No complex analytic tori in complex deformations of Kummer varieties.
problem Existence of complex analytic tori in Kummer varieties.
method Proving non-existence through complex deformation analysis.
result Generic complex deformations of Kummer varieties contain no complex analytic tori.
In this paper, we study asymptotic behavior of projective embeddings of Kummer varieties given by theta functions, and their amoebas. We prove that a Lagrangian fibration of the Kummer variety can be approximated by moment maps of the projective spaces.
Let X be a hyperkaehler manifold. Trianalytic subvarieties of X are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a 2-dimensional complex torus T, the Hilbert scheme T[n] classifying zero-dimensional subschemes of T admits a hype…
Recent studies show Kummer surfaces in P5 have fewer osculating spaces than expected.
problem Understanding the osculating spaces of Kummer surfaces in P5. method Analyzing geometric properties and recent results about varieties with hypo-osculating behavior.
result Osculating spaces of Kummer surfaces have dimension less than 5.
Character variety of a 6-punctured sphere mapped to CP^3 with specific branch points.
problem Characterizing the traceless SU(2) character variety of a 6-punctured 2-sphere.
method Analysis of 2-fold branched cover and application of Narasimhan-Ramanan theorem.
result Character variety corresponds to a 2-fold branched cover of CP^3 over a singular Kummer surface.
Study calculates cohomology of G2-manifolds using Kummer construction.
problem Calculating the cohomology of G2-manifolds.
method Intersection theory applied to generalized Kummer construction.
result Generators and product structure of rational cohomology ring identified.
Alternative proof shows Kummer rigidity for K3 surfaces.
problem Proving automorphisms of K3 surfaces are Kummer examples.
method Exploits Ricci-flat metrics on K3 surfaces.
result Shows measure of maximal entropy automorphisms are Kummer.
By carrying out a rational transformation on the base curve CP1 of the Seiberg-Witten curve for N=2 supersymmetric pure SU(2)-gauge theory, we obtain a family of Jacobian elliptic K3 surfaces of Picard rank 17. The isogeny relating the Seiberg-Witten curve for pure SU(2)-ga…
Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.
problem Characterizing holomorphic automorphisms with high entropy on hyperkähler manifolds.
method Using Jensen's inequality and properties of stable and unstable distributions, the authors show uniform contraction and expansion, leading to the conclusion that the manifold is birational to a torus quotient.
result Holomorphic automorphisms with high entropy on hyperkähler manifolds are Kummer examples.
Existence of Ricci flat metric on Kummer K3 surface proven.
problem Proving existence of Ricci flat metric on Kummer K3 surface.
method General strategy of Donaldson's gluing construction, compact elliptic theory on usual Hölder and Sobolev spaces, explicit isometry to Gibbons-Hawking ansatz.
result Existence of a Ricci flat metric on the Kummer K3 surface.
Article constructs coassociative submanifolds in Joyce's G2-manifolds.
problem Constructing coassociative submanifolds in Joyce's G2-manifolds. method Using Joyce's generalised Kummer construction, focusing on the critical region where the metric degenerates.
result The volume of the coassociatives shrinks to zero as the orbifold-limit is approached.
Extends Masur's divergence theorem to complex tori and Kummer surfaces.
problem Establishing uniquely ergodic horizontal foliations for geodesic flows on moduli spaces.
method Defined and calculated horizontal foliations and geodesic flows on moduli spaces of Kähler metrics.
result Proved that horizontal foliations are uniquely ergodic if geodesic flows are recurrent.
In this work we describe a method to reconstruct the braid monodromy of the preimage of a curve by a Kummer cover. This method is interesting, since it combines two techniques, namely, the reconstruction of a highly non-generic braid monodromy with a systematic method to go from a non-generic to a generic braid monodro…
In this paper we study a functional equation associated to the Kummer's equation (K) of the trilogarithm. Then we apply our results to web geometry and to characterize the functions solution of (K).
The paper explores anti-hyperbolicity for hyperkähler varieties.
problem Anti-hyperbolicity of hyperkähler varieties.
method Exploring various examples and criteria for meromorphic and holomorphic dominability by C^m.
result Generalizing known results about K3 surfaces to hyperkähler manifolds.
