Geometriae Dedicata最新文献

An R-link is an n-component link L in (S^3) such that Dehn surgery on L yields (#^n(S^1 times S^2)). Every R-link L gives rise to a geometrically simply-connected homotopy 4-sphere (X_L), which in turn can be used to produce a balanced presentation of the trivial group. Adapting work of Gompf, Scharlemann, and Thompson, Meier and Zupan produced a family of R-links L(p, q; c/d), where the pairs (p, q) and (c, d) are relatively prime and c is even. Within this family, (L(3,2;2n/(2n+1))) induces the infamous trivial group presentation (langle x,y , | , xyx=yxy, x^{n+1}=y^n rangle ), a popular collection of potential counterexamples to the Andrews–Curtis conjecture for (n ge 3). In this paper, we use 4-manifold trisections to show that the group presentations corresponding to a different subfamily, L(3, 2; 4/d), are Andrews–Curtis trivial for all d.

We exhibit an infinite family of discrete subgroups of ({{,mathrm{textbf{Sp}},}}_4(mathbb {R})) which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters (left( tfrac{N-3}{2N},tfrac{N-1}{2N}, tfrac{N+1}{2N}, tfrac{N+3}{2N}right) ) at infinity and maximal unipotent monodromy at zero, for any integer (Nge 4). Additionally, we relate the cones used for ping-pong in (mathbb {R}^4) with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian. These domains of discontinuity lead to uniformizations of variations of Hodge structure with Hodge numbers (1, 1, 1, 1).

We construct complete Calabi–Yau metrics on non-compact manifolds that are smoothings of an initial complete intersection (V_0) that is a Calabi–Yau cone, extending the work of Székelyhidi (Duke Math J 168(14):2651–2700, 2019). The constructed Calabi–Yau manifold has tangent cone at infinity given by ({mathbb {C}}times V_0). This construction produces Calabi–Yau metrics with fibers having varying complex structures and possibly isolated singularities.

Branched coverings boast a rich history, ranging from the ramification of Riemann surfaces to the realization of 3-manifolds as coverings branched over knots and spanning both geometric topology and algebraic geometry. This work delves into branched coverings “à la Fox” of (G, X)-manifolds, encompassing three main avenues: Firstly, we introduce a comprehensive class of singular (G, X)-manifolds, elucidating elementary theory paired with illustrative examples to showcase its efficacy and universality. Secondly, building on Montesinos’ work, we revisit and augment the prevailing knowledge, formulating a Galois theory tailored for such branched coverings. This includes a detailed portrayal of the fiber above branching points. Lastly, we identify local attributes that guarantee the existence of developing maps for singular (G, X)-manifolds within the branched coverings framework. Notably, we pinpoint conditions that ensure the existence of developing maps for these singular manifolds. This research proves especially pertinent for non-metric singular (G, X)-manifolds like those of Lorentzian or projective nature, as discussed by Barbot, Bonsante, Suhyoung Choi, Danciger, Seppi, Schlenker, and the author, among others. While examples are sprinkled throughout, a standout application presented is a uniformization theorem “à la Mess” for singular locally Minkowski manifolds exhibiting BTZ-like singularities.

In this note we establish the weak unique continuation theorem for caloric functions on compact RCD(K, 2) spaces and show that there exists an RCD(K, 4) space on which there exist non-trivial eigenfunctions of the Laplacian and non-stationary solutions of the heat equation which vanish up to infinite order at one point . We also establish frequency estimates for eigenfunctions and caloric functions on the metric horn. In particular, this gives a strong unique continuation type result on the metric horn for harmonic functions with a high rate of decay at the horn tip, where it is known that the standard strong unique continuation property fails.

In this work we analysed the validity of a type of Borsuk-Ulam theorem for multimaps between surfaces. We developed an algebraic technique involving braid groups to study this problem for n-valued maps. As a first application we described when the Borsuk-Ulam theorem holds for split and non-split multimaps (phi :X multimap Y) in the following two cases: (i) X is the 2-sphere equipped with the antipodal involution and Y is either a closed surface or the Euclidean plane; (ii) X is a closed surface different from the 2-sphere equipped with a free involution (tau ) and Y is the Euclidean plane. The results are exhaustive and in the case (ii) are described in terms of an algebraic condition involving the first integral homology group of the orbit space (X / tau ).

Abstract

The absolute regularity of a Fano variety, denoted by (hat{textrm{reg}}(X)) , is the largest dimension of the dual complex of a log Calabi–Yau structure on X. The absolute coregularity is defined to be $$begin{aligned} hat{textrm{coreg}}(X):= dim X - hat{textrm{reg}}(X)-1. end{aligned}$$ The coregularity is the complementary dimension of the regularity. We expect that the coregularity of a Fano variety governs, to a large extent, the geometry of X. In this note, we review the history of Fano varieties, give some examples, survey some theorems, introduce the coregularity, and propose several problems regarding this invariant of Fano varieties.

Putman introduced a notion of a partitioned surface which is a surface with boundary with decoration restricting how the surface can be embedded into larger surfaces, and defined the Torelli group of the partitioned surfaces. In this paper, we study some topological and dynamical aspects of the Torelli groups of partitioned surfaces. More precisely, first we obtain upper and lower bounds on the cohomological dimension of Torelli groups of partitioned surfaces and show that those two bounds coincide when at most three boundary components are grouped together in the partition of the boundary. Second, we study the asymptotic translation lengths of Torelli groups of partitioned surfaces on the corresponding curve complexes. We show that the minimal asymptotic translation length asymptotically behaves almost like the reciprocal of the Euler characteristic of the surface. This generalizes the previous result of the first and second authors on Torelli groups for closed surfaces.

Inspired by results of Eskin and Mirzakhani (J Mod Dyn 5(1):71–105, 2011) counting closed geodesics of length (le L) in the moduli space of a fixed closed surface, we consider a similar question in the (Out (F_r)) setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping class group) of pseudo-Anosovs whose dilatations have natural logarithm (le L). Let ({mathfrak {N}}_r(L)) denote the number of (Out (F_r))-conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is (le L). We prove for (rge 3) that as (Lrightarrow infty ), the number ({mathfrak {N}}_r(L)) has double exponential (in L) lower and upper bounds. These bounds reveal behavior not present in the surface setting or in classical hyperbolic dynamical systems.

Let D be a reduced effective strict normal crossing divisor on a smooth complex variety X, and let (mathfrak {X}_D) be the associated root stack over (mathbb C). Suppose that X admits an anti-holomorphic involution (real structure) that keeps D invariant. We show that the root stack (mathfrak {X}_D) naturally admits a real structure compatible with X. We also establish an equivalence of categories between the category of real logarithmic connections on this root stack and the category of real parabolic connections on X.