LMS Durham Symposium

# Special Structures in Differential Geometry

University of Durham
Monday 30th July to Thursday 9th August, 2001

## Programme

### Monday, 30th July

14:00-19:00 Registration Dinner

### Tuesday, 31st July

9:00-10:00 10:15-11:15 Bär 1 Lecture room CG93 Schwachhöfer Lecture room CG93 Coffee Acharya Lecture room CG93 Lunch Tea Thomas Lecture room CG93 Wang Lecture room CG93 Wine reception Dinner

### Wednesday, 1st August

9:00-10:00 10:15-11:15 Ivanov Lecture room CG93 LeBrun 1 Lecture room CG93 Coffee Gibbons Lecture room CG93 Lunch Tea Herrera Lecture room CG93 Calderbank Lecture room CG93 Wood Lecture room CG93 Pedersen Lecture room CM107 Dinner

### Thursday, 2nd August

9:00-10:00 10:15-11:15 Biquard Lecture room CG93 Candelas Lecture room CG93 Coffee Gauntlett Lecture room CG93 Lunch Conference Photograph Cathedral Visit Dinner

### Friday, 3rd August

9:00-10:00 10:15-11:15 Pope Lecture room CG93 Singer Lecture room CG93 Coffee Tian 1 Lecture room CG93 Lunch Tea Spence Lecture room CG93 Galicki Lecture room CG93 Poon Lecture room CM107 Cortés Lecture room CG93 Dinner

### Saturday, 4th August

9:00-10:00 10:15-11:15 Atiyah Lecture room CG93 Tian 2 Lecture room CG93 Coffee LeBrun 2 Lecture room CG93 Lunch Colding 1 Lecture room CG93 Tea Apostolov Lecture room CG93 Verbitsky Lecture room CM107 Boyer Lecture room CG93 García Prada Lecture room CG93 Dinner

### Sunday, 5th August

Day trip to Harewood House and Harrogate

### Monday, 6th August

9:00-10:00 10:15-11:15 Ward Lecture room CG93 Hitchin Lecture room CG93 Coffee Merkulov Lecture room CG93 Lunch Tea Figueroa-O'Farrill Lecture room CG93 Berndt Lecture room CM107 Bielawski Lecture room CG93 Gutowski Lecture room CM107 Dancer Lecture room CG93 Vukmirovic Lecture room CM107 Dinner

### Tuesday, 7th August

9:00-10:00 10:15-11:15 Guan Lecture room CG93 Bär 2 Lecture room CG93 Coffee Gauduchon Lecture room CG93 Lunch Colding 2 Lecture room CG93 Tea de Bartolomeis Lecture room CG93 Ferapontov Lecture room CM107 Strachan Lecture room CG93 Dotti Lecture room CM107 Dunajski Lecture room CG93 Nagatomo Lecture room CM107 Wine reception followed by Conference Dinner

### Wednesday, 8th August

9:00-10:00 10:15-11:15 Kovalev Lecture room CG93 Böhm Lecture room CG93 Coffee Pidstrigatch Lecture room CG93 Lunch Informal talk: Acharya CG93 Tea Tod Lecture room CG93 Belgun Lecture room CG93 Rawnsley Lecture room CM107 Pantilie Lecture room CG93 Semmelmann Lecture room CM107 Dinner

