Correspondence Theorem Research Articles

A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings

We give a one-to-one correspondence between ideals in the Steinberg algebra of a Hausdorff ample groupoid G, and certain families of ideals in the group algebras of isotropy groups in G. This generalises a known ideal correspondence theorem for Steinberg algebras of strongly effective groupoids. We use this to give a complete graph-theoretic description of the ideal lattice of Leavitt path algebras over arbitrary commutative rings, generalising the classification of ideals in Leavitt path algebras over fields.

Counting tropical rational space curves with cross-ratio constraints

This is a follow-up paper of Goldner (Math Z 297(1–2):133–174, 2021), where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in Tyomkin (Adv Math 305:1356–1383, 2017) allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. The multiplicities of such cross-ratio floor diagrams can be calculated by enumerating certain rational tropical curves in the plane.

Refined floor diagrams from higher genera and lambda classes

We show that, after the change of variables q=e^{iu}, refined floor diagrams for {mathbb {P}}^2 and Hirzebruch surfaces compute generating series of higher genus relative Gromov–Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov–Witten theory and an explicit result in relative Gromov–Witten theory of {mathbb {P}}^1. Combining this result with the similar looking refined tropical correspondence theorem for log Gromov–Witten invariants, we obtain a non-trivial relation between relative and log Gromov–Witten invariants for {mathbb {P}}^2 and Hirzebruch surfaces. We also prove that the Block–Göttsche invariants of {mathbb {F}}_0 and {mathbb {F}}_2 are related by the Abramovich–Bertram formula.

A remark on the freeness condition of Suzuki's correspondence theorem for intermediate C$^*$-algebras

Let $\Gamma$ be a discrete group satisfying the approximation property (AP). Let $X$, $Y$ be $\Gamma$-spaces and $\pi \colon Y \to X$ be a proper factor map which is injective on the non-free part. We prove the one-to-one correspondence between intermediate C$^*$-algebras of $C_0(X) \rtimes_r \Gamma \subset C_0(Y) \rtimes_r \Gamma$ and intermediate $\Gamma$-${\rm C}^\ast$-algebras of $C_0(X) \subset C_0(Y)$. This is a generalization of Suzuki's theorem that proves the statement for free actions.

Bulk-Deformed Potentials for Toric Fano Surfaces, Wall-Crossing, and Period

Abstract We provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces via wall-crossing techniques and a tropical–holomorphic correspondence theorem for holomorphic discs. As an application of the correspondence theorem, we also prove a big quantum period theorem for toric Fano surfaces, which relates the log descendant Gromov–Witten invariants with the oscillatory integrals of the bulk-deformed potentials.

A comparison of positivity in complex and tropical toric geometry

Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford–Taylor theory of plurisubharmonic functions on the tropicalization.

Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are showno coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttsche, Cooper–Pandharipande and Block–Gathmann–Markwig.

Correspondence theorem between holomorphic discs and tropical discs on K3 surfaces

In this paper, we prove that the open Gromov–Witten invariants defined in [20] on K3 surfaces satisfy the Kontsevich–Soibelman wall-crossing formula. One hand, this gives a geometric interpretation of the slab functions in Gross–Siebert program. On the other hands, the open Gromov–Witten invariants coincide with the weighted counting of tropical discs. This is an analog of the corresponding theorem on toric varieties [26][27] but on compact Calabi–Yau surfaces.

The near-boundary geometry of Einstein-vacuum asymptotically anti-de Sitter spacetimes

We study the geometry of a general class of vacuum asymptotically anti-de Sitter spacetimes near the conformal boundary. In particular, the spacetime is only assumed to have finite regularity, and it is allowed to have arbitrary boundary topology and geometry. For the main results, we derive limits at the conformal boundary of various geometric quantities, and we use these limits to construct partial Fefferman–Graham expansions from the boundary. The results of this article will be applied, in upcoming papers, toward proving symmetry extension and gravity–boundary correspondence theorems for vacuum asymptotically anti-de Sitter spacetimes.

Generalizing Tropical Kontsevich's Formula to Multiple Cross-Ratios

Kontsevich's formula is a recursion that calculates the number of rational degree $d$ curves in $\mathbb{P}_{\mathbb{C}}^2$ passing through $3d-1$ points in general position. Kontsevich proved it by considering curves that satisfy extra conditions besides the given point conditions. These crucial extra conditions are two line conditions and a condition called cross-ratio.
 This paper addresses the question whether there is a general Kontsevich's formula which holds for more than one cross-ratio. Using tropical geometry, we obtain such a recursive formula. For that, we use a correspondence theorem of Tyomkin that relates the algebro-geometric numbers in question to tropical ones. It turns out that the general tropical Kontsevich's formula we obtain is capable of not only computing the algebro-geometric numbers we are looking for, but also of computing further tropical numbers for which there is no correspondence theorem yet.
 We show that our recursive general Kontsevich's formula implies the original Kontsevich's formula and that the initial values are the numbers Kontsevich's fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities.

Contact Dual Pairs

Abstract We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condition. Contact groupoids and contact reduction are the main sources of examples. Among other properties, we prove the characteristic leaf correspondence theorem for contact dual pairs that parallels the analogous result of Weinstein for symplectic dual pairs.

