arXiv subjects
Publications and source records attributed to Nigel Hitchin.
We calculate the characteristic classes for flat SL(n,R) and Sp(2m,R)-bundles over a compact surface as functions of the spectral data in the Higgs bundle description, which consists of the points of order 2 in an abelian variety. Using similar methods we produce a conjectural mirror for the moduli space of flat U(m,m)-bundles with a fixed characteristic class. This is a hyperholomorphic bundle over the Sp(2m,C)-moduli space.
The character varieties of representations of a surface group into the Lie groups SL(m,H), SO(2m,H) and Sp(m,m) have a holomorphic description in terms of the moduli space of Higgs bundles. We show that the fibres of the integrable system in these cases are not abelian varieties, but are instead moduli spaces of rank 2 bundles on a spectral curve, satisfying natural stability conditions.
We study the hyperholomorphic line bundle on a hyperkaehler manifold with circle action introduced by A Haydys, and in particular show how it transforms under a hyperkaehler quotient. Applications include ALE spaces and coadjoint orbits.
A hyperkaehler manifold with a circle action fixing just one complex structure admits a natural a hyperholomorphic line bundle. This forms the basis for the construction of a corresponding quaternionic Kaehler manifold in the work of A.Haydys. We construct in this paper the corresponding holomorphic line bundle on twistor space and compute many examples, including monopole and Higgs bundle moduli spaces. We also show that the bundle on twistor space has a natural meromorphic connection which realizes it as the quantum line bundle for the hyperkaehler family of holomorphic symplectic structures. Finally we give a twistor version of the HK/QK correspondence.
An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in particular the Hilbert schemes of the projective plane and show that a generic deformation is determined by two parameters -- an elliptic curve and a translation on it.
On a generalized complex manifold there is an associated definition of a generalized holomorphic bundle, introduced by Gualtieri. This notion in the case of an ordinary complex structure yields an object which we call a co-Higgs bundle and we consider the B-field action of a closed form of type (1,1), both local and global. The effect makes contact with both Nahm's equations and holomorphic gerbes.
These are lecture notes mainly aimed at graduate students on selected aspects of generalized geometry: in particular generalized complex and Kaehler structures and generalized holomorphic bundles. They are based on lectures given in March 2010 at the Chinese University of Hong Kong.
We introduce an algebra of Schouten-commuting holomorphic polyvector fields on the moduli space of stable G-bundles over a curve by using invariant forms on the Lie algebra. The generators begin in degree three -- we prove a vanishing theorem for degree two in the case of G=GL(n).
A plane curve C defined by a homogeneous polynomial satisfying Laplace's equation appears canonically as the vanishing of the Pfaffian of a skew-symmetric matrix of linear forms. As a consequence there is a natural semi-stable rank two vector bundle defined on C. We consider the case of degree 3, and apply Atiyah's classification of bundles to determine various invariant strata in the space of harmonic cubics. We encounter the Mukai-Umemura threefold and link up with the classical geometry of the Clebsch diagonal cubic surface, certain distinguished rational curves on it, and the action of the icosahedral group.
The Serre construction of rank two holomorphic bundles with a section is adapted to construct generalized holomorphic bundles on a generalized complex 4-manifold from the data of a set of points on an elliptic curve. The motivation is the special case of rank two Poisson modules on a complex surface with a holomorphic Poisson structure.
We define a sextic invariant J on the seven-dimensional space of degree three spherical harmonics and show that J is positive if and only if the nodal set of the spherical harmonic contains the vertices of exactly two regular icosahedra. The proof uses the geometry of the Clebsch diagonal cubic surface, Atiyah's classification of vector bundles on an elliptic curve and a Fano threefold introduced by Mukai.
We first demonstrate how duality for the fibres of the so-called Hitchin fibration works for the Langlands dual groups Sp(2m) and SO(2m+1). We then show that duality for G2 is implemented by an involution on the base space which takes one fibre to its dual. A formula for the natural cubic form is given and shown to be invariant under the involution.
From a hermitian metric on the anticanonical bundle on a Del Pezzo surface, and a holomorphic section of it, we construct a one parameter family of bihermitian metrics (or equivalently generalized Kaehler structures). The construction appears to be linked to noncommutative geometry.
In the context of generalized geometry we first show how the Courant bracket helps to define connections with skew torsion and then investigate a five-dimensional invariant functional and its associated geometry. A Hamiltonian flow arising from this corresponds to a version of the Nahm equations using the Courant bracket, and we investigate the six-dimensional geometrical structure this describes.
Using the idea of a generalized Kaehler structure, which is a pair of commuting generalized complex structures, we construct bihermitian metrics on the projective plane and the product of two projective lines, and show that any such structure on a compact 4-manifold M defines one on the moduli space of anti-self-dual connections on a fixed principal bundle over M. We highlight the role of holomorphic Poisson structures in all these constructions.
A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphisms and closed 2-forms. In the special case of six dimensions we characterize them as critical points of a natural variational problem on closed forms, and prove that a local moduli space is provided by an open set in either the odd or even cohomology.
We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general formalism becomes a practical tool for calculating homogeneous or cohomogeneity one examples, and we illustrate this with some newly discovered cases of Spin(7) and G2 metrics.
We study the special algebraic properties of alternating 3-forms in 6 and 7 dimensions and introduce a diffeomorphism-invariant functional on the space of differential 3-forms on a closed manifold M in these dimensions. Restricting the functional to closed forms in a fixed cohomology class, we find that a critical point which is generic in a suitable sense defines in the 6-dimensional case a complex threefold with trivial canonical bundle and in 7 dimensions a Riemannian manifold with holonomy G2. This approach gives a direct method of finding a local moduli space, with its special geometry, for these structures.