Extends Kummer's theory to singular surfaces for line congruences.
problem Applying Kummer's theory to singular surfaces for line congruences.
method Analyzing the equation of principal surfaces and developable surfaces for normal congruences.
result The multiplicative factor for the principal surfaces is associated with the singular set of ξ. New 5-manifold found with zero Ricci curvature.
problem Finding almost Ricci-flat manifolds with specific properties.
method Kummer-type constructions using Riemannian metrics.
result Constructed a simply connected, nonspin 5-manifold with zero Ricci curvature.
Smooth family of G2-instantons over a Kummer construction.
problem Constructing a smooth family of G2-instantons over a Kummer construction.
method Extending gluing construction for G2-instantons to Kummer constructions, using a Z2-action to overcome obstructions.
result Proves the existence of a smooth 1-parameter family of G2-instantons, showing their infinitesimal rigidity and non-flatness.
New hyperKähler orbifolds of Kummer type discovered.
problem Identifying new compact hyperKähler orbifolds.
method Construction of orbifolds and analysis of singularities.
result Only orbifolds with known holomorphic symplectic resolutions lead to known hyperKähler 8-manifolds.
In this article we introduce a method to construct G2-instantons on G2-manifolds arising from Joyce's generalised Kummer construction. The method is based on gluing ASD instantons over ALE spaces to flat bundles on G2-orbifolds of the form T7/Γ. We use this construction to produce non-trivia…
The paper explores the geometry of the Spence-Kummer trilogarithm equation and its Galois analogue.
problem Investigating the geometry and functional equation of the Spence-Kummer trilogarithm.
method Using algebraic relations between polylogarithm generating series and path systems, along with tensor and homotopy criteria for functional equations.
result Derives a precise form of the Spence-Kummer equation and its Galois analogue.
Study constructs associative submanifolds in G2-manifolds from orbifolds.
problem Constructing associative submanifolds in G2-manifolds from orbifolds. method Using Joyce's generalised Kummer construction.
result The volume of associative submanifolds tends to zero as they approach orbifolds.
The paper explores the multi-asset Black-Scholes model, focusing on the correlation matrix and its impact on option pricing.
problem The multi-asset Black-Scholes model's validity is compromised by the determinant of the correlation matrix becoming zero or negative.
method The authors use the Wei-Norman theorem to compute the propagator over the variable rank surface ΣK for the general N asset case. result The option pricing model is not well-defined for regions outside the Kummer surface ΣK where the determinant of the correlation matrix becomes negative. Generalizes Kummer construction for Kaehler metrics.
problem Finding conditions for Kaehler metrics on orbifolds.
method Analyzes singular points and holomorphic vector fields for Kaehler desingularization.
result Sufficient conditions for existence of global Kaehler metrics.
Researchers found new functions for spherical clothoids using special functions.
problem Developing new mathematical functions for spherical clothoids.
method Used confluent hypergeometric functions and Meixner-Pollaczek polynomials.
result Presented Cartesian coordinate functions and stereographic projections.
Formulates Hilbert reciprocity law on 3-manifolds.
problem Developing arithmetic topology on 3-manifolds.
method Formulated an analogue of Hilbert reciprocity law using intersection forms and Kummer extensions.
result Cyclic covers of links are analogues of Kummer extensions.
The D-groupoid of symmetries is minimal under specific conditions.
problem Conditions for the minimality of the D-groupoid of symmetries of a projective structure. method Analyzing the D-groupoid and its sub-groupoids, and relating it to the non-integrability of certain equations. result The minimality of the D-groupoid is equivalent to the non-integrability of specific equations. We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space V by an action of a finite group G of linear transforms. Under some additional assumptions, we prove that the partial desingularization of Hilbert type is smooth only if t…
The 4-dimensional abstract Kummer variety K^4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of K^4 we resolve its 16 isolated singularities - step by step - by simplicial blowu…
We construct F-structures on a Bott manifold and on some other manifolds obtained by Kummer-type constructions. We also prove that if M=E#X, where E is a fiber bundle with structure group G and a fiber admitting a G-invariant metric of non-negative sectional curvature and X admits an F-structure with one trivial coveri…
This is an expository paper which aims to give a simple proof of the existence of Ricci-flat metrics on certain K3 surfaces, as an illustration of general "glueing" techniques.
