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. 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.
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.
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 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.
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.
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.
Let S be a K3 surface that admits a non-symplectic automorphism ρ of order 3. We divide S×P1 by ρ×ψ where ψ is an automorphism of order 3 of P1. There exists a threefold ramified cover of a partial crepant resolution of the quotient that is a Calabi-Yau orbifold. We compute the …
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.
We construct metrics with the holonomy group SU(2) on the tangent bundles of weighted complex projective lines and give a geometric description of the moduli space of special Kahler metrics on a K3-surface in the neighborhood of the flat orbifold T4/Z3.
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 show that K3 surfaces with non-symplectic automorphisms of prime order can be used to construct new compact irreducible G2-manifolds. This technique was carried out in detail by Kovalev and Lee for non-symplectic involutions. We use Chen-Ruan orbifold cohomology to determine the Hodge diamonds of certain complex thr…
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,…
The Berglund-Hübsch rule connects Calabi-Yau orbifolds to Sasakian manifolds.
problem Connecting Calabi-Yau orbifolds to Sasakian manifolds.
method Applying the Berglund-Hübsch transpose rule to associate Sasaki manifolds.
result Four seven-dimensional Sasakian manifolds of positive Ricci curvature are associated with a K3 orbifold.
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.
We construct new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for 13≤k≤22 and 14≤r≤22, the smooth manifolds S1×♯k(S2×S3)…
We describe a class of compact G2 orbifolds constructed from non-symplectic involutions of K3 surfaces. Within this class, we identify a model for which there are infinitely many associative submanifolds contributing to the effective superpotential of M-theory compactifications. Under a chain of dualities, these can…
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.
Construct G2-instantons on orbifold resolutions using specific geometric ingredients.
problem Constructing G2-instantons on orbifold resolutions. method Combining G2-instantons on orbifolds with Fueter sections over singular sets. result Construction of numerous G2-instantons on resolutions of T7/Γ. We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kahler parameters. We show that Kahler geometry can be exploited for significant gains in computational efficiency. As a proof of principle, we apply our methods to a one-paramete…
To pave the way for the journey from geometry to conformal field theory (CFT), these notes present the background for some basic CFT constructions from Calabi-Yau geometry. Topics include the complex and Kaehler geometry of Calabi-Yau manifolds and their classification in low dimensions. I furthermore discuss CFT const…
Study of D2 ALF manifolds via hyperkahler quotients of affine spaces.
problem Resolving flat orbifold quotients of R4. method Infinite-dimensional generalization of Kronheimer's construction, hyperkahler quotients of affine spaces, singular equivariant instantons, stability of Nahm data.
result Construction of the family of D2 ALF manifolds as a deformation of the flat orbifold (R3×S1)/Z2. Study symplectic 4-orbifolds with vanishing canonical class, finding new structures and resolutions.
problem Investigate symplectic 4-orbifolds with vanishing canonical class.
method Introduced a cyclic orbifold covering and a successive symplectic blowing-down procedure.
result Minimal resolution of symplectic 4-orbifolds yields symplectic Calabi-Yau manifolds.
Paper proves structure of compact Kähler 3-folds with specific bundles.
problem Characterizing compact Kähler 3-folds with nef anti-canonical bundles.
method Minimal Model Program, positivity of direct image sheaves, Q-conic bundles, orbifold vector bundles.
result Compact Kähler 3-folds with nef anti-canonical bundles are essentially one of three types.
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.
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…
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.
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.
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.
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.
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…
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. 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.
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.
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.
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.
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.
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.
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.
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.
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…