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We study base-point-freeness for big and nef line bundles on hyperkähler manifolds of generalized Kummer type: For n ∈ { 2 , 3 , 4 } , we show that, generically in all but a finite number of irreducible components of the moduli space of polarized Kum n -type varieties, the polarization is base-point-free. We also prove generic base-point-freeness in the moduli space in all dimensions if the polarization has divisibility one. [ABSTRACT FROM AUTHOR]

As in the case of irreducible holomorphic symplectic manifolds, the perioddomain $Compl$ of compact complex tori of even dimension $2n$ contains twistorlines. These are special $2$-spheres parametrizing complex tori whose complexstructures arise from a given quaternionic structure. In analogy with the caseof irreducible holomorphic symplectic manifolds, we show that the periods ofany two complex tori can be joined by a {\em generic} chain of twistor lines.We also prove a criterion of twistor path connectivity of loci in $Compl$ wherea fixed second cohomology class stays of Hodge type (1,1). Furthermore, we showthat twistor lines are holomorphic submanifolds of $Compl$, of degree $2n$ inthe Pl\"ucker embedding of $Compl$.

We propose a new method to construct degenerate 3-(α, δ)-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperkähler manifolds. Subsequently, we study homogeneous degenerate 3-(α, δ)-Sasakian manifolds and prove that no non-trivial compact examples exist aswell as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.

A hyperkähler manifold M has a family of induced complex structures indexed by a two-dimensional sphere S 2 ≅ CP 1 . The twistor space of M is a complex manifold Tw (M) together with a natural holomorphic projection Tw (M) → CP 1 , whose fiber over each point of CP 1 is a copy of M with the corresponding induced complex structure. We remove one point from this sphere (corresponding to one fiber in the twistor space), and for the case of M = R 4 n , n ∈ N , equipped with the standard hyperkähler structure, we construct one quantization that replaces the family of Berezin–Toeplitz quantizations parametrized by S 2 - { p t } . We provide semiclassical asymptotics for this quantization. [ABSTRACT FROM AUTHOR]

We show that given an embedding of an Enriques manifold of index d in a large enough projective space, there will exist embedded multiple structures with conormal bundle isomorphic to the trace zero module of the universal covering map, the universal cover being either a hyperkähler or a Calabi–Yau manifold. We then show that these multiple structures (also known as d-ropes) can be smoothed to smooth hyperkähler or Calabi–Yau manifolds respectively. Hence we obtain a flat family of hyperkähler (or Calabi–Yau) manifolds embedded in the same projective space which degenerates to an embedded d-rope structure on the given Enriques manifold of index d. The above shows that these d-rope structures on the embedded Enriques manifold are points of the Hilbert scheme containing the fibres of the above family. We show that they are smooth points of the Hilbert scheme when d = 2 . [ABSTRACT FROM AUTHOR]

We describe a barrier argument for compact minimal submanifolds in the multi-Eguchi--Hanson and in the multi-Taub--NUT spaces, which are hyperkähler 4-manifolds given by the Gibbons--Hawking ansatz. This approach is used to obtain results towards a classification of compact minimal submanifolds in this setting. We also prove a converse of Tsai and Wang's result that relates the strong stability condition to the convexity of the distance function. [ABSTRACT FROM AUTHOR]

Recently, T. Bridgeland defined a complex hyperkähler metric on the tangent bundle over the space of stability conditions of a triangulated category, based on a Riemann–Hilbert problem determined by the Donaldson–Thomas invariants. This metric is encoded in a function W (z , θ) satisfying a heavenly equation, or a potential F (z , θ) satisfying an isomonodromy equation. After recasting the RH problem into a system of TBA-type equations, we obtain integral expressions for both W and F in terms of solutions of that system. These expressions are recognized as conformal limits of the ‘instanton generating potential’ and ‘contact potential’ appearing in studies of D-instantons and BPS black holes. By solving the TBA equations iteratively, we reproduce Joyce’s original construction of F as a formal series in the rational DT invariants. Furthermore, we produce similar solutions to deformed versions of the heavenly and isomonodromy equations involving a non-commutative star product. In the case of a finite uncoupled BPS structure, we rederive the results previously obtained by Bridgeland and obtain the so-called τ function for arbitrary values of the fiber coordinates θ , in terms of a suitable two-variable generalization of Barnes’ G function. [ABSTRACT FROM AUTHOR]