### Thursday, 9th August

Departure after breakfast

## Titles and Abstracts

Nigel Hitchin (Oxford)
Variational approaches to special holonomy
Christian Bär (Hamburg)
1. & 2. Surgery and the Dirac spectrum
Tobias Colding (Courant Institute)
1. & 2. The space of embedded minimal surfaces of fixed genus in a 3-manifold
Claude LeBrun (Stony Brook)
1. Einstein Manifolds, Weyl Curvature, and Differential Topology
2. Minimal Volumes and Seiberg-Witten Theory
Gang Tian (MIT)
1. & 2. Extremal Metrics and Geometric Stability
Roger Bielawski (Glasgow)
Hyperkähler metrics with large symmetry
Philip Candelas (Oxford)
Calabi-Yau Manifolds over Finite Fields
Andrew Dancer (Oxford)
Einstein equations and integrability
The full Einstein equations and their reductions to ODEs are typically not integrable. We discuss some work on identifying cases where the equations can in fact be solved.
Gary Gibbons (DAMPT, Cambridge)
Some Recent work on metrics with special holonomy
Sergei Merkulov (Glasgow)
On the derived differential geometry
Michael Singer (Edinburgh)
Complex Surface Singularities and Complete Einstein Metrics
Paul Tod (Oxford)
D'Atri spaces
Richard Ward (Durham)
Hopf Solitons from Instanton Holonomy
The holonomy of an SU(2) N-instanton in the x^4-direction gives a map from R^3 into SU(2), which provides a good model of an N-Skyrmion. Combining this map with the standard Hopf map then gives a configuration for a Hopf soliton of charge N. In this way, one may define a collective-coordinates manifold for Hopf solitons. This talk will deal with instanton approximations to Hopf solitons in the Skyrme-Faddeev system, focussing in particular on the the two-soliton sector.
Vestislav Apostolov (Montréal)
The curvature and the integrability of almost Kaehler manifolds
Olivier Biquard (Strasbourg)
New complete quaternion-Kähler metrics
Charles Boyer (Albuquerque)
Sasakian-Einstein Geometry via Algebraic Geometry
The relationship between Sasakian-Einstein geometry and algebraic geometry is described. It is then shown how methods from Mori theory in algebraic geometry can be used to construct families of Einstein metrics on the k-fold connected sum of $S^2\times S^3$. The moduli problem for Sasakian-Einstein structures on these 5-manifolds is also discussed.
Christoph Böhm (Kiel)
The scalar curvature functional on compact homogeneous spaces
For a compact homogeneous space G/H the space of G-invariant metrics of volume 1 is a symmetric space of non-compact type. We will describe the asymptotic behaviour of the scalar curvature functional. This will enable us to formulate existence theorems for G-invariant Einstein metrics. Existence is guaranteed if certain invariants of G/H, defined by Lie theoretic data, are "non-trivial".
Vicente Cortés (Freiburg)
Special Kähler manifolds and affine differential geometry
Paolo de Bartolomeis (Florence)
Frobenius Structures in Symplectic Geometry
Krystoff Galicki (Albuquerque)
3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients
Using 3-Sasakian reduction techniques we obtain infinite families of new 3-Sasakian manifolds in dimension 11 and 15 respectively. The metric cones on these are generalizations of the Kronheimer hyperk\"ahler metric on the regular 12-dimensional maximal nilpotent orbit of ${\bf s}{\bf l}(3,{\bf C})}$ and the 16-dimensional orbit of ${\bf s}{\bf o}(6,{\bf C})}. Ours are the first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some more general torus reductions which yield examples of non-toric 3-Sasakian orbifold metrics in dimensions 7. As one of the results we obtain explicit families of compact self-dual positive scalar curvature Einstein metrics with orbifold singularities and with only one Killing vector field. We further discuss the classcation of all toric reductions of classical 3-Sasakian homogeneous spces. Paul Gauduchon (Palaiseau) Compact Weakly Self-Dual Kaehler Surfaces A Kaehler complex surface is called {\it weakly self-dual} is its self-dual Weyl tensor is harmonic. It turns out that weakly self-dual Kaehler surfaces are extremal, in the sense of Calabi; moreover, they generically admit a$T ^2$-torus action and are explicitely integrable (in the self-dual case, this has been discovered independently by Apostolov-Gauduchon and by Bryant, who also proved similar