arXiv research

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 $π_{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb{Z}))$ has a section over its image. The second is that, there exists a subgroup $G$ of $π_{0}(Diff(K3))$ of order two over which…

Researchers found non-smoothable surfaces in a 4-sphere, solving K3 problems.

Researchers find geodesics on K3 surfaces using electrostatics.

The study of special Lagrangian classes and semistable Mukai vectors on K3 surfaces.

Constructs Ricci-flat K3 metrics using D-geometry.

Dehn twists on K3-type 4-manifolds are not homotopy coherently Nielsen realizable.

The paper solves a geometric problem related to K3 surfaces and complex-hyperkähler metrics.

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.

New solutions found for $G_2$ system using K3 orbifolds.

Study on K3 surfaces' collapsing and special Kähler structures.

We address the problem of classification of hyper-Kähler fourfolds with $b_2=23$. In particular we prove some special cases of the Conjecture of O'Grady about hyper-Kähler $4$-folds numerically equivalent to the Hilbert scheme of two points on a $K3$ surface.

Study on automorphisms of K3 and Enriques surfaces, proving entropy gaps and achirality.

Solves Nielsen realization problem for hyper-Kähler manifolds.

Nontrivial boundary Dehn twist found on K3#K3 manifold.

Compactifies metrics on K3 surfaces with algebraic description.

Simply connected moduli space of Ricci flat metrics on 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.

Special Lagrangian submanifolds emerge from K3 surface collapse.

Boundary Dehn twist on $K3$ surfaces becomes trivial after abelianization.

Constructs entropy-minimizing pseudo-Anosov diffeomorphisms on K3 surfaces.

Study finds chirally cosmetic surgeries on knots and manifolds, contradicting previous conjectures.

Study describes limits of non-collapsing K3 surfaces using algebraic 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.

New K3 metrics derived from torus orbifold loci.

Study shows symplectic mapping groups of K3 surfaces are infinitely generated.

Simplified proof of K3 surface period map surjectivity.

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.

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.

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.

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 $\mathbb{Z}^2_2$-actions for the construction of $G_2$-manifolds. We find a large class …

Researchers solve Nielsen Realization problems for K3 surfaces in various categories.

An integral hyperbolic lattice is called reflective if its automorphism group is generated by reflections, up to finite index. Since 1981, it is known that their number is essentially finite. We show that K3 surfaces over C with reflective Picard lattices can be characterized in terms of compositions of their self-corr…

Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability…

Study geodesics on K3 surfaces near orbifold limit.

Complete Calabi-Yau metrics made on special 3D spaces.

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.

We give an alternative proof of a result of Cantat and Dupont, showing that any automorphism of a K3 surface with measure of maximal entropy in the Lebesgue class must be a Kummer example. Our method exploits the existence of Ricci-flat metrics on K3s and also covers the non-projective case.

Stability conditions on K3 surfaces are linked to the masses of spherical objects.

Certain six-dimensional (1,0) supersymmetric little string theories, when compactified on $T^3$, have moduli spaces of vacua given by smooth K3 surfaces. Using ideas of Gaiotto-Moore-Neitzke, we show that this provides a systematic procedure for determining the Ricci-flat metric on a smooth K3 surface in terms of BPS d…

New K3 surfaces with two involutions and low Picard number constructed.

Study shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.