Let p : M → B be a Lagrangian fibration on a hyperkähler manifold of maximal holonomy (also known as IHS), and H be the generator of the Picard group of B. Assume that p has no multiple fibers in codimension 1. We prove that p ∗ (H) is a primitive class on M. [ABSTRACT FROM AUTHOR]

Let M be a simple hyperkähler manifold. Kuga-Satake construction gives an embedding of H 2 (M , C) into the second cohomology of a torus, compatible with the Hodge structure. We construct a torus T and an embedding of the graded cohomology space H • (M , C) → H • + l (T , C) for some l , which is compatible with the Hodge structures and the Poincaré pairing. Moreover, this embedding is compatible with an action of the Lie algebra generated by all Lefschetz sl (2) -triples on M. [ABSTRACT FROM AUTHOR]

We consider triholomorphic maps u from an almost hyper-Hermitian manifold M4m into a (simply connected) hyperKähler manifold N4n. This entails that the map u ε W1,2 satisfies a quaternionic del-bar equation. We work under the assumption that u is locally strongly approximable in W1,2 by smooth maps; such maps are almost stationary harmonic in a suitable sense (in the important special case thatMis hyperKähler as well, they are stationary harmonic). We show, by means of the bmo-h¹ duality, that in this more general situation the classical "-regularity result still holds and we establish the validity, for triholomorphic maps, of the W2,1-conjecture (i.e. an a priori W2,1-estimate in terms of the energy). We then address compactness issues for a weakly converging sequence uℓ - u∞ of strongly approximable triholomorphic maps uℓ: M - 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 Σ of codimension 2, away from which the sequence converges strongly. The defect measure 2.x/H4m-2 6 encodes the loss of energy in the limit and we prove that for a.e. point on 6 the value of 2 is given by the sum of the energies of a (finite) number of smooth non-constant holomorphic bubbles (here the holomorphy is understood with respect to a complex structure on N that depends on the chosen point on 6). When M is hyperKähler this quantization result was established by C. Y. Wang [41] with a different proof; our arguments rely on Lorentz spaces estimates. By means of a calibration argument and a homological argument we further prove that whenever the restriction of 6 to an open ball B = Mn Singu1 is covered by a connected (4m - 2)-dimensional Lipschitz graph, then actually 6 \ B is a smooth submanifold and it is pseudo-holomorphic for a (unique) almost complex structure onM (with 2 constant on this portion); moreover the bubbles originating at points of 6 \ B are all holomorphic for a common complex structure on N. [ABSTRACT FROM AUTHOR]

Abstract In this article, we study a generalisation of the Seiberg–Witten equations, replacing the spinor representation with a hyperKähler manifold equipped with certain symmetries. Central to this is the construction of a (non-linear) Dirac operator acting on the sections of the non-linear fibre-bundle. For hyperKähler manifolds admitting a hyperKähler potential, we derive a transformation formula for the Dirac operator under the conformal change of metric on the base manifold. As an application, we show that when the hyperKähler manifold is of dimension four, then, away from a singular set, the equations can be expressed as a second order PDE in terms of almost-complex structure on the base manifold, and a conformal factor. This extends a result of Donaldson to generalised Seiberg–Witten equations. [ABSTRACT FROM AUTHOR]

The Beauville-Fujiki relation for a compact Hyperkähler manifold of dimension allows to equip the symmetric power with a symmetric bilinear form induced by the Beauville-Bogomolov form. We study some of its properties and compare it to the form given by the Poincaré pairing. The construction generalizes to a definition for an induced symmetric bilinear form on the symmetric power of any free module equipped with a symmetric bilinear form. We point out how the situation is related to the theory of orthogonal polynomials in several variables. Finally, we construct a basis of homogeneous polynomials that are orthogonal when integrated over the unit sphere , or equivalently, over with a Gaussian kernel. [ABSTRACT FROM AUTHOR]

We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori. [ABSTRACT FROM AUTHOR]

The definition and structure of hyperkähler structure preserving transformations (invariance group) for quaternionic structures have been recently studied and some preliminary results on the Euclidean case discussed. In this work we present the whole structure of the invariance Lie algebra in the Euclidean case for any dimension. [ABSTRACT FROM AUTHOR]

