Hyperkahler Research Articles

Special Joyce structures and hyperkähler metrics

Joyce structures were introduced by T. Bridgeland in the context of the space of stability conditions of a three-dimensional Calabi–Yau category and its associated Donaldson–Thomas invariants. In subsequent work, T. Bridgeland and I. Strachan showed that Joyce structures satisfying a certain non-degeneracy condition encode a complex hyperkähler structure on the tangent bundle of the base of the Joyce structure. In this work we give a definition of an analogous structure over an affine special Kähler (ASK) manifold, which we call a special Joyce structure. Furthermore, we show that it encodes a real hyperkähler (HK) structure on the tangent bundle of the ASK manifold, possibly of indefinite signature. Particular examples include the semi-flat HK metric associated to an ASK manifold (also known as the rigid c-map metric) and the HK metrics associated to certain uncoupled variations of BPS structures over the ASK manifold. Finally, we relate the HK metrics coming from special Joyce structures to HK metrics on the total space of algebraic integrable systems.

Asymptotic base loci on hyper-Kahler manifolds

Asymptotic base loci on hyper-Kahler manifolds

RIGID STABLE VECTOR BUNDLES ON HYPERKÄHLER VARIETIES OF TYPE $K3^{[n]}$

AbstractWe prove existence and unicity of slope-stable vector bundles on a general polarized hyperkähler (HK) variety of type $K3^{[n]}$ with certain discrete invariants, provided the rank and the first two Chern classes of the vector bundle satisfy certain equalities. The latter hypotheses at first glance appear to be quite restrictive, but, in fact, we might have listed almost all slope-stable rigid projectively hyperholomorphic vector bundles on polarized HK varieties of type $K3^{[n]}$ with $20$ moduli.

A hyperkähler geometry associated to the BPS structure of the resolved conifold

We associate to the resolved conifold an affine special Kähler (ASK) manifold of complex dimension 1, and an instanton corrected hyperkähler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like smoothing of the semi-flat HK metric associated to the ASK geometry. On the other hand, the instanton corrected HK geometry associated to the resolved conifold can be described in terms of a twistor family of two holomorphic Darboux coordinates. We study a certain conformal limit of the twistor coordinates, and conjecture a relation to a solution of a Riemann-Hilbert problem previously considered by T. Bridgeland.

On the Brauer Group of an Arithmetic Model of Strictly Complete Intersection in a HyperkÄhler Variety Over a Number Field

We prove the Artin conjecture asserting the finiteness of the Brauer group of an arithmetic model of a smooth projective variety V over a number field k under the assumptions that V admits a k-embedding V ↪ W to a smooth projective hyperkahler ksubvariety $$ W\to {\mathbb{P}}_k^N $$ such that V = H1 ∩ … ∩ Hr ∩ W in the sense of the theory of schemes for some k-hypersurfaces H1, . . .,Hr in the space $$ {\mathbb{P}}_k^N $$ ; moreover, for any i ≤ r the variety H1 ∩ . . . ∩ Hi ∩ W is smooth, has codimension i in the variety W, and dimkV ≥ 3.

Local projectivity of Lagrangian fibrations on Hyperkähler manifolds

We show that if $$f:X\rightarrow B$$ is a Lagrangian fibration from a compact connected hyperkahler (and thus Kahler) manifold X onto a projective normal variety B, then f is locally projective. This answers a question raised by L. Kamenova and strengthens a former result (see Campana in Math Z 252:147–156, 2005, Proposition 2.1), according to which the smooth fibres of f are projective.

K-Theory of Toric HyperKähler Manifolds

Let X be a toric hyperKahler manifold. The purpose of this note is to describe the topological K-ring K*(X) of X. We give a presentation for the topological K-ring in terms of generators and relations similar to the known description of the cohomology ring of these manifolds.

