Torus actions and Ricci-flat metrics

Size: px
Start display at page:

Download "Torus actions and Ricci-flat metrics"

Transcription

1 Department of Mathematics, University of Aarhus November 2016 / Trondheim To Eldar Straume on his 70th birthday DFF

2 Delzant HyperKähler G2

3 Outline Delzant HyperKähler G2 1 The Delzant picture 2 HyperKähler manifolds Dimension four Toric hyperkähler 3 G 2 manifolds

4 Delzant HyperKähler G2 The Delzant picture (M 2n, ω) symplectic with a Hamiltonian action of T n = The moment map µ : M R n = t identifies the orbit space M/T n with a convex polytope in R n. Each such (M 2n, ω) may be constructed as a symplectic quotient of R 2m by an Abelian subgroup of T m.

5 Delzant HyperKähler G2 Symplectic moment maps Let (M, ω) be symplectic: dω = 0. If X X(M) preserves ω, then Cartan s formula gives 0 = L X ω = d(x ω), so (locally) X ω = dµ X for some µ X : M R. For G Abelian acting on M preserving ω, the action is Hamiltonian if there is a G-invariant map µ : M g such that d µ, X = X ω for each X g. M simply-connected and G connected Abelian = Hamiltonian all orbits are isotropic ω(x, Y) = 0 for all X, Y g.

6 Delzant HyperKähler G2 The Delzant polytope Faces of = µ(m) are of the form F k = { µ, u k = λ k } and the Delzant polytope is = {a R n a, u k λ k k}. u k R n with points over (F k ) having stabiliser the subtorus with Lie algebra {v R n = t v, u k = 0}, so u k Q n. Smoothness of M is equivalent to: F k1 F kr = = the corresponding u k1,..., u kr are part of a Z-basis. This restricts the possible u i s locally.

7 Delzant HyperKähler G2 4d toric HyperKähler manifolds (M, ω I, ω J, ω K ) is hyperkähler if: 1 each ω A is a symplectic two-form, 2 the tangent bundle endomorphisms I = ω 1 K ω J, etc., satisfy I 2 = 1 = J 2 = K 2, IJ = K = JI, etc., and g = ω A (A, ) is independent of A and positive definite. Consequences dim M = 4n, g is Ricci-flat, with holonomy contained in Sp(n) SU(2n).

8 Delzant HyperKähler G2 4d toric Symmetry considerations Ricci-flatness implies: if M is compact, then any Killing vector field is parallel, so the holonomy of M reduces and M splits as a product, if M is homogeneous then g is flat, so M is a quotient of flat R 4n by a discrete group (Alekseevskiĭ and Kimel fel d 1975). Take (M, g) complete and G Abelian group of tri-holomorphic isometries. Assume the action is tri-hamiltonian, so there is a hyperkähler moment map: a G-invariant map with d µ A, X = X ω A. This forces 4 dim G dim M. µ = (µ I, µ J, µ K ) : M R 3 g

9 Delzant HyperKähler G2 4d toric Gibbons-Hawking Ansatz in 4d X a tri-hamiltonian vector field on hyperkähler M 4 Away from M X, locally g = 1 V (dt + ω)2 + V(dx 2 + dy 2 + dz 2 ) where V = 1/g(X, X), dx = X ω I = dµ I, etc., and on R 3. In particular, dω = 3 dv µ = (µ I, µ J, µ K ) is locally a conformal submersion to (R 3, dx 2 + dy 2 + dz 2 ), V is locally a harmonic function on R 3.

10 Examples Delzant HyperKähler G2 4d toric V(p) = c i Z 1 p p i, c 0, p i R 3 distinct c = 0, Z < : multi-euguchi Hanson metrics Z space flat R 4 T CP(1)... c > 0, Z < : multi-taub-nut metrics Z space flat S 1 R 3 Taub-NUT R 4 T CP(1)... Z countably infinite: require V(p) to converge at some p R 3, get A metrics (Anderson et al. 1989; Goto 1994), e.g. Z = N >0, p n = (n 2, 0, 0), and their Taub-NUT deformations.