Given a hyperkahler manifold M , the hyperkahler structure defines a triple of symplectic structures on M ; with these, a triple of Hamiltonians defines a so-called hyperHamiltonian dynamical system on M . These systems are integrable when can be mapped to a system of quaternionic oscillators. We discuss the symmetry of integrable hyperHamiltonian systems, i.e. quaternionic oscillators, and conversely how these symmetries characterize, at least in the Euclidean case, integrable hyperHamiltonian systems. [ABSTRACT FROM AUTHOR]

Abstract: Let M be a compact hyperkähler manifold, and W the coarse moduli of complex deformations of M. Every positive integer class v in defines a divisor in W consisting of all algebraic manifolds polarized by v. We prove that every connected component of this divisor is dense in W. [Copyright &y& Elsevier]

We generalize and refine the hyperkähler Arnold conjecture, which was originally established, in the non-degenerate case, for three-dimensional time by Hohloch, Noetzel and Salamon by means of hyperkähler Floer theory. In particular, we prove the conjecture in the case where the time manifold is a multidimensional torus and also establish the degenerate version of the conjecture. Our method relies on Morse theory for generating functions and a finite-dimensional reduction along the lines of the Conley-Zehnder proof of the Arnold conjecture for the torus. [ABSTRACT FROM AUTHOR]

Let S[n] be the Hilbert scheme of length n subschemes of a K3 surface S. H2(S[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL[H*(S[n],ℤ)] generated by monodromy operators, and let Mon2 be its image in OH2(S[n],ℤ). We prove that Mon2 is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon2 does not surject onto OH2(S[n],ℤ)/(±1), when n - 1 is not a prime power. As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S[n]. The weight 2 Hodge structure on H2(X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2ρ(n - 1) - 1 distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1. The second main result states, that if a monodromy operator acts as the identity on H2(S[n],ℤ), then it acts as the identity on Hk(S[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon2, if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6). [ABSTRACT FROM AUTHOR]

In this article, we define a new class of middle dimensional submanifolds of a Hyperkähler manifold which contains the class of complex Lagrangian submanifolds, and show that this larger class is invariant under the mean curvature flow. Along the flow, the complex phase map satisfies the generalized harmonic map heat equation. It is also related to the mean curvature vector via a first order differential equation. Moreover, we proved a result on nonexistence of Type I singularity. [ABSTRACT FROM AUTHOR]

In this paper we construct an abelian fibration over P3 on the Hilbert cube of the primitive K3 surface of genus 9. After the abelian fibration constructed by Hassett and Tschinkel on the Hilbert square on the primitive K3 surface of genus 5, this is the second example where the abelian fibration is constructed directly on $\mathcal{H}ilb_{n}S$. The recent more general result of Sawon proves the existence of an abelian fibration on the Hilbert scheme $\mathcal{H}ilb_{n}S$ of a primitive K3 surface S of degree 2g - 2 = m2(2n - 2). Our example provides an alternative proof in the case m = 2, n = 3. Furthermore we identify the general fiber with the Hilbert scheme of twisted cubic curves in a Fano 3-fold of genus 9, and interpret the addition law on this variety. [ABSTRACT FROM AUTHOR]

Building on a recent paper [8], here we argue that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperkähler varieties. We show that just like toric varieties occupy a central role in Stanley’s proof for the necessity of McMullen’s conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperkähler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we will give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperkähler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari [3], leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz [20]. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley. [ABSTRACT FROM AUTHOR]

We shall describe the twistor space of a hyperka¨hler 4n-manifold with an isometric S1-action which is holomorphic for one of the complex structures, scales the corresponding holomorphic symplectic form and whose fixed point set has complex dimension n.We deduce that any hyperka¨hler metric on the cotangent bundle of a real-analytic Ka¨hler manifold which is compatible with the canonical holomorphic symplectic structure, extends the given Ka¨hler metric and for which the S1-action by scalar multiplication in the fibres is isometric is unique in a neighbourhood of the zero section. These metrics have been constructed independently by the author and Kaledin. [Copyright &y& Elsevier]

The original version of the article was published in Central European Journal of Mathematics, 2008, 6(2), 191–203, DOI: 10.2478/s11533-008-0026-8. Unfortunately, the original version of this article contains a mistake, which we correct here. [ABSTRACT FROM AUTHOR]