A transfer principle: from periods to isoperiodic foliations

Genus g Torelli space is the moduli space of genus g curves of compact type equipped with a homology framing. The hyperelliptic locus is a closed analytic subvariety consisting of finitely many mutually isomorphic components. We use properties of the hyperelliptic Torelli group to show that when \(g\ge 3\) these components do not have the homotopy type of a finite CW complex. Specifically, we show that the second rational homology of each component is infinite-dimensional. We give a more detailed description of the topological features of these components when \(g=3\) using properties of genus 3 theta functions.

Let f : X → C be a family of semi-stable curves of genus g over a smooth projective C of genus q, and S ⊂ C the degeneration locus of f. The so-called Arakelov inequality states that deg f∗ωX/C ≤ g2 deg Ω 1 C(log S) = g 2 (2q − 2 + #S). When g ≥ 2 and #S = 0, the Miyaoka-Yau inequality for surfaces implies a much stronger inequality deg f∗ωX/C ≤ g − 1 6 deg Ω 1 C . In general, Tan [28] proved that the Arakelov inequality for a family f : X → C of semi-stable curves of genus ≥ 2 holds strictly. If g = 1, then deg f∗ωX/C can reach the upper bound in the inequality. Beauville has classified such families over C = P with #S = 4. More precisely, there are exactly 6 non-isotrivial families of semi-stable elliptic curves over P with 4 singular fibres. All of them are modular families of elliptic curves [2]. In this paper, we will consider the similar question for families of higher dimensional varieties. The Arakelov inequality is a special case of some more general inequalities for Hodge bundles. To state them, let V denote a polarized real variation of Hodge structure on a smooth projective curve C S such that the local monodromies around S are all unipotent, let (⊕p+q=kE, θ) denote the corresponding Hodge bundles. In [8] the following Arakelov-Yau type inequality was proven (also see [18] for a similar inequality): If k = 2l + 1, then

The notion of fuzzy sets initiated to overcome the uncertainty of an object. Fuzzy topological space, in- tuitionistic fuzzy sets in topological structure space, vagueness in topological structure space, rough sets in topological space, theory of hesitancy and neutrosophic topological space, etc. are the extension of fuzzy sets. Soft set is a family of parameters which is also a set. Fuzzy soft topological space, intuitionistic fuzzy soft and neutrosophic soft topological space are obtained by incorporating soft sets with various topological structures. This motivates to write a review and study on various soft set concepts. This paper shows the detailed review of soft topological spaces in various sets like fuzzy, Intuitionistic fuzzy set and neutrosophy. Eventually, we compared some of the existing tools in the literature for easy understanding and exhibited their advantages and limitations.

S. Kondō defined a birational period map between the moduli space of genus three curves and a moduli space of polarized K3 surfaces. In this paper we give a resolution of the period map, providing a surjective morphism from a suitable compactification of 3 to the Baily-Borel compactification of a six dimensional ball quotient.

This paper is devoted to a proof of the rationality of the moduli space of those genus four smooth complex projective curves which are double covers of some elliptic curve. The study of the canonical model of a genus four curve as above allows to reduce the initial moduli problem to a simple one in plane projective geometry; this last formulation leads to compute an explicit representation of a certain group on a vector space and its corresponding field of invariants.

We prove that the existence of a divisor of degree 3 and Baker-Norine rank at least 1 on a 3 -edge connected tropical curve is equivalent to the existence of a non-degenerate harmonic morphism of degree 3 from a tropical modification of it to a tropical rational curve. Using the second description, we define the moduli spaces of 3 -edge connected tropical trigonal covers and of 3 -edge connected tropical trigonal curves, the latter as a locus in the moduli space of tropical curves. Finally, we prove that the moduli space of 3 -edge connected genus g tropical trigonal curves has the same dimension as the moduli space of genus g algebraic trigonal curves.

