Rozansky–Witten invariants via Atiyah classes

Recently, L. Rozansky and E. Witten associated to any hyper-Kähler manifold X a system of weights (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to the existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyper-Kähler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work at the level of cohomology and for any Kähler manifold, if we work at the level of Dolbeault cochains. As an outcome of our considerations, we give a formula for Rozansky–Witten classes using any Kähler metric on a holomorphic symplectic manifold.     

On hyper-Kähler manifolds of typeA ∞

In this paper we shall construct new families of 4m dimensional non-compact complete hyper-Kähler manifolds on whichm dimensional torus acts. In the 4 dimensional case our manifolds should be considered as hyper-Kähler manifolds which correspond to the extended Dynkin diagram of typeA .     

Almost Hermitian structures and quaternionic geometries

Gray and Hervella gave a classification of almost Hermitian structures (g,I) into 16 classes. We systematically study the interaction between these classes when one has an almost hyper-Hermitian structure (g,I,J,K). In general dimension we find at most 167 different almost hyper-Hermitian structures. In particular, we obtain a number of relations that give hyper-Kähler or locally conformal hyper-Kähler structures, thus generalising a result of Hitchin. We also study the types of almost quaternion-Hermitian geometries that arise and tabulate the results.     

The topological types of some irregular Kähler surfaces

The topological type of generalized Kummer surfaces is described in terms of sphere bundles over Riemann surfaces and the complex projective plane. Explicit examples of sets of pairwise non-diffeomorphic K?hler surfaces of the same topological type are given. Received: 5 January 2000     

Divisorial Zariski decompositions on compact complex manifolds

Using currents with minimal singularities, we introduce pointwise minimal multiplicities for a real pseudo-effective (1,1)-cohomology class α on a compact complex manifold X, which are the local obstructions to the numerical effectivity of α. The negative part of α is then defined as the real effective divisor N(α) whose multiplicity along a prime divisor D is just the generic multiplicity of α along D, and we get in that way a divisorial Zariski decomposition of α into the sum of a class Z(α) which is nef in codimension 1 and the class of its negative part N(α), which is an exceptional divisor in the sense that it is very rigidly embedded in X. The positive parts Z(α) generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kähler manifold in some detail. Using the intersection form (respectively the Beauville-Bogomolov form), we characterize the modified nef cone and the exceptional divisors. The divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kähler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series |kL| as k→∞.     

Pluriharmonic maps into Kähler symmetric spaces and Sym’s formula

A construction due to Sym and Bobenko recovers constant mean curvature surfaces in euclidean 3-space from their harmonic Gauss maps. We generalize this construction to higher dimensions and codimensions replacing the surface by a complex manifold and the sphere (the target space of the Gauss map) by a Kähler symmetric space of compact type with its standard embedding into the Lie algebra ${\mathfrak{g}}A construction due to Sym and Bobenko recovers constant mean curvature surfaces in euclidean 3-space from their harmonic Gauss maps. We generalize this construction to higher dimensions and codimensions replacing the surface by a complex manifold and the sphere (the target space of the Gauss map) by a K?hler symmetric space of compact type with its standard embedding into the Lie algebra \mathfrakg{\mathfrak{g}} of its transvection group. Thus we obtain a new class of immersed K?hler submanifolds of \mathfrakg{\mathfrak{g}} and we derive their properties.     

Compact Kähler Surfaces with Trivial Canonical Bundle

The classical conjectures of Weil on K3 surfaces – that the set of suchsurfaces is connected; that a version of the Torelli theorem holds; thateach such surface is Kähler; and that the period map issurjective – are reconsidered in the light of a generalisation of theNakai–Moishezon criterion, and short proofs of all the conjectures aregiven. Most of the proofs apply equally or with minor variation tocomplex 2-tori, the only other compact Kähler surfaces with trivialcanonical bundle.     

<Emphasis Type="Italic">K</Emphasis>3 surfaces with involution,equivariant analytic torsion,and automorphic forms on the moduli space

In this paper, we introduce an invariant of a K3 surface with ₂-action equipped with a ₂-invariant Kähler metric, which we obtain using the equivariant analytic torsion of the trivial line bundle. This invariant is shown to be independent of the choice of the Kähler metric. It can be viewed as a function on the moduli space of K3 surfaces with involution. The main result of this paper is that this function can be identified with an automorphic form, which characterizes the discriminant locus. In particular, we show that the Ray–Singer analytic torsion of the trivial line bundle on an Enriques surface with Ricci-flat Kähler metric is given by the value of the norm of the Borcherds -function at its period point. Mathematics Subject Classification (1991) 58G26, 14J28, 14J15, 32G20, 32N10, 32N15     

Relations between the Kähler cone and the balanced cone of a Kähler manifold

In this paper, we consider a natural map from the Kähler cone of a compact Kähler manifold to its balanced cone. We study its injectivity and surjectivity. We also give an analytic characterization theorem on a nef class being Kähler.     

