Compactifies metrics on K3 surfaces with algebraic description.
problem Classify Gromov-Hausdorff limits of K3 surfaces with fixed structures or polarizations.
method Algebraic description of Gromov-Hausdorff compactification.
result Classification of Gromov-Hausdorff limits of K3 surfaces.
Study shows isolated singularities in maps to K3 surfaces.
problem Characterizing singularities of triholomorphic maps.
method Analyzes weakly triholomorphic maps from hyperkähler to K3 surfaces.
result Maps have only isolated singularities.
We show that for complex analytic K3 surfaces any torsion class in H^2(X,O_X^*) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surfa…
Nontrivial boundary Dehn twist found on K3#K3 manifold.
problem Proving nontriviality of a Dehn twist on a specific 4-manifold.
method Algebraic criterion and equivariant topological K-theory to show non-isotopy.
result Boundary Dehn twist is nontrivial in the smooth mapping class group.
Researchers prove smoothings for surfaces with triple points.
problem Smoothings of surfaces with triple points.
method Differential geometric proof.
result Proves existence of smoothings for surfaces satisfying suitable conditions.
Study describes limits of non-collapsing K3 surfaces using algebraic data.
problem Understanding limits of non-collapsing polarized K3 surfaces.
method Explicit description via period mapping and algebro-geometric data.
result Bubbling limits depend solely on algebro-geometric data.
The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.
problem Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces.
method Analyzing flat surfaces and K3 surfaces, using asymptotics and stability conditions.
result Exact leading term in the asymptotics of the number of semistable Mukai vectors.
New K3 surfaces with two involutions and low Picard number constructed.
problem Finding K3 surfaces with specific properties and low Picard numbers.
method Construction of K3 surfaces over the rational numbers with low Picard numbers and two involutions.
result Explicit examples of K3 surfaces over the rational numbers with minimum Picard number 2 for various degrees.
Study Dehn-Seidel twists on Lagrangian spheres in K3 surfaces.
problem Understanding configurations of Lagrangian spheres in symplectic K3 surfaces.
method Use Seiberg-Witten theory and tools from symplectic mapping class groups.
result Proves algebraic independence of certain twists and generation results.
Constructs non-Kähler Calabi-Yau manifolds with large Betti numbers.
problem Finding non-Kähler Calabi-Yau manifolds with high Betti numbers.
method Smoothing normal crossing varieties to create K3 fibrations over smooth projective varieties.
result Examples of non-Kähler Calabi-Yau manifolds with arbitrarily large 2nd Betti numbers.
Unified boundary conditions lead to supergroup WZW models.
problem Boundary conditions in topologically twisted Chern-Simons matter theories.
method Unified complexified gauge invariant boundary conditions.
result Low-energy effective field theories on M2-M5 intersections are supergroup WZW models.
Undecidability proved for DG algebras problems.
problem Stable isomorphism, quasi-isomorphism, and Morita equivalence problems for semifree DG algebras.
method Essentially autonomous solutions by Gemini Deep Think and Aletheia.
result Proved undecidability of problems for semifree DG algebras.
We consider an equivariant analogue of a conjecture of Borcherds. Let Y be a real K3 surface without real points. Let g be a Ricci-flat Kaehler metric on Y invariant under the complex conjugation. We shall prove that the equivariant determinant of the Laplacian of (Y,g) with respect to the complex conjugation…
We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds …
The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on D…
Three theorems on K3 surfaces' diffeomorphism and homeomorphism groups.
problem Properties of diffeomorphism and homeomorphism groups of K3 surfaces.
method Seiberg-Witten theory and global Torelli theorem.
result The natural map π₀(Diff(K3)) → Aut(H²(K3; ℤ)) has a section over its image.
Study on moduli spaces of sextic curves with simple singularities and their compactifications.
problem Understanding moduli spaces of sextic curves with simple singularities.
method Using period maps of K3 surfaces with ADE singularities, algebraic open embeddings into arithmetic quotients of type IV domains, and GIT and Looijenga compactifications.
result Identifications of GIT and Looijenga compactifications for all cases.