Formula derived for G2-manifolds, showing moduli spaces are incomplete.
problem Incompleteness of moduli spaces for G2-manifolds. method Derived a formula for the energy of paths in moduli spaces, provided conditions for finite energy and length.
result Compact G2-manifolds produced by the generalised Kummer construction have incomplete moduli spaces. Two new proofs provide Eguchi-Hanson metrics as ALE bubbles for Kummer constructions of K3 metrics.
problem Constructing Ricci-flat Kähler metrics on the K3 surface with special holonomy.
method Singular perturbation and weighted function space analysis.
result Large families of compact hyper-Kähler orbifolds as volume non-collapsed limits of Kummer constructions.
Investigates webs related to cluster algebras and polylogarithms.
problem Understanding webs associated with cluster algebras and polylogarithms.
method Introducing AMP webs and analyzing their properties, proving results and conjectures.
result Many webs associated with polylogarithms and cluster algebras are AMP webs.
We give a simple and uniform construction of essentially all known deformation classes of gravitational instantons with ALF, ALG or ALH asymptotics and nonzero injectivity radius. We also construct new ALH Ricci flat metrics asymptotic to the product of a real line with a flat 3-manifold.
We classify compact Kähler manifolds M of dimension n≥3 on which acts a lattice of an almost simple real Lie group of rank ≥n−1. This provides a new line in the so-called Zimmer program, and characterizes certain type of complex tori by a property of their automorphisms groups.
Computes a ν-invariant for Joyce's G2-manifolds.
problem Computing an invariant for Joyce's compact G2-manifolds. method Using spectral description of Crowley, Goette and Nordström, computes invariant for Joyce's manifolds.
result Computed invariant for many Joyce's G2-manifolds. Consider a family of K3 surfaces over a hyperbolic curve (i.e. Riemann surface). Their second cohomology groups form a local system, and we show that its top Lyapunov exponent is a rational number. One proof uses the Kuga-Satake construction, which reduces the question to Hodge structures of weight 1. A second proof us…
Study connects K3 surfaces to holomorphic metrics, solving complex structure variation.
problem Understanding complex structure variation on K3 surfaces.
method Using Picard-Fuchs equations and lattice polarizations.
result Explicit example of locally conformally flat holomorphic metric.
We study the moduli space M(G,A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, M(G,A) is the (coarse) moduli space of s-equivalence classes of holomorphic semi-stable G_C-bundles with trivial Chern classes where G_C is the complexified grou…
Study geodesics on K3 surfaces near orbifold limit.
problem Understanding geodesics on K3 surfaces near the orbifold limit.
method Improves metric estimates for K3 surfaces, uses hyperkähler identities.
result Restrictions and existence conditions for stable geodesics.
We study complex surfaces with locally CAT(0) polyhedral Kahler metrics and construct such metrics on CP^2 with various orbifold structures. In particular, in relation to questions of Gromov and Davis-Moussong we construct such metrics on a compact quotient of the two-dimensional unite complex ball. In the course of th…
Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this notion and other related ones to systems of higher-order differential equations …
The paper constructs flat metrics on orbifolds and resolutions.
problem Finding flat metrics on orbifolds and their resolutions.
method Gluing construction and analysis of singularities.
result All crepant resolutions of non-Kähler Calabi-Yau orbifolds with Chern-Ricci flat balanced metrics admit such metrics.
For a congruence of straight lines defined by a hypersurface in Rn+1,n≥1, and a field of reflected directions created by a point source we define the notion of intensity in a tangent direction and introduce elementary symmetric functions Sm,m=1,2,...,n, of {\it principal intensities}. The problem of exi…
Smooths local volatility model calibration with piecewise linear variance.
problem Calibrating local volatility models to option prices.
method Extends LiptonSepp2011 approach with piecewise linear variance and non-zero interest rates/dividends.
result Analytical tractability with Kummer's hypergeometric functions.
It is demonstrated that the stationary Veselov-Novikov (VN) and the stationary modified Veselov-Novikov (mVN) equations describe one and the same class of surfaces in projective differential geometry: the so-called isothermally asymptotic surfaces, examples of which include arbitrary quadrics and cubics, quartics of Ku…
The paper shows non-aspherical path components in G2-moduli spaces.
problem Characterizing non-aspherical path components in G2-moduli spaces.
method Using generalised Kummer construction and resolving singularities with Eguchi-Hanson spaces, the authors establish a fibration over each path component with Eilenberg Mac Lane spaces as fibres.
result The comparison map is a fibration over each path component with Eilenberg Mac Lane spaces as fibres, indicating non-trivial families of G2-manifolds.