terms of how badly the Cayley graph fails the fellow traveler property. The first author and Trakuldit showed that if it fails by at most a constant amount, then the group is in fact automatic. In this paper, we show that for a large class of non-automatic Cayley automatic groups this function is bounded below by a linear function in a precise sense defined herein. In fact, for all Cayley automatic groups which have super-quadratic Dehn function, or which are not finitely presented, we can construct a non-decreasing function which (1) depends only on the group and (2) bounds from below the distance function for any Cayley automatic structure on the group.

Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated.

Shortcut graphs and groups

We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory, including: the 1-skeletons of systolic and quadric complexes (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), 1-skeletons of finite dimensional CAT(0) cube complexes, hyperbolic graphs, standard Cayley graphs of finitely generated Coxeter groups and the standard Cayley graph of the Baumslag-Solitar group BS(1,2). Most of these examples satisfy a strong form of the shortcut property. The shortcut properties also have important geometric group theoretic consequences. We show that shortcut groups are finitely presented and have exponential isoperimetric and isodiametric functions. We show that groups satisfying the strong form of the shortcut property have polynomial isoperimetric and isodiametric functions.

The Tangle-Free Hypothesis on Random Hyperbolic Surfaces

Abstract This article introduces the notion of $L$-tangle-free compact hyperbolic surfaces, inspired by the identically named property for regular graphs. Random surfaces of genus $g$, picked with the Weil–Petersson probability measure, are $(a \log g)$-tangle-free for any $a<1$. This is almost optimal, for any surface is $(4 \log g + O (1))$-tangled. We establish various geometric consequences of the tangle-free hypothesis at a scale $L$, among which the fact that closed geodesics of length $< \frac L 4$ are simple, disjoint, and embedded in disjoint hyperbolic cylinders of width $\geq \frac{L}{4}$.

On Immersions of Surfaces into SL(2, ℂ) and Geometric Consequences

Abstract We study immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1, including $SL(2,\mathbb{C})$ and the space of geodesics of $\mathbb{H}^3$, and we prove a Gauss–Codazzi theorem in this setting. This approach has some interesting geometric consequences: (1) it provides a model for the transitioning of hypersurfaces among $\mathbb{H}^n$, ${\mathbb{A}}\textrm{d}{\mathbb{S}}^n$, $\textrm{d}{\mathbb{S}}^n$, and ${\mathbb{S}}^n$; (2) it provides an effective tool to construct holomorphic maps to the $\textrm{SO}(n,\mathbb{C})$-character variety, bringing to a simpler proof of the holomorphicity of the complex landslide; and (3) it leads to a correspondence, under certain hypotheses, between complex metrics on a surface (i.e., complex bilinear forms of its complexified tangent bundle) and pairs of projective structures with the same holonomy. Through Bers theorem, we prove a uniformization theorem for complex metrics.

The six books of Diophantus’ Arithmetic increased and reduced to specious: the lost manuscript of Jacques Ozanam (1640–1718)

The introduction of a new analytical method, due fundamentally to Francois Viete and Rene Descartes and the later dissemination of their works, resulted in a profound change in the way of thinking and doing mathematics. This change, known as process of algebrization, occurred during the seventeenth and early eighteenth centuries and led to a great transformation in mathematics. Among many other consequences, this process gave rise to the treatment of the results in the classic treatises with the new analytical method, which allowed new visions of such treatises and the obtaining of new results. Among those treatises is the Arithmetic of Diophantus of Alexandria (approx. 200–284) which was written, using the new algebraic language, by the French mathematician Jacques Ozanam (1640–1718), who in addition to profusely increasing the original problems of Diophantus, solved them in a general way, thus obtaining many geometric consequences. The work is handwritten, it has never been published, it has been lost for almost 300 years, and the known references show its importance. We will show that Ozanam’s manuscript was quoted as an important work on several occasions by others mathematicians of the time, among whom G. W. Leibniz stands out. Once the manuscript has been located, our aim in this article is to show and analyze this work of Ozanam, its content, its notation and its structure and how, through the new algebraic method, he not only solved and expanded the questions proposed by Diophantus, but also introduced a connection between the algebraic solutions and what he called geometric determinations by obtaining loci from the solutions.

Similar curvature-to-width ratios for channels and channel belts: Implications for paleo-hydraulics of fluvial ridges on Mars

AbstractThe surface of Mars contains abundant sinuous ridges that appear similar to river channels in planform, but they stand as topographic highs. Ridges have similar curvature-to-width ratios as terrestrial meandering rivers, which has been used to support the hypothesis that ridges are inverted channels that directly reflect channel geometry. Anomalously wide ridges, in turn, have been interpreted as evidence for larger rivers on Mars compared to Earth. However, an alternate hypothesis is that ridges are exhumed channel-belt deposits— a larger zone of relatively coarse-grained deposits formed from channel lateral migration and aggradation. Here, we measured landform wavelength, radius of curvature, and width to compare terrestrial channels, terrestrial channel belts, and martian ridges. We found that all three landforms follow similar scaling relations, in which ratios of radius of curvature to width range from 1.7 to 7.3, and wavelength-to-width ratios range from 5.8 to 13. We interpret this similarity to be a geometric consequence of a sinuous curved line of finite width. Combined with observations of ridge-stacking patterns, our results suggest that wide ridges on Mars could indicate fluvial channel belts that formed over significant time rather than anomalously large rivers.

Singularity categories via the derived quotient

Given a noncommutative partial resolution A=EndR(R⊕M) of a Gorenstein singularity R, we show that the relative singularity category ΔR(A) of Kalck–Yang is controlled by a certain connective dga A/LAeA, the derived quotient of Braun–Chuang–Lazarev. We think of A/LAeA as a kind of ‘derived exceptional locus’ of the partial resolution A, as we show that it can be thought of as the universal dga fitting into a suitable recollement. This theoretical result has geometric consequences. When R is an isolated hypersurface singularity, it follows that the singularity category Dsg(R) is determined completely by A/LAeA, even when A has infinite global dimension. Thus our derived contraction algebra classifies threefold flops, even those X→Spec(R) where X has only terminal singularities. This gives a solution to the strongest form of the derived Donovan–Wemyss conjecture, which we further show is the best possible classification result in this singular setting.

Hasse diagrams with large chromatic number

For every positive integer n, we construct a Hasse diagram with n vertices and independence number O ( n 3 / 4 ) . Such graphs have chromatic number Ω ( n 1 / 4 ) , which significantly improves the previously best-known constructions of Hasse diagrams having chromatic number Θ ( log n ) . In addition, if we also require girth of at least k ⩾ 5 , we construct such Hasse diagrams with independence number at most n 1 − 1 2 k − 4 + o ( 1 ) . The proofs are based on the existence of point-line arrangements in the plane with many incidences and avoids certain forbidden subconfigurations, which we find of independent interest. These results also have the following surprising geometric consequence. They imply the existence of a family C of n curves in the plane such that the disjointness graph G of C is triangle-free (or has high girth), but the chromatic number of G is polynomial in n. Again, the previously best-known construction, due to Pach, Tardos and Tóth, had only logarithmic chromatic number.