INDEFINITE HYPERKÄHLER METRICS ON LIE GROUPS WITH ABELIAN COMPLEX STRUCTURES

We consider Lie groups G endowed with a pair of anticommuting left-invariant abelian complex structures (J1, J2) and a left-invariant (possibly indefinite) metric g such that (G, J1, J2, g) results to be a hyperkahler manifold. We give a classification of their Lie algebras up to dimension 12 and study some of their geometric properties. In particular, we show that all such groups are locally symmetric and complete and that the metric is flat if and only if the group is 2-step nilpotent.

A conjectural bound on the second Betti number for hyper-Kaehler manifolds

A conjectural bound on the second Betti number for hyper-Kaehler manifolds

The Euler number of hyper-K¨ahler manifolds of OG10 type

Using the Laza-Sacca-Voisin construction, we give a simple proof for the fact that the Euler characteristic of a hyper-Kahler manifold of OG10 type is 176,904, a result previously established by Mozgovoy.

A note on the collapsing geometry of hyperkähler four manifolds

We make some observations concerning the one-dimensional collapsing geometry of four-dimensional hyperkahler metrics.

On the Homeomorphism Type of Smooth Projective Fourfolds

In this paper, we study smooth complex projective 4-folds which are topologically equivalent. First we show that Fano fourfolds are never oriented homeomorphic to Ricci-flat projective fourfolds and that Calabi-Yau manifolds and hyperkahler manifolds in dimension ≥ 4 are never oriented homeomorphic. Finally, we give a coarse classification of smooth projective fourfolds which are oriented homeomorphic to a hyperkahler fourfold which is deformation equivalent to the Hilbert scheme S[2] of two points of a projective K3 surface S.

Finite groups of symplectic automorphisms of hyperkahler manifolds of type K3

We determine the possible finite groups $G$ of symplectic automorphisms of hyperkahler manifolds which are deformation equivalent to the second Hilbert scheme of a K3 surface. We prove that $G$ has such an action if, and only if, it is isomorphic to a subgroup of either the Mathieu group $M_{23}$ having at least four orbits in its natural permutation representation on $24$ elements, or one of two groups $3^{1+4}{:}2.2^2$ and $3^4{:}A_6$ associated to $\mathcal{S}$-lattices in the Leech lattice. We describe in detail those $G$ which are maximal with respect to these properties, and (in most cases) we determine all deformation equivalence classes of such group actions. We also compare our results with the predictions of Mathieu Moonshine.

ALF gravitational instantons and collapsing Ricci-flat metrics on the $K3$ surface

We construct large families of new collapsing hyperkahler metrics on the $K3$ surface. The limit space is a flat Riemannian $3$-orbifold $T^3 / \mathbb{Z}_2$. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most $24$ exceptional points where the curvature concentrates, which always contains the 8 fixed points of the involution on $T^3$. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type ($D_k$) for the fixed points of the involution on T3 and of cyclic type ($A_k$) otherwise. The collapsing metrics are constructed by deforming approximately hyperkahler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) $S^1$–invariant hyperkahler metric arising from the Gibbons–Hawking ansatz over a punctured $3$-torus. As an immediate application to submanifold geometry, we exhibit hyperkahler metrics on the $K3$ surface that admit a strictly stable minimal sphere which cannot be holomorphic with respect to any complex structure compatible with the metric.

A characterization of compact locally conformally hyperkähler manifolds

We give an equivalent definition of compact locally conformally hyperkahler manifolds in terms of the existence of a non-degenerate complex two-form with natural properties. This is a conformal analogue of Beauville’s theorem stating that a compact Kahler manifold admitting a holomorphic symplectic form is hyperkahler.