On the existence and nonexistence of extremal metrics on toric Kähler surfaces

In this paper we study the existence of extremal metrics on toric Kähler surfaces. We show that on every toric Kähler surface, there exists a Kähler class in which the surface admits an extremal metric of Calabi. We found a toric Kähler surface of 9 -fixed points which admits an unstable Kähler class and there is no extremal metric of Calabi in it. Moreover, we prove a characterization of the K-stability of toric surfaces by simple piecewise linear functions. As an application, we show that among all toric Kähler surfaces with 5 or 6 -fixed points, is the only one which allows vanishing Futaki invariant and admits extremal metrics of constant scalar curvature.     

Deformations of twisted harmonic maps and variation of the energy

We study the deformations of twisted harmonic maps \(f\) with respect to the representation \(\rho \). After constructing a continuous “universal” twisted harmonic map, we give a construction of every first order deformation of \(f\) in terms of Hodge theory; we apply this result to the moduli space of reductive representations of a Kähler group, to show that the critical points of the energy functional \(E\) coincide with the monodromy representations of polarized complex variations of Hodge structure. We then proceed to second order deformations, where obstructions arise; we investigate the existence of such deformations, and give a method for constructing them, as well. Applying this to the energy functional as above, we prove (for every finitely presented group) that the energy functional is a potential for the Kähler form of the “Betti” moduli space; assuming furthermore that the group is Kähler, we study the eigenvalues of the Hessian of \(E\) at critical points.     

On the geometry of moduli spaces

We construct a Kähler metric on the moduli spaces of compact complex manifolds with c₁,<0 and of polarized compact Kähler manifolds with c₁=0, which is a generalization of the Petersson-Well metric. It is induced by the variation of the Kähler-Einstein metrics on the fibers that exist according to the Calabi-Yau theorem. We compute the above metric on the moduli spaces of polarized tori and symplectic manifolds. It turns out to be the Maaß metric on the Siegel upper half space and the Bergmann metric on a symmetric space of type III resp. In particular it is Kähler-Einstein with negative curvature.Dedicated to Karl SteinHeisenberg-Stipendiat der Deutschen Forschungsgemeinschaft     

Extremal Kähler metrics and complex deformation theory

Let (M, J, g) be a compact Kähler manifold of constant scalar curvature. Then the Kähler class [] has an open neighborhood inH ^1,1 (M, ) consisting of classes which are represented by Kähler forms of extremal Kähler metrics; a class in this neighborhood is represented by the Kähler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [] is nondegenerate, every small deformation of the complex manifold (M, J) also carries Kähler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kähler metrics on certain compact complex surfaces.The first author is supported in part by NSF grant DMS 92-04093.     

On Principally Polarized Adler-van Moerbeke Surfaces

In the present paper we study Kummer surfaces in IP⁶ arising as projections of a certain family of principally polarized abelian surfaces in IP⁶. This family was introduced by Adler and van Moerbeke in connection with some Hamiltonian dynamical systems. In this paper we give explicit equations of the associated Kummer surfaces. This enables us to deacribe their moduli space and to give the full list of degenerations.     

Remarks on the Segre cubic

The Igusa quartic 3-fold is the moduli space of Abelian surfaces with the level two structure, and it is known that the tangent hyperplanes cut out Kummer surfaces. In this note, we show similar results for the Segre cubic.Received: 18 March 2002     

Strongly Extremal Kähler Metrics

On a compact complex manifold (M, J) of the Kähler type, we consider the functional defined by the L²-norm of the scalar curvature with its domain the space of Kähler metrics of fixed total volume. We calculate its critical points, and derive a formula that relates the Kähler and Ricci forms of such metrics on surfaces. If these metrics have a nonzero constant scalar curvature, then they must be Einstein. For surfaces, if the scalar curvature is nonconstant, these critical metrics are conformally equivalent to non-Kähler Einstein metrics on an open dense subset of the manifold. We also calculate the Hessian of the lower bound of the functional at a critical extremal class, and show that, in low dimensions, these classes are weakly stable minima for the said bound. We use this result to discuss some applications concerning the two-points blow-up of CP².     

Aspherical Kähler manifolds with solvable fundamental group

We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of $\mathbb{C}^{n}We survey recent developments which led to the proof of the Benson-Gordon conjecture on K?hler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a K?hler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical K?hler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of \mathbbCⁿ\mathbb{C}^{n} by a discrete group of complex isometries.     

Hilbert schemes of surfaces are dense in moduli

We prove that the locus of Hilbert schemes of n points on a projective K 3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well. Along the way we prove an integral constraint on the monodromy group of generalized Kummer manifolds.     

On hyper Kähler manifolds associated to Lagrangian Kähler submanifolds of

For any Lagrangian Kähler submanifold , there exists a canonical hyper Kähler metric on . A Kähler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper Kähler manifolds with large automorphism group. Using it, an interesting class of pseudo hyper Kähler manifolds of complex signature is constructed. For any manifold in this class a group of automorphisms with a codimension one orbit on is specified. Finally, it is shown that the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper Kähler metric of complex signature .