A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…
Study K3 surfaces and their metrics, focusing on dynamics.
problem Understanding dynamics on K3 surfaces.
method Interactions between K3 surface geometry and Ricci-flat metrics, dynamical study of automorphisms.
result Positive entropy automorphisms on K3 surfaces.
Differential geometry applied to Mukai duality on K3 surfaces.
problem Understanding Mukai duality on K3 surfaces.
method Differential geometric approach.
result New insights into Mukai duality on K3 surfaces.
New solutions found for G2 system using K3 orbifolds.
problem Finding smooth solutions to the G2 Hull-Strominger system. method Torus fibrations over K3 orbifolds, adapted Serre construction for singular settings.
result Constructed new smooth solutions to the G2 Hull-Strominger system. New characteristic classes for manifold bundles with specific forms.
problem Characteristic classes for manifold bundles with indefinite intersection forms.
method Using a technique of Millson--Raghunathan, cohomology for non-uniform arithmetic lattices is produced.
result New characteristic classes defined on a finite cover of BDiff(M), nontrivial for specific manifolds. Study on K3 surfaces' collapsing and special Kähler structures.
problem Understanding the structure of K3 surfaces' collapsing metrics.
method Analyzing M2 and establishing connections to SKSs and Jacobian elliptic K3 surfaces. result Established a bijection between integral singular SKSs on P1 and Jacobian elliptic K3 surfaces. Study on automorphisms of K3 and Enriques surfaces, proving entropy gaps and achirality.
problem Entropy norms and achirality of automorphisms on K3 and Enriques surfaces.
method Proves gap theorems for entropy norms and studies achirality in terms of genus-one fibrations.
result Entropy gaps and achirality results for automorphisms of K3 and Enriques surfaces.
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.
Simply connected moduli space of Ricci flat metrics on K3 surfaces.
problem Understanding the topology of Ricci flat metrics on K3 surfaces.
method Analyzing the moduli space of metrics with unit volume.
result The moduli space is simply connected and has cohomology matching the automorphism group.
S.Bauer and M.Furuta defined a stable cohomotopy refinement of the Seiberg-Witten invariants. In this paper, we prove a vanishing theorem of Bauer-Furuta invariants for 4-manifolds with smooth Z/2-actions. As an application, we give a constraint on smooth Z/2-actions on homotopy K3#K3, and construct a nonsmoothable loc…
Compact K3 metrics derived from string theory vacua.
problem Finding Ricci-flat metrics on K3 surfaces.
method Using little string theory and BPS degeneracies.
result Systematic procedure for determining K3 metrics.
Study on K3 surfaces with collapsing metrics and nilpotent structures.
problem Exploring Ricci-flat metrics on K3 surfaces that collapse.
method Continuous map from K3 surface to an interval, with specific fiber types.
result Bubbles of Tian-Yau and Taub-NUT metrics appear as fibers.
Study of K3 surfaces with special involutions and singularities.
problem Characterizing K3 surfaces with specific symmetries.
method Analysis of K3 surfaces with commuting non-symplectic involutions.
result Identification of 320 examples of K3 surfaces with various singularities.
Researchers use gluing method to describe collapsing Calabi-Yau metrics on K3 fibred 3-folds.
problem Describing collapsing Calabi-Yau metrics on K3 fibred 3-folds.
method Gluing method
result Refined description of collapsing Calabi-Yau metrics.
Study geometric quantization on K3 surfaces, showing spectral convergence.
problem Quantization of K3 surfaces from spectral perspective.
method Special Lagrangian fibrations and hyper-Kähler structures.
result Spectral convergence of ∂ˉ-Laplacians on prequantum line bundles. K3 surfaces get a rational curve when a divisor is big and positive enough.
problem Finding rational curves on K3 surfaces with specific conditions.
method Degeneration technique to prove existence of integral nodal rational curves.
result Generic Λ-polarised K3 surface has an integral nodal rational curve in the linear system ∣L∣. Researchers found a special basis for cycles on a K3 surface.
problem Understanding the structure of two-cycles on K3 surfaces.
method Constructed a canonical basis of two-cycles using formal sums of smooth submanifolds.
result The intersection form of the basis takes a specific canonical form.