facts for Bochner-flat Kaehler manifolds in all dimensions). While compact self-dual Kaehler surfaces are known to be locally symmetric or the product of two Riemann surfaces of opposite constant curvatures, compact weakly self-dual Kaehler surfaces with non-constant scalar curvature do appear as members of the familly of extremal Kaehler metrics constructed by Calabi on the first Hirzebruch surface. Conversely, all compact weakly self-dual Kaehler surfaces with non-constant curvature are of this type (joint work with V. Apostolov and D. Calderbank). Stefan Ivanov (Sofia) Parallel spinors, special holonomy and string backgrounds Henrik Pedersen (Odense) Neutral Ricci-flat nilmanifolds Victor Pidstrigatch (Göttingen) Symplectomorphisms of noncommutative complex plane and instantons on S^4 This is part of a bigger project about an action of the mentioned group on the moduli space of based instantons on S^4. Comparing this action with the action on Hilb scheme, one can show that generic instantons are not "particle-like", i.e. there is no natural way to assocociate to a generic instanton a point on a Hilb. Yat Sun Poon (Riverside) Moduli of complex structures associated to Heisenberg groups Chris Pope (Texas A&M) New cohomogeneity one metrics of Spin(7) holonomy Lorenz Schwachhöfer (Leipzig) Cohomogeneity-one manifolds of almost-nonnegative curvature McKenzie Wang (McMaster) The cohomogeneity one Ricci-flat equations and conserved quantities Zhuang-Dan Guan (Riverside) (Either) Toward a classification of compact hyperkähler manifolds (or) Geodesic stability and extremal Kähler metrics on compact almost homogeneous manifolds of cohomogeneity one Rafael Herrera (Riverside) A-hat-genus on non-spin manifolds and quaternion-Kähler 12-manifolds Uwe Semmelmann (Munich) An upper bound for a Hilbert polynomial on quaternionic Kähler manifolds On a quaternionic Kähler manifold of positive scalar curvature one can define a Hilbert polynomial. This is done using the contact line bundle on the twistor space or the index of a twisted Dirac operator. S. Salamon showed that this polynomial contains interesting information on the quaternionic Kähler manifold, e.g., the dimension of the isometry group. In this talk we discuss a sharp upper bound. As corollaries we obtain bounds on the volume and the degree of the associated twistor space. The proof is based on representation theory and a interpretation of the kernel of twisted Dirac operators as minimal eigenspaces of certain natural 2nd order differential operators. Bobby Acharya (Rutgers) String theory and Special Holonomy Manifolds José M. Figueroa-O'Farrill (Edinburgh) A theorem in eleven-dimensional supergravity Sir Michael Atiyah (Edinburgh) The Riemannian Geometry of Manifolds in dimensions 6 and 7 There has been much interest by physicists recently in 7-manifolds of G_2 holonomy and of actions by the circle on them. This is connected with M-theory and its reduction to string theory. I will describe some aspects of this story and its relation to the deformation of special Lagrangian 3-folds in C^3, due to Joyce. There are still many geometric problems to be solved. Jürgen Berndt (Hull) Cohomogeneity One Actions Abstract: ps, pdf David Calderbank (Edinburgh) Selfdual Einstein metrics with torus symmetry I Maciej Dunajski (Oxford) Four-manifolds with a parallel real spinor I consider anti-self-dual metrics in the$(++--)$signature which admit a covariantly constant real spinor are studied. It is shown that finding such metrics reduces to solving a fourth order integrable PDE, and some examples are given. The corresponding twistor space is characterised by existence of a canonical rank-one divisor line bundle. Jerome Gauntlett (QMW, London) Branes wrapping calibrated cycles in supergravity Jan Bernard Gutowski (QMW, London) Moduli spaces and brane solitons for M-theory compactifications on holonomy G_2 manifolds The moduli space of geometry of G_2 structures which arise in M-theory compactifications on holonomy G_2 manifolds is examined, and brane solitons of N=1, D=4 supergravity are found. Alexei Kovalev (Edinburgh) Riemannian 7-manifolds with holonomy G_2 We give a `generalized connected sum' construction of compact Riemannian 7-manifolds with special holonomy G_2. We also find families