11 Delzant HyperKähler G2 4d toric Classification Theorem (Swann 2016) The above potentials V(p) = c i Z 1 p p i, c 0, p i R 3 distinct, with 0 < V(p) < for some p R 3, classify all complete hyperkähler four-manifolds with tri-hamiltonian circle action. When Z <, this is due to Bielawski (1999), and the first parts of the proof are essentially the same.

12 Delzant HyperKähler G2 4d toric Proof structure The only special orbits are fixed points µ : M/S 1 R 3 is a local homeomorphism near a fixed point x, V(µ(y)) = 1 2 µ(y) µ(x) 1 + φ(µ(y)) with φ positive harmonic (Bôcher s Theorem; a Chern class) µ : N 3 = (M \ M X )/S 1 R 3 is conformal: conformal factor V, positive harmonic can adjust to V superharmonic, so that N becomes complete and use Schoen and Yau (1994) to show µ : N R 3 is injective with image Ω having boundary that is polar V is then given by a Martin integral representation supported on Ω; completeness of M forces Ω to be discrete.

13 Delzant HyperKähler G2 4d toric Toric hyperkähler (Dancer and Swann 2016) M 4n complete hyperkähler with tri-hamiltonian action of T n. Is given locally by the Pedersen-Poon Ansatz: g = (V 1 ) ij (dt + ω i )(dt + ω j ) + V ij (dx i dx j + dy i dy j + dz i dz j ), with V ij = 2 F x i x j with F a positive function on R 3 R n harmonic on every affine three-plane X a,v = a + R 3 v. For generic X a,v, then Y = µ 1 (X a,v ) is smooth with free T n 1 action, Y/T n 1 is complete hyperkähler of dimension 4 with S 1 -action. Above analysis then fixes V on X a,v, and F, providing a classification. All examples may be constructed as hyperkähler quotients of flat affine subspaces of Hilbert spaces, cf. Goto 1994; Hattori 2011.

14 Delzant HyperKähler G2 4d toric Hypertoric configuration data In the hypertoric situation µ is surjective: µ(m 4n ) = R 3n = R 3 R n = Im H R n. Polytope faces are replaced by affine flats of codimension 3: H k = H(u k, λ k ) = {a Im H R n a, u k = λ k }, u k R n, λ k Im H. Again stabilisers of points mapping to H k are contained in the subtorus with Lie algebra orthogonal to u k, forcing u k Q n. This time H(u k1, λ k ) H(u kn, λ kr ) = whenever u k1,..., u kr are linearly independent. Smoothness implies each such set u k1,..., u kr is part of a Z-basis for Z n, giving a global restriction on the u k s. Get only finitely many distinct vectors u k, but possibly infinitely many λ k s.

15 Delzant HyperKähler G2 G 2 manifolds M 7 with ϕ Ω 3 (M) pointwise of the form ϕ = e e e e 246 e 257 e 356 e 347, e ijk = e i e j e k. ϕ specifies a metric g and an orientation The holonomy of g lies in G 2 when dϕ = 0 = d ϕ g is then Ricci-flat

16 Delzant HyperKähler G2 Multi-moment maps: T 2 symmetry (Madsen and Swann 2012) Suppose T 2 acts preserving (M, ϕ), holonomy in G 2, with generating vector fields U, V. The Cartan formula implies U V ϕ is closed. A function ν : M R with dν = U V ϕ is called a multi-moment map. At regular values X 4 = ν 1 (x)/t 2 is a four manifold carrying three symplectic forms of the same orientation induced by U ϕ, V ϕ, U V ϕ. These do not form a hyperkähler structure in general. (M, ϕ) may be recovered from the four-manifold X 4 by building a T 2 -bundle Y 6, constructing an SU(3) geometry (σ, ψ + ) on this bundle and then using an adaptation of the Hitchin flow for time derivatives of these forms.

17 Delzant HyperKähler G2 Multi-moment maps: T 3 symmetry Suppose T 3 acts preserving (M, ϕ), holonomy in G 2, with generating vector fields U, V, W. Multi-moment maps ν U, ν V, ν W given by dν U = V W ϕ, etc. Hamiltonian condition is that this should be T 3 -invariant, is equivalent to ϕ(u, V, W) = 0. There is a fourth multi-moment µ associated to ϕ via dµ = U V W ϕ. (ν U, ν V, ν W, µ) : M 7 R 4 has generic fibre T 3. What is the analogue of the Gibbons-Hawking Ansatz?