Special Lagrangian submanifolds emerge from K3 surface collapse.
problem Understanding special Lagrangian submanifolds in K3 surface collapse.
method Lifting affine lines to degenerating sequences of special Lagrangian submanifolds.
result Constructing special Lagrangian two-spheres connecting Taub-NUT bubbles.
Boundary Dehn twist on K3 surfaces becomes trivial after abelianization.
problem Understanding the boundary Dehn twist on K3 surfaces. method Obstruction from Baraglia-Konno and global Torelli theorem of K3 surfaces. result Boundary Dehn twist becomes trivial after abelianization.
Study decomposability of Lagrangian classes on K3 surfaces.
problem Decomposability of Lagrangian classes on K3 surfaces.
method Lattice theory and special Lagrangian submanifolds.
result Prove decomposability of Kähler classes on dense subsets of the Kähler cone.
Constructs entropy-minimizing pseudo-Anosov diffeomorphisms on K3 surfaces.
problem Finding minimal entropy diffeomorphisms on K3 surfaces.
method Constructs pseudo-Anosov diffeomorphisms minimizing entropy.
result Obtains infinitely many entropy-minimizing diffeomorphisms.
Researchers find geodesics on K3 surfaces using electrostatics.
problem Locating closed geodesics on K3 surfaces.
method Using Foscolo's construction of Ricci-flat Kahler metrics.
result Computed indices and lengths of geodesics with high precision.
Researchers found non-smoothable surfaces in a 4-sphere, solving K3 problems.
problem Non-smoothable surfaces in the 4-sphere.
method Constructed non-orientable surfaces with specific knot groups.
result Found surfaces that are non-smoothable and answered K3 problems.
Example of non-smooth isotopy using K3 surfaces.
problem Non-smooth isotopy examples in 4-manifolds.
method Dehn twist along a 3-sphere in K3 surfaces' connected sum.
result First example of self-diffeomorphisms isotopic to identity but not smoothly.
It is shown that there exists a twistor space on the n-fold connected sum of complex projective planes nCP2, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed ove…
Complete conjecture on rational curves on K3 surfaces.
problem Existence of infinitely many rational curves on K3 surfaces.
method Two new techniques: regeneration and marked point trick.
result Existence of integral curves of unbounded degree for any projective K3 surface.
Constructs Ricci-flat K3 metrics using D-geometry.
problem Computing the BPS index of a heterotic string theory.
method Hyper-Kähler quotient and D-geometry technology.
result Contains solution to BPS state counting problem.
New K3 metrics derived from torus orbifold loci.
problem Finding new K3 metrics near torus orbifold loci.
method Hyper-Kähler quotients and BPS spectra analysis.
result Infinitely many constraints on BPS spectra of SCFTs.
Study of K3 surfaces' limits using moduli theory.
problem Understanding Gromov-Hausdorff limits of K3 surfaces.
method Explicit moduli-theoretic framework for Ricci-flat Kahler metrics.
result Proof of Kontsevich-Soibelman conjecture for K3 surfaces.
Study Gromov-Hausdorff limits of K3 surface metrics via moduli compactification.
problem Understanding limits of K3 surface metrics.
method Moduli-theoretic framework for collapsing Ricci-flat Kahler metrics.
result Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters.
Study shows symplectic mapping groups of K3 surfaces are infinitely generated.
problem Understanding symplectic mapping class groups of K3 surfaces.
method Uses Kronheimer's approach and Seiberg-Witten invariants.
result Symplectic mapping class groups of many K3 surfaces are infinitely generated.