of K3 surfaces embedded as calibrated (coassociative) submanifolds and discuss the existence of calibrated K3 fibrations of the ambient G_2 manifolds. Radu Pantilie (Leeds) Harmonic morphisms from self-dual and Einstein manifolds We present the classification of harmonic morphisms with one-dimensional fibres from self-dual and from Einstein manifolds of dimension greater than or equal to four. The result can be viewed as the local classification results for self-dual and Einstein metrics which admit a certain normal form of R. L. Bryant. John Rawnsley (Warwick) The curvature of symplectic connections Bill Spence (QMW, London) Conformal Symmetry and Twisted N=4 SYM Ian Strachan (Hull) Frobenius submanifolds Richard Thomas (Oxford) Lagrangians and special Lagrangians I will discuss a Hitchin-Kobayashi-type correspondence for producing SLags from Lags John C. Wood (Leeds) Jacobi Fields Along Harmonic Maps A Jacobi field along a harmonic map is said to be \emph{integrable} if it is tangent to a smooth deformation through harmonic maps. Integrability gives information on the space of harmonic maps as well as on the singular set of energy minimizing maps. Well-known examples for closed geodesics show that Jacobi fields are not always integrable and integrability has only been established in a small number of cases. We show that several properties of a harmonic map are preserved to first order under variations along a Jacobi field and this allows us to establish that any Jacobi field along any harmonic 2-sphere in the complex projective plane$CP^2$is integrable, with implications for the moduli space and for energy minimizing maps in$CP^2\$.
Misha Verbitsky (Glasgow)
Hodge theory on hyperKähler manifolds with torsion
HKT manifolds are non-Kähler; however, harmonic analysis gives marvellous results
Isabel Dotti (Córdoba)
Invariant Abelian Complex Structures
We will discuss results on the construction of Abelian hypercomplex structures on homogeneous spaces and the construction of HKT structures
Eugene Ferapontov (Loughborough)
Systems of conservation laws and projective theory of congruences
In this talk I will discuss a correspondenc between hyperbolic systems of conservation laws and congruences of lines in projective space. In the particular case of linearly degenerate systems of Temple s class the correspondence proposed leads to congruences whose developable surfaces are planar pencils of lines. The example of the system of equations of associativity will be discussed, the properties of which are closely related to projective differential geometry of the Veronese variety
Yasuyuki Nagatomo (Tsukuba, Japan)
Twistor sections and their application to quaternion instantons
It is shown that the zero locus of a solution of the twistor equation coupled to a quaternion ASD connection is a quaternionic submanifold of a hyper- or quaternion-Kähler manifold
Representations of the fundamental group of a surface in PU(p,q) and holomorphic triples
We count the connected components in the moduli space of PU(p,q)-representations of the fundamental group for a closed oriented surface. The components are labelled by pairs of integers which arise as topological invariants of the flat bundles associated to the representations. Our results show that for each allowed value of these invariants, which are bounded by a Milnor--Wood type inequality, there is a unique non-empty connected component. Interpreting the moduli space of representations as a moduli space of Higgs bundles, we take a Morse theoretic approach using a certain smooth proper function on the Higgs moduli space. A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. We prove that these moduli spaces of triples are non-empty and irreducible.
Joint work with S. Bradlow and P. Gothen.
Florin Belgun (Humboldt)
Sasakian and normal CR structures on compact 3-manifolds
We classify all Sasakian structures on compact 3-manifolds, and their CR automorphism groups.
Srdjan Vukmirovic (Hull)
Para-quaternionic Kähler reduction
Para-quaternions are a real Clifford algebra is isomorphic to the algebra of real 2x2 matrices. We say that a pseudo-Riemannian manifold M is para-quaternionic if Hol(M) is contained in Sp(n,R)Sp(1,r). For these manifolds we develop the reduction technique and give examples.