18 References I References Alekseevskiĭ, D. V. and B. N. Kimel fel d (1975), Structure of homogeneous Riemannian spaces with zero Ricci curvature, Funktsional. Anal. i Prilozhen. 9:2, pp. 5 11, trans. as Structure of homogeneous Riemannian spaces with zero Ricci curvature, Functional Anal. Appl. 9:2, pp Anderson, M. T., P. B. Kronheimer and C. LeBrun (1989), Complete Ricci-flat Kähler manifolds of infinite topological type, Comm. Math. Phys. 125:4, pp Bielawski, R. (1999), Complete hyper-kähler 4n-manifolds with a local tri-hamiltonian R n -action, Math. Ann. 314:3, pp Dancer, A. S. and A. F. Swann (2016), Hypertoric manifolds and hyperkähler moment maps, arxiv: [math.dg].

19 References II References Goto, R. (1994), On hyper-kähler manifolds of type A, Geometric and Funct. Anal. 4, pp Hattori, K. (2011), The volume growth of hyper-kähler manifolds of type A, Journal of Geometric Analysis 21:4, pp Madsen, T. B. and A. F. Swann (2012), Multi-moment maps, Adv. Math. 229, pp Schoen, R. and S.-T. Yau (1994), Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu, Translated from the Chinese by Ding and S. Y. Cheng, Preface translated from the Chinese by Kaising Tso, International Press, Cambridge, MA, pp. v+235.

20 References III References Swann, A. F. (2016), Twists versus modifications, Adv. Math. 303, pp

Moment Maps and Toric Special Holonomy

Moment Maps and Toric Special Holonomy Department of Mathematics, University of Aarhus PADGE, August 2017, Leuven DFF - 6108-00358 Delzant HyperKähler G 2 Outline The Delzant picture HyperKähler manifolds Hypertoric Construction G 2 manifolds

More information

DIFFERENTIAL GEOMETRY AND THE QUATERNIONS. Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013

DIFFERENTIAL GEOMETRY AND THE QUATERNIONS. Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013 DIFFERENTIAL GEOMETRY AND THE QUATERNIONS Nigel Hitchin (Oxford) The Chern Lectures Berkeley April 9th-18th 2013 16th October 1843 ON RIEMANNIAN MANIFOLDS OF FOUR DIMENSIONS 1 SHIING-SHEN CHERN Introduction.

More information

Holonomy groups. Thomas Leistner. School of Mathematical Sciences Colloquium University of Adelaide, May 7, /15

Holonomy groups. Thomas Leistner. School of Mathematical Sciences Colloquium University of Adelaide, May 7, /15 Holonomy groups Thomas Leistner School of Mathematical Sciences Colloquium University of Adelaide, May 7, 2010 1/15 The notion of holonomy groups is based on Parallel translation Let γ : [0, 1] R 2 be

More information

Holonomy groups. Thomas Leistner. Mathematics Colloquium School of Mathematics and Physics The University of Queensland. October 31, 2011 May 28, 2012

Holonomy groups. Thomas Leistner. Mathematics Colloquium School of Mathematics and Physics The University of Queensland. October 31, 2011 May 28, 2012 Holonomy groups Thomas Leistner Mathematics Colloquium School of Mathematics and Physics The University of Queensland October 31, 2011 May 28, 2012 1/17 The notion of holonomy groups is based on Parallel

More information

Stable bundles on CP 3 and special holonomies

Stable bundles on CP 3 and special holonomies Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M

More information

MODIFYING HYPERKÄHLER MANIFOLDS WITH CIRCLE SYMMETRY

MODIFYING HYPERKÄHLER MANIFOLDS WITH CIRCLE SYMMETRY ASIAN J. MATH. c 2006 International Press Vol. 10, No. 4, pp. 815 826, December 2006 010 MODIFYING HYPERKÄHLER MANIFOLDS WITH CIRCLE SYMMETRY ANDREW DANCER AND ANDREW SWANN Abstract. A construction is

