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. 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. 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.
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 …
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…
For any elliptic K3 surface F:K→P1, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to P1 equipped with the McLean metric. There are well-known e…
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.
The study constructs G2-manifolds from K3 surfaces with a specific action.
problem Creating G2-manifolds from K3 surfaces with a Z22-action. method Assuming a K3 surface with a Z22-action, extending this action to SimesT3, resolving singularities, and computing Betti numbers. result Several new values of (b2,b3) for G2-manifolds are found. New solutions found for complex structures on specific manifolds.
problem Constructing smooth solutions to the Hull-Strominger system.
method Using fibrations over K3 orbisurfaces.
result Proved existence of solutions for certain manifolds.
Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.
problem Understanding the boundary of K-moduli of prime Fano threefolds of genus twelve.
method Developed a modular relation between Fano threefolds and their anticanonical K3 surfaces, proving forgetful morphism is an open immersion.
result Proved the boundary of K-moduli of V22 is purely divisorial and consists of four irreducible components. 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.
Study contact instantons on Sasakian 5-manifolds with Calabi-Yau structures.
problem Anti-self-dual contact instantons on Sasakian 5-manifolds with transverse Calabi-Yau structures.
method Singularity data of leaf spaces and computation of moduli spaces.
result Explicit computation of complex dimensions of moduli spaces for 95 orbifold K3 surfaces.
Study shows Lefschetz fibrations on Milnor fibers of certain singularities.
problem Understanding Lefschetz fibrations on Milnor fibers of specific singularities.
method Analyzes Milnor fibers of cusp and simple elliptic singularities to construct Lefschetz fibrations.
result Milnor fibers of cusp and simple elliptic singularities admit genus-one Lefschetz fibrations.
Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kähler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsev…
We construct compact G2-orbifolds with ADE-singularities that carry exactly one parallel spinor. Our examples are related to certain quotients of C2×T3 that have been investigated in arXiv:hep-th/9812205. We shortly discuss the physical applications of our examples.
We improve Gross-Wilson's local estimates to global ones. As an application, we study the blow-up limits of the degenerating Calabi-Yau metrics on singular fibers.
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 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 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.
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.
We will show the following three theorems on the diffeomorphism and homeomorphism groups of a K3 surface. The first theorem is that the natural map π0(Diff(K3))→Aut(H2(K3;Z)) has a section over its image. The second is that, there exists a subgroup G of π0(Diff(K3)) of order two over which…
Model for associative submanifolds in K3 fibrations.
problem Understanding singularity formation in associative submanifolds.
method Graphs in a 3-manifold with locally gradient flow lines.
result Produces analogues of known singularity formation phenomena.
A class of examples of Riemannian metrics with holonomy G_2 on compact 7-manifolds was constructed by the author in arXiv:math.DG/0012189 and later in a joint work with N.-H. Lee in arXiv:0810.0957, using a certain `generalized connected sum' of two asymptotically cylindrical manifolds with holonomy SU(3). We consider,…
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.
Researchers resolve a SYZ conjecture for A_n singularities using quantum-corrected T-duality.
problem Resolving a mathematically precise SYZ conjecture for A_n singularities.
method Building a quantum-corrected T-duality between two singular torus fibrations.
result Constructing a parameter-dependent SYZ mirror fibration partner with matching singular loci and integral affine structure.
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.
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. 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.
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.
We prove that a Lefschetz fibration over the disc that, after compactification, has the same singular fibers as an extremal rational elliptic surface can be obtained by deleting a singular fiber and a section from the rational extremal elliptic surface, i.e. such a Lefschetz fibration is determined up to topological eq…
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.
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.
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.
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.
On an affine flat manifold with coordinates x^j and convex local potential function f, we call the affine Kahler metric f_{ij} dx^i dx^j semi-flat Calabi-Yau if it satisfies det f_{ij} = 1. Recently Gross-Wilson have constructed many such metrics on S^2 minus 24 singularities, as degenerate limits of Calabi-Yau metrics…
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 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.
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.
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…
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…
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.
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 \(\mathbb{P}^3\).
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.
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.
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.
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.