The theory of topological vector spaces (TVS), being a foundation of modern functional analysis, is now considered as a completely mature, or, to be more specific, dead mathematical discipline. This pessimistic view is based on a picture, where, on the one hand, a well-known system of facts is stated, facts that have been considered classical since the times of Mackey and Grothendieck, and, on the other hand, an opinion exists explicitly or implicitly, that any deviation from this system inevitably tends to a disappointment. “Do not seek any other patterns or connections but those that we have described here, because any of your assumptions will find a counterexample in Nature”; this phrase should have been cited as an afterward to the three textbooks on topological vector spaces translated into Russian [12, 41, 43], which represent the face of this science from the sixties until now. Indeed, after the famous Enflo counterexample [22], the development of the theory of topological vector spaces (which was planned originally as an extension of a clear and simple discipline, linear algebra) was transformed into a sequence of reports on the oddities one can encounter in modern mathematics. The result of this process is a general skepticism about and a loss of interest in this field. The number of papers on the theory of TVS is declining, while the attitude of other mathematicians now consists of polite perplexity or lenient irony. And add to that the fact that the vast majority of specialists in functional analysis do not concern themselves with topological vector spaces but consciously concentrate on Banach theory. (According to Compumath, for example, in 1996 only 7 papers were devoted to locally convex and topological vector spaces, while 612 were devoted to Banach spaces. In the theory of topological algebras, the ratio is 6:90.) The dominant position of Banach theory in functional analysis justifies a natural question: what are the advantages of the Banach case over the more general topological situation? At the conceptual level, the difference becomes most apparent in the theory of topological algebras. Let us cast a critical look. The first impression we will get when comparing Banach and non-Banach theories of topological algebras is as follows. While the concept of Banach algebra emerges naturally from intuitive expectations, faciliated by an acquaintance with general algebra and the theory of Banach spaces (and leads to the profound theory of Banach algebras), the situation with general topological algebras looks completely different. We find here unexpectedly that the very attempts to define a topological algebra (and topological module) lead to results that contradict intuition. Indeed, intuitively it is clear that a “good” definition of topological algebra (and topological module) must fulfill at least the following minimal list of requirements:

We extend earlier work of Waldhausen which defines operations on the algebraic $K$-theory of the one-point space. For a connected simplicial abelian group $X$ and symmetric groups $\Sigma_n$, we define operations $\theta^n \colon A(X) \rightarrow A(X{\times}B\Sigma_n)$ in the algebraic $K$-theory of spaces. We show that our operations can be given the structure of $E_{\infty}$-maps. Let $\phi_n \colon A(X{\times}B\Sigma_n) \rightarrow A(X{\times}E\Sigma_n) \simeq A(X)$ be the $\Sigma_n$-transfer. We also develop an inductive procedure to compute the compositions $\phi_n \circ \theta^n$, and outline some applications.

This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.

We define the affine stratification number asn X of a scheme X. For X equidimensional, it is the minimal number k such that there is a stratification of X by locally closed affine subschemes of codimension at most k. We show that the affine stratification number is well-behaved, and bounds many aspects of the topological complexity of the scheme, such as vanishing of cohomology groups of quasicoherent, constructible, and l-adic sheaves. We explain how to bound asn X in practice. We give a series of conjectures (the first by E. Looijenga) bounding the affine stratification number of various moduli spaces of pointed curves. For example, the philosophy of [GV, Theorem *] yields: the moduli space of genus g, n-pointed complex curves of compact type (resp. with rational tails) should have the homotopy type of a finite complex of dimension at most 5g-6+2n (resp. 4g-5+2n). This investigation is based on work and questions of Looijenga. One relevant example turns out to be a proper integral variety with no embeddings in a smooth algebraic space. This one-paragraph construction appears to be simpler and more elementary than the earlier examples, due to Horrocks and Nori.

We construct an infinite collection of universal—independent of $(g,n)$—polynomials in the Miller–Morita–Mumford classes $\kappa _m\in H^{2m}( \overline{\mathcal{M}}_{g,n},{\mathbb{Q}})$, defined over the moduli space of genus $g$ stable curves with $n$ labeled points. We conjecture vanishing of these polynomials in a range depending on $g$ and $n$.