More information

HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

More information

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9

More information

Linear connections on Lie groups

Linear connections on Lie groups Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)

More information

SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

More information

Delzant s Garden. A one-hour tour to symplectic toric geometry

Delzant s Garden. A one-hour tour to symplectic toric geometry Delzant s Garden A one-hour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem

More information

Hyperkähler geometry lecture 3

Hyperkähler geometry lecture 3 Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843

More information

The Strominger Yau Zaslow conjecture

The Strominger Yau Zaslow conjecture The Strominger Yau Zaslow conjecture Paul Hacking 10/16/09 1 Background 1.1 Kähler metrics Let X be a complex manifold of dimension n, and M the underlying smooth manifold with (integrable) almost complex

More information

ALF spaces and collapsing Ricci-flat metrics on the K3 surface

ALF spaces and collapsing Ricci-flat metrics on the K3 surface ALF spaces and collapsing Ricci-flat metrics on the K3 surface Lorenzo Foscolo Stony Brook University Simons Collaboration on Special Holonomy Workshop, SCGP, Stony Brook, September 8 2016 The Kummer construction

More information

Higgs Bundles and Character Varieties

Higgs Bundles and Character Varieties Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character

More information

Marsden-Weinstein Reductions for Kähler, Hyperkähler and Quaternionic Kähler Manifolds

Marsden-Weinstein Reductions for Kähler, Hyperkähler and Quaternionic Kähler Manifolds Marsden-Weinstein Reductions for Kähler, Hyperkähler and Quaternionic Kähler Manifolds Chenchang Zhu Nov, 29th 2000 1 Introduction If a Lie group G acts on a symplectic manifold (M, ω) and preserves the

More information

Two simple ideas from calculus applied to Riemannian geometry

Two simple ideas from calculus applied to Riemannian geometry Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University

More information

Global aspects of Lorentzian manifolds with special holonomy

Global aspects of Lorentzian manifolds with special holonomy 1/13 Global aspects of Lorentzian manifolds with special holonomy Thomas Leistner International Fall Workshop on Geometry and Physics Évora, September 2 5, 2013 Outline 2/13 1 Lorentzian holonomy Holonomy

More information

The topology of asymptotically locally flat gravitational instantons

The topology of asymptotically locally flat gravitational instantons The topology of asymptotically locally flat gravitational instantons arxiv:hep-th/0602053v4 19 Jan 2009 Gábor Etesi Department of Geometry, Mathematical Institute, Faculty of Science, Budapest University

More information

ALF spaces and collapsing Ricci-flat metrics on the K3 surface

ALF spaces and collapsing Ricci-flat metrics on the K3 surface ALF spaces and collapsing Ricci-flat metrics on the K3 surface Lorenzo Foscolo Stony Brook University Recent Advances in Complex Differential Geometry, Toulouse, June 2016 The Kummer construction Gibbons

More information

Citation Osaka Journal of Mathematics. 49(3)

Citation Osaka Journal of Mathematics. 49(3) Title ON POSITIVE QUATERNIONIC KÄHLER MAN WITH b_4=1 Author(s) Kim, Jin Hong; Lee, Hee Kwon Citation Osaka Journal of Mathematics. 49(3) Issue 2012-09 Date Text Version publisher URL http://hdl.handle.net/11094/23146

More information

K-stability and Kähler metrics, I

K-stability and Kähler metrics, I K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates

More information

HAMILTONIAN ACTIONS IN GENERALIZED COMPLEX GEOMETRY

HAMILTONIAN ACTIONS IN GENERALIZED COMPLEX GEOMETRY HAMILTONIAN ACTIONS IN GENERALIZED COMPLEX GEOMETRY TIMOTHY E. GOLDBERG These are notes for a talk given in the Lie Groups Seminar at Cornell University on Friday, September 25, 2009. In retrospect, perhaps

More information

Solvable Lie groups and the shear construction

Solvable Lie groups and the shear construction Solvable Lie groups and the shear construction Marco Freibert jt. with Andrew Swann Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel 19.05.2016 1 Swann s twist 2 The shear construction The