Some classes of abstract simplicial complexes motivated by module theory

In this paper we analyze some classes of abstract simplicial complexes relying on algebraic models arising from module theory. To this regard, we consider a left-module on a unitary ring and find models of abstract complexes and related set operators having specific regularity properties, which are strictly interrelated to the algebraic properties of both the module and the ring.Next, taking inspiration from the aforementioned models, we carry out our analysis from modules to arbitrary sets. In such a more general perspective, we start with an abstract simplicial complex and an associated set operator. Endowing such a set operator with the corresponding properties obtained in our module instances, we investigate in detail and prove several properties of three subclasses of abstract complexes.More specifically, we provide uniformity conditions in relation to the cardinality of the maximal members of such classes. By means of the notion of OSS-bijection, we prove a correspondence theorem between a subclass of closure operators and one of the aforementioned families of abstract complexes, which is similar to the classic correspondence theorem between closure operators and Moore systems. Next, we show an extension property of a binary relation induced by set systems when they belong to one of the above families.Finally, we provide a representation result in terms of pairings between sets for one of the three classes of abstract simplicial complexes studied in this work.

Tropical Geometry: new directions

The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative geometry (using positive integer q -numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress in more traditional areas of tropical research, such as tropical moduli spaces, tropical homology and tropical correspondence theorems.

Counting tropical rational curves with cross-ratio constraints

We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is (Tyomkin in Adv Math 305:1356–1383, 2017), where a tropical-algebraic correspondence theorem was proved that relates counts of rational curves in toric varieties that satisfy point conditions and cross-ratio constraints to the analogous tropical counts. We proceed in two steps: based on tropical intersection theory we first study tropical cross-ratios and introduce degenerated cross-ratios. Second we provide a lattice path algorithm that produces all rational tropical curves satisfying such degenerated conditions explicitly. In a special case simpler combinatorial objects, so-called cross-ratio floor diagrams, are introduced which can be used to determine these enumerative numbers as well.

On the tropical discs counting on elliptic K3 surfaces with general singular fibres

Using Lagrangian Floer theory, we study the tropical geometry of K3 surfaces with more general singular fibres other than simple nodal curves. In particular, we give the local models for the types I n I_n , I I II , I I I III and I V IV singular fibres in the Kodaira classification. This generalizes the correspondence theorem in “Correspondence theorem between holomorphic discs and tropical discs on K3 surfaces”, arXiv:1703.00411 between open Gromov-Witten invariants/tropical discs counting to these cases.

Subgroups of 3-Factor Direct Products

Abstract Extending Goursat’s Lemma we investigate the structure of subdirect products of 3-factor direct products. We construct several examples and then provide a structure theorem showing that every such group is essentially obtained by a combination of the examples. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat’s Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphisms between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.

Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely describe all intermediate operator algebras: the Galois Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the third, new complete description theorem which gives a canonical bijective correspondence between intermediate operator algebras and intermediate extensions of dynamical systems. One can also regard this theorem as a crossed product splitting theorem, analogous to the Tensor Splitting Theorem. We then give concrete applications, particularly to maximal amenability problem and a new realization result of intermediate operator algebra lattice.

Weak convergence of probability measures to Choquet capacity functionals

In the definition of weak convergence of probability measures it is assumed that the limit is a probability measure as well. We get rid of this assumption and require that the limit merely needs to be a Choquet-capacity functional. In terms of random variables this means that the distributional limit no longer is a random point, but a random closed set, namely one uniquely determined by the Choquet capacity. For our extended notion of weak convergence there is a counterpart of the portmanteau theorem. Moreover, we demonstrate basic relations to the theory of random closed sets with emphasis on weak convergence in hyperspace topologies including two correspondence theorems. Finally, the approach carries over to sequences of Choquet capacities.

The Galois correspondence theorem for groupoid actions

A theory about groupoids will be developed, emphasizing the notion of normal subgroupoid and quotient groupoid. It will be also constructed a version of the Galois correspondence theorem for groupoids acting on commutative rings.

An algebraic ordered extension of vector space

In this paper we have discussed an algebraic ordered extension of vector space. This new structure comprises a semigroup structure, a scalar multiplication and a compatible partial order. It is an algebraic axiomatisation of topological hyperspace; also it can be thought of as a generalisation of vector space in the sense that, it always contains a vector space and conversely, every vector space can be embedded maximally into such a structure. Initially the idea of this structure was given by S. Ganguly et al. with the name “quasi-vector space” in “An Associated Structure Of A Topological Vector Space; Bull. Cal. Math. Soc; Vol-96, No.6 (2004), 489-498”. The axioms of this structure evolve a very rapid growth of its elements with respect to the partial order and also evoke some sort of positiveness in each element. Meanwhile, a vector space is evolved within this structure and positivity of each element of the new structure is judged with respect to the elements of the vector space generated. Considering the exponential behaviour of its elements, we have studied this structure in the present paper with a new nomenclature —“exponential vector space” in short ‘evs’. We have developed a quotient structure on an evs by defining ‘congruence’ on it and have shown that the quotient structure also forms an evs with respect to suitably defined operations and partial order. We have obtained an isomorphism theorem and a correspondence theorem in the context of exponential vector space. Further, we have topologised the quotient evs by defining compatibility of the associated congruence with the topology of the base evs. A necessary and sufficient condition has been deduced so that the order-isomorphism stated under the isomorphism theorem becomes topological. Also, we have constructed order-morphisms on a quotient evs corresponding to that on the base evs.