Moduli spaces of algebraic curves are closely related to Hurwitz spaces, that is, spaces of meromorphic functions on curves. All of these spaces naturally arise in numerous problems of algebraic geometry and mathematical physics, especially in connection with string theory and Gromov-Witten invariants. In particular, the classical Hurwitz problem of enumerating the topologically distinct ramified coverings of the sphere with prescribed ramification type reduces to the study of the geometry and topology of these spaces. The cohomology rings of such spaces are complicated even in the simple case of rational curves and functions. However, the cohomology classes most important for applications (namely, the classes Poincaré dual to the strata of functions with given singularities) can be expressed in terms of relatively simple “basic” classes (which are, in a sense, tautological). The aim of the present paper is to identify these basic classes, to describe relations between them, and to find expressions for the strata in terms of them. Our approach is based on Thom's theory of universal polynomials of singularities, which has been extended to the case of multisingularities by the first author. Although the general Hurwitz problem still remains open, our approach enables one to achieve significant progress towards its solution and an understanding of the geometry and topology of Hurwitz spaces.

The concept of moduli space was introduced by Riemann in his study of the conformal (or equivalently, complex) structures on a Riemann surface. Let us consider the simplest non-trivial case, namely, that of a Riemann surface of genus 1 or the torus T 2. The set of all complex structures \(\mathcal{C}({T}^{2})\) on the torus is an infinite-dimensional space acted on by the infinite-dimensional group Diff(T 2). The quotient space $$\mathcal{M}({T}^{2}) := \mathcal{C}({T}^{2})/\mathrm{Diff}({T}^{2})$$ is the moduli space of complex structures on T 2. Since T 2 with a given complex structure defines an elliptic curve, \(\mathcal{M}({T}^{2})\) is, in fact, the moduli space of elliptic curves. It is well known that a point ω = ω1 + iω2 in the upper half-plane H (ω2 > 0) determines a complex structure and is called the modulus of the corresponding elliptic curve. The modular group SL(2, Z) acts on H by modular transformations and we can identify \(\mathcal{M}({T}^{2})\) with H ∕ SL(2, Z). This is the reason for calling \(\mathcal{M}({T}^{2})\) the space of moduli of elliptic curves or simply the moduli space. The topology and geometry of the moduli space has rich structure. The natural boundary ω2 = 0 of the upper half plane corresponds to singular structures. Several important aspects of this classical example are also found in the moduli spaces of other geometric structures. Typically, there is an infinite-dimensional group acting on an infinite-dimensional space of geometric structures with quotient a “nice space” (for example, a finite-dimensional manifold with singularities). For a general discussion of moduli spaces arising in various applications see, for example, [193].

Over the moduli space of pointed smooth algebraic curves, the projectivized [Formula: see text]th Hodge bundle is the space of [Formula: see text]-canonical divisors. The incidence loci are defined by requiring the [Formula: see text]-canonical divisors to have prescribed multiplicities at the marked points. We compute the classes of the closure of the incidence loci in the projectivized [Formula: see text]th Hodge bundle over the moduli space of curves with rational tails. The classes are expressed as a linear combination of tautological classes indexed by decorated stable graphs with coefficients enumerating appropriate weightings. As a consequence, we obtain an explicit expression for some relations in tautological rings of moduli of curves with rational tails.

Contemporary augmented spaces are situated in-between digital-analog, material-immaterial, online-offline, and real-virtual binaries. Relatedly, postdigital as a concept suggests that digital technologies are now integrated with almost all aspects of the individual and social atmosphere. Therefore, we should engage digitality through a critical approach by focusing on its intermingled situation. For this reason, reading augmented spaces with a postdigital perspective is essential to understanding and evaluating the potential of digital technologies in the context of the current line of vision. This paper aims to provide a better understanding of contemporary debates on digital technologies in the context of design. For this purpose, the article briefly reviews the definition of augmented space, and later postdigital as a concept is discussed with its fundamental characteristics. In what follows selected projects amongst the 2023 Media Architecture Biennale finalists are studied concerning the postdigital augmented spaces. As an outcome of this research, we suggest a conceptual framework that can be effective for the theory and practice of postdigital augmented spaces.