More information

LIST OF PUBLICATIONS. Mu-Tao Wang. March 2017

LIST OF PUBLICATIONS. Mu-Tao Wang. March 2017 LIST OF PUBLICATIONS Mu-Tao Wang Publications March 2017 1. (with P.-K. Hung, J. Keller) Linear stability of Schwarzschild spacetime: the Cauchy problem of metric coefficients. arxiv: 1702.02843v2 2. (with

More information

The Explicit Construction of Einstein Finsler Metrics with Non-Constant Flag Curvature

The Explicit Construction of Einstein Finsler Metrics with Non-Constant Flag Curvature Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 2009, 045, 7 pages The Explicit Construction of Einstein Finsler Metrics with Non-Constant Flag Curvature Enli GUO, Xiaohuan MO and

More information

Algebraic geometry over quaternions

Algebraic geometry over quaternions Algebraic geometry over quaternions Misha Verbitsky November 26, 2007 Durham University 1 History of algebraic geometry. 1. XIX centrury: Riemann, Klein, Poincaré. Study of elliptic integrals and elliptic

More information

The Calabi Conjecture

The Calabi Conjecture The Calabi Conjecture notes by Aleksander Doan These are notes to the talk given on 9th March 2012 at the Graduate Topology and Geometry Seminar at the University of Warsaw. They are based almost entirely

More information

How to recognise a conformally Einstein metric?

How to recognise a conformally Einstein metric? How to recognise a conformally Einstein metric? Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:1304.7772., Comm. Math. Phys. (2014).

More information

Self-dual conformal gravity

Self-dual conformal gravity Self-dual conformal gravity Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:1304.7772., Comm. Math. Phys. (2014). Dunajski (DAMTP, Cambridge)

More information

arxiv:math/ v1 [math.dg] 16 Mar 2003

arxiv:math/ v1 [math.dg] 16 Mar 2003 arxiv:math/0303191v1 [math.dg] 16 Mar 2003 Gravitational Instantons, Confocal Quadrics and Separability of the Schrödinger and Hamilton-Jacobi equations. G. W. Gibbons C.M.S, Cambridge University, Wilberforce

More information

AFFINE SPHERES AND KÄHLER-EINSTEIN METRICS. metric makes sense under projective coordinate changes. See e.g. [10]. Form a cone (1) C = s>0

AFFINE SPHERES AND KÄHLER-EINSTEIN METRICS. metric makes sense under projective coordinate changes. See e.g. [10]. Form a cone (1) C = s>0 AFFINE SPHERES AND KÄHLER-EINSTEIN METRICS JOHN C. LOFTIN 1. Introduction In this note, we introduce a straightforward correspondence between some natural affine Kähler metrics on convex cones and natural

More information

Conification of Kähler and hyper-kähler manifolds and supergr

Conification of Kähler and hyper-kähler manifolds and supergr Conification of Kähler and hyper-kähler manifolds and supergravity c-map Masaryk University, Brno, Czech Republic and Institute for Information Transmission Problems, Moscow, Russia Villasimius, September

More information

Questions arising in open problem sessions in AIM workshop on L 2 -harmonic forms in geometry and string theory

Questions arising in open problem sessions in AIM workshop on L 2 -harmonic forms in geometry and string theory Questions arising in open problem sessions in AIM workshop on L 2 -harmonic forms in geometry and string theory Notes by: Anda Degeratu, Mark Haskins 1 March, 17, 2004 L. Saper as moderator Question [Sobolev-Kondrakov

More information

THE GEOMETRY OF B-FIELDS. Nigel Hitchin (Oxford) Odense November 26th 2009

THE GEOMETRY OF B-FIELDS. Nigel Hitchin (Oxford) Odense November 26th 2009 THE GEOMETRY OF B-FIELDS Nigel Hitchin (Oxford) Odense November 26th 2009 THE B-FIELD IN PHYSICS B = i,j B ij dx i dx j flux: db = H a closed three-form Born-Infeld action: det(g ij + B ij ) complexified

More information

H-projective structures and their applications

H-projective structures and their applications 1 H-projective structures and their applications David M. J. Calderbank University of Bath Based largely on: Marburg, July 2012 hamiltonian 2-forms papers with Vestislav Apostolov (UQAM), Paul Gauduchon

More information

Moduli Space of Higgs Bundles Instructor: Marco Gualtieri

Moduli Space of Higgs Bundles Instructor: Marco Gualtieri Moduli Space of Higgs Bundles Instructor: Lecture Notes for MAT1305 Taught Spring of 2017 Typeset by Travis Ens Last edited January 30, 2017 Contents Lecture Guide Lecture 1 [11.01.2017] 1 History Hyperkähler

More information

Twistor constructions of quaternionic manifolds and asymptotically hyperbolic Einstein Weyl spaces. Aleksandra Borówka. University of Bath

Twistor constructions of quaternionic manifolds and asymptotically hyperbolic Einstein Weyl spaces. Aleksandra Borówka. University of Bath Twistor constructions of quaternionic manifolds and asymptotically hyperbolic Einstein Weyl spaces submitted by Aleksandra Borówka for the degree of Doctor of Philosophy of the University of Bath Department

More information

Morse theory and stable pairs

Morse theory and stable pairs Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction

More information

arxiv: v1 [math.sg] 26 Jan 2015

arxiv: v1 [math.sg] 26 Jan 2015 SYMPLECTIC ACTIONS OF NON-HAMILTONIAN TYPE ÁLVARO PELAYO arxiv:1501.06480v1 [math.sg] 26 Jan 2015 In memory of Professor Johannes (Hans) J. Duistermaat (1942 2010) Abstract. Hamiltonian symplectic actions

More information

Infinitesimal Einstein Deformations. Kähler Manifolds

Infinitesimal Einstein Deformations. Kähler Manifolds on Nearly Kähler Manifolds (joint work with P.-A. Nagy and U. Semmelmann) Gemeinsame Jahrestagung DMV GDM Berlin, March 30, 2007 Nearly Kähler manifolds Definition and first properties Examples of NK manifolds

More information

arxiv: v1 [math.dg] 29 Oct 2011

arxiv: v1 [math.dg] 29 Oct 2011 Thomas Bruun Madsen and Andrew Swann Abstract arxiv:1110.6541v1 [math.dg] 9 Oct 011 We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental

More information

Multi-moment maps. CP 3 Journal Club. Thomas Bruun Madsen. 20th November 2009

Multi-moment maps. CP 3 Journal Club. Thomas Bruun Madsen. 20th November 2009 Multi-moment maps CP 3 Journal Club Thomas Bruun Madsen 20th November 2009 Geometry with torsion Strong KT manifolds Strong HKT geometry Strong KT manifolds: a new classification result Multi-moment maps

More information

Higher codimension CR structures, Levi-Kähler reduction and toric geometry

Higher codimension CR structures, Levi-Kähler reduction and toric geometry Higher codimension CR structures, Levi-Kähler reduction and toric geometry Vestislav Apostolov (Université du Québec à Montréal) May, 2015 joint work in progress with David Calderbank (University of Bath);

More information

A NOTE ON RIGIDITY OF 3-SASAKIAN MANIFOLDS

A NOTE ON RIGIDITY OF 3-SASAKIAN MANIFOLDS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 10, Pages 3027 3034 S 0002-9939(99)04889-3 Article electronically published on April 23, 1999 A NOTE ON RIGIDITY OF 3-SASAKIAN MANIFOLDS

More information

Complete integrability of geodesic motion in Sasaki-Einstein toric spaces

Complete integrability of geodesic motion in Sasaki-Einstein toric spaces Complete integrability of geodesic motion in Sasaki-Einstein toric spaces Mihai Visinescu Department of Theoretical Physics National Institute for Physics and Nuclear Engineering Horia Hulubei Bucharest,

More information

Twisted Poisson manifolds and their almost symplectically complete isotropic realizations

Twisted Poisson manifolds and their almost symplectically complete isotropic realizations Twisted Poisson manifolds and their almost symplectically complete isotropic realizations Chi-