New insights into algebraic geometry of a conjecture, leading to origami curves.
problem Algebraic and geometric perspectives on the Putman-Wieland conjecture.
method Algebraic and geometric constructions of origami curves.
result Origami curves with high-dimensional isotrivial isogeny factors.
The Frey--Mazur conjecture states that an elliptic curve over Q is determined up to isogeny by its p-torsion Galois representation for p≥17. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multip…
We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …
Quantum codes linked to abelian varieties, providing mathematical rigor.
problem Quantum error correction through complex abelian varieties.
method Mathematical formulation of Gottesman-Kitaev-Preskill codes using abelian varieties.
result Asymptotic isometry of encoding, precise gate realizations, and failure probability optimization.
Derives criteria for Kähler structures on holomorphic submersions.
problem Criteria for Kähler structures on holomorphic submersions.
method Derives a criterion for Kähler structures using holomorphic submersions.
result Proves Kähler structures for certain holomorphic submersions.
Survey on minimal rational curves and their geometric structures.
problem Germ-equivalence problem of minimal rational curves on uniruled projective manifolds.
method Analysis of isotrivial families of projective varieties and G-structures.
result Natural G-structure on Zariski-open subset of uniruled projective manifolds.
Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient of the group of complex unipotent 3×3 matrices by a cocompact lattice. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic torus. We also prove that any Kähler surface in an Iwasawa mani…
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…
Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fibre bundles. They are known to have positive signature divisible by 4. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent…
We construct examples of non-isotrivial algebraic families of smooth complex projective curves over a curve of genus 2. This solves a problem from Kirby's list of problems in low-dimensional topology. Namely, we show that 2 is the smallest possible base genus that can occur in a 4-manifold of non-zero signature which i…
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefo…
Proves properties of complex algebraic varieties and local systems.
problem Properties of complex algebraic varieties and local systems.
method Analytic Zariski open subsets and algebraic maps.
result Trivializing covering spaces of complex algebraic varieties.
The paper provides estimates for higher-order Ricci curvature along Kähler-Ricci flows.
problem Estimating higher-order curvature along Kähler-Ricci flows on compact Kähler manifolds.
method Proving uniform bounds for Ricci curvature and scalar curvature in various orders and norms.
result A geometric obstruction causes a specific third-order derivative of Ricci curvature to blow up at rate et/2. The paper studies cyclic covers of rational surfaces and their Hodge structures.
problem Understanding the Hodge structures of cyclic covers of rational surfaces.
method Generalization of Esnault-Viehweg method to analyze monodromy actions.
result The monodromy action splits into direct sums for specific cyclic covers.
Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.
problem Understanding foliations with numerically flat tangent bundles on Kähler manifolds.
method Analyzing the structure of foliations on compact Kähler manifolds, extending earlier results.
result Smooth foliations with numerically flat tangent bundles induce a decomposition of the ambient manifold's tangent bundle.
The article describes canonical metrics on holomorphic fibre bundles.
problem Existence of canonical metrics on isotrivial Kähler fibrations.
method Induced from Hermite--Einstein connections on holomorphic principal bundles.
result Existence of optimal symplectic connections when principal bundles are polystable.
Geodesics in Kähler metrics connect metrics with constant scalar curvature.
problem Deriving geodesics for relatively Kähler metrics on fibrations.
method Deriving geodesic equation, proving uniqueness, convexity of log-norm functional.
result Fibrations with optimal symplectic connections are polystable.
The thesis explores stability conditions and metrics in differential geometry.
problem Understanding extremal objects in differential geometry.
method Introduces and analyzes Z-critical metrics and optimal symplectic connections. result Proves a correspondence between existence of metrics and stability conditions.
The Helon model identifies Standard Model quarks and leptons with certain framed braids joined together at both ends by a connecting node (disk). These surfaces with boundary are called braided 3-belts (or simply belts). Twisting and braiding of ribbons composing braided 3-belts are interchangeable, and it was shown in…
This work defines a categorical notion of principal bundles.
problem Different definitions of principal bundles in various categories.
method Formulated in join-restriction categories, which generalize partial maps.
result Shows the tangent bundle as the product of tangent space and group object.
The paper constructs Ricci-flat Kähler manifolds with specific decay properties.
problem Finding Ricci-flat Kähler manifolds with controlled decay rates.
method Geometric existence proof and construction of ansatz.
result Existence of 39 distinct Ricci-flat Kähler 3-folds with specific asymptotic angles.
The article studies conic connections on complex manifolds and their geometric properties.
problem Characterizing and understanding conic connections on complex manifolds.
method Develops a new approach to the cubic torsion and studies the geometric conditions for its vanishing.
result Provides a geometric condition characterizing the vanishing of cubic torsion.