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. 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 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.
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 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. 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.
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.
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.
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.
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.
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.
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.
Generalized Calabi-Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are b…
Constructs currents and heights on K3 surfaces.
problem Understanding the geometry and arithmetic of K3 surfaces.
method Constructs canonical positive currents and heights on K3 surfaces, equivariant for automorphism group.
result Continuous family of currents and heights defined over an enlarged boundary of the ample cone.
The aim of this paper is to show the rigidity of homologically trivial actions of prime order on K3 surfaces. To be precise, we show that homotopy K3 surfaces do not admit a periodic diffeomorphism of odd prime order 3 acting trivially on cohomology. Moreover, we give an obstruction in terms of the rationality and sign…
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…
Study on energy of maps from K3 surface to flat orbifold.
problem Energy of maps from K3 surface to flat orbifold.
method Investigate Dirichlet energy of smooth maps and introduce an invariant.
result Ratio of energy to invariant converges to 1 for Foscolo's collapsing families.
In this paper, we give a weak classification of locally linear pseudofree actions of the cyclic group of order 3 on a K3 surface, and prove the existence of such an action which can not be realized as a smooth action on the standard smooth K3 surface.
We compute the genus zero family Gromov-Witten invariants for K3 surfaces using the topological recursion formula and the symplectic sum formula for a degeneration of elliptic K3 surfaces. In particular we verify the Yau-Zaslow formula for non-primitive classes of index two.
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 K3 surfaces with a pair of commuting involutions that are non-symplectic with respect to two anti-commuting complex structures that are determined by a hyper-Kähler metric. One motivation for this paper is the role of such Z22-actions for the construction of G2-manifolds. We find a large class …
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.
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.
We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.
Stability conditions on K3 surfaces are linked to the masses of spherical objects.
problem Determining stability conditions on K3 surfaces.
method Using the masses of spherical objects and lax stability conditions associated to spherical bundles.
result Stability conditions on K3 surfaces are determined by the masses of spherical objects up to a natural C-action. 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 shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.
problem Understanding the moduli spaces of quartic K3 surfaces and their birational models.
method Interpolates between GIT and Baily-Borel moduli spaces, describes wall crossings, and classifies degenerations.
result Verifies Laza-O'Grady's prediction and classifies Gorenstein canonical Fano degenerations of \(\mathbb{P}^3\).
Study shows complex K3 surfaces have infinite free abelian subgroup in their diffeomorphism group.
problem Understanding the structure of diffeomorphism groups of complex K3 surfaces.
method Used families of Seiberg-Witten invariants and moduli spaces of Einstein metrics.
result Proved the existence of a free abelian subgroup of countably infinite rank in the identity component of the diffeomorphism group.
We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the decomposability on an arbitrary K3 surface with respect to the Kähler classes in dense subse…
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.
The paper proves geometrical finiteness for automorphism groups of K3 surfaces and related varieties.
problem Establishing geometrical finiteness for automorphism groups of K3 surfaces and related varieties.
method Using cone conjecture, the paper establishes geometrical finiteness for the natural isometric actions of automorphism groups on hyperbolic spaces.
result Automorphism groups of K3 surfaces and related varieties are non-positively curved and relatively hyperbolic.
New geometric examples of involutions on Hilbert square K3 surfaces.
problem Existence of non-symplectic involutions on Hilbert square K3 surfaces.
method Geometric constructions of involutions on IHS manifolds.
result New examples of involutions proven to exist.
We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously known cases. To achieve this, we introduce two new techniques in the deformation the…
Constructs stable bundles on K3 surfaces using monad construction.
problem Stability of bundles on K3 surfaces.
method Monad construction, Generalised Hoppe Criterion, computer aid.
result Examples of real stable bundles constructed on K3 surfaces.
Simplified proof of K3 surface period map surjectivity.
problem Surjectivity of period map on K3 surfaces.
method Utilizes hyperkähler geometry and collapsing techniques.
result Simple proof of Todorov's result on K3 surfaces.
Study on harmonic forms on K3 surfaces converging to a flat 4D orbifold.
problem Behavior of harmonic 2-forms on K3 surfaces with Ricci-flat metrics.
method Analysis of convergence of harmonic forms to flat 4D orbifold.
result Decomposition of harmonic 2-forms into converging subspaces.
4-manifolds can be broken down into pairs-of-pants and K3 surfaces.
problem Decomposing 4-manifolds diffeomorphic to complex hypersurfaces.
method Pair-of-pants and K3 surfaces decomposition.
result 4-manifolds diffeomorphic to complex hypersurfaces can be decomposed into a specific number of pair-of-pants and K3 surfaces.
Rational configurations in K3 surfaces and simply-connected pg=1 surfaces for K2=1,2,3,4,5,6,7,8,9math.AG The study finds families of surfaces with specific properties and connects them to rational curve configurations in K3 surfaces.
problem Existence and properties of surfaces with specific canonical and geometric genus conditions.
method Study of rational curve configurations and use of Q-Gorenstein smoothings. result Existence of (20−2K2)-dimensional families of simply-connected surfaces with pg=1 and K2=1,2,3,4,5,6,7,8,9. The paper constructs moduli spaces for genus one fibered K3 surfaces.
problem Understanding the moduli spaces and period mappings of genus one fibered K3 surfaces.
method Constructing various moduli spaces and period mappings related to locally symmetric spaces.
result Computed fundamental groups of moduli spaces and applied results to mapping class groups.
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 shows K-moduli spaces of curves on quadrics and K3 surfaces match with VGIT quotients.
problem Understanding K-moduli spaces of curves on quadrics and K3 surfaces.
method Using log Fano pairs and VGIT quotients, the study compares K-moduli spaces of curves on P1imesP1 and quartic hyperelliptic K3 surfaces. result K-moduli spaces of curves on quadrics and K3 surfaces form a natural interpolation.
This note is a summary of our work [OO] which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not nec…
This paper treats the theory of Mukai duality on K3 surfaces from the differential geometric perspective, taylored to the need of the author's companion paper about Mukai duality of adiabatic coassociative K3 fibrations.
We prove that any weakly triholomorphic map from a compact hyperkähler surface to an algebraic K3 surface defined by a homogeneous polynomial of degree 4 in CP3 has only isolated singularities.
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…
Let M be a K3 surface or an even-dimensional compact torus. We show that the category of coherent sheaves on M is independent from the choice of the complex structure, if this complex structure is generic.
New PL invariant classifies K3 surface degenerations.
problem Classifying type II degenerations of K3 surfaces.
method Explicit PL convex function from interval, differential geometric viewpoint.
result Function classifies degenerations into combinatorial types.