Compactness results for triholomorphic maps

We consider triholomorphic maps from an almost hyper-Hermitian manifold $\mathcal{M}^{4m}$ into a hyperKahler manifold $\mathcal{N}^{4n}$. This means that $u \in W^{1,2}$ satisfies a quaternionic del-bar equation. We work under the assumption that $u$ is locally strongly approximable in $W^{1,2}$ by smooth maps: then such maps are almost stationary harmonic (when $\mathcal{M}$ is hyperKahler as well, then stationary harmonic). We show that in this more general situation the classical $\epsilon$-regularity result still holds. We then address compactness issues for a weakly converging sequence $u_\ell \rightharpoonup u_\infty$ of strongly approximable triholomorphic maps $u_\ell:\mathcal{M} \to \mathcal{N}$ with uniformly bounded Dirichlet energies. The blow up analysis leads, as in the usual stationary setting, to the existence of a rectifiable blow-up set $\Sigma$ of codimension $2$, away from which the sequence converges strongly. The defect measure $\Theta(x) {\mathcal{H}}^{4m-2} \llcorner \Sigma$ encodes the loss of energy in the limit; we prove that for a.e. point on $\Sigma$ the value of $\Theta$ is given by the sum of energies of a (finite) number of smooth non-constant holomorphic bubbles (here the holomorphicity is understood w.r.t. a complex structure on $\mathcal{N}$ that depends on the chosen point on $\Sigma$). In the case that $\mathcal{M}$ is hyperKahler this result was established by C. Y. Wang (2003) with a different proof; we rely on Lorentz space estimates. By means of a calibration and a homological argument we further prove that for each portion of $\Sigma \setminus \text{Sing}_{u_\infty}$ contained in a Lipschitz graph we find a unique alm. compl. st. on $\mathcal{M}$ that makes the portion pseudoholomorphic and smooth, with $\Theta$ constant; moreover the bubbles originating at points of such a smooth piece are holomorphic for a common complex structure.

Jumps, folds and singularities of Kodaira moduli spaces

For any integer k we construct an explicit example of a twistor space which contains a one-parameter family of jumping rational curves, where the normal bundle changes from O(1) � O(1) to O(k) � O(2 k). For k > 3 the resulting anti-self-dual Ricci-flat manifold is a Zariski cone in the space of holomorphic sections of O(k). In the case k = 2 we recover the canonical example of Hitchin's folded hyper-Kahler manifold, where the jumping lines form a three-parameter family. We show that in this case there exist normalisable solutions to the Schrodinger equation which extend through the fold.

Collapsing hyperkähler manifolds

Given a projective hyperkahler manifold with a holomorphic Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers shrinking to zero collapse in the Gromov-Hausdorff sense (and smoothly away from the singular fibers) to a compact metric space which is a half-dimensional special Kahler manifold outside a singular set of real Hausdorff codimension 2 and is homeomorphic to the base projective space.

From ALE to ALF gravitational instantons

In this article, we give an analytic construction of ALF hyperkähler metrics on smooth deformations of the Kleinian singularity $\mathbb{C}^{2}/{\mathcal{D}}_{k}$, with ${\mathcal{D}}_{k}$ the binary dihedral group of order $4k$, $k\geqslant 2$. More precisely, we start from the ALE hyperkähler metrics constructed on these spaces by Kronheimer, and use analytic methods, e.g. resolution of a Monge–Ampère equation, to produce ALF hyperkähler metrics with the same associated Kähler classes.

AN EQUIVARIANT DESCRIPTION OF CERTAIN HOLOMORPHIC SYMPLECTIC VARIETIES

Varieties of the form$G\times S_{\!\text{reg}}$, where$G$is a complex semisimple group and$S_{\!\text{reg}}$is a regular Slodowy slice in the Lie algebra of$G$, arise naturally in hyperkähler geometry, theoretical physics and the theory of abstract integrable systems. Crooks and Rayan [‘Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices’,Math. Res. Let., to appear] use a Hamiltonian$G$-action to endow$G\times S_{\!\text{reg}}$with a canonical abstract integrable system. To understand examples of abstract integrable systems arising from Hamiltonian$G$-actions, we consider a holomorphic symplectic variety$X$carrying an abstract integrable system induced by a Hamiltonian$G$-action. Under certain hypotheses, we show that there must exist a$G$-equivariant variety isomorphism$X\cong G\times S_{\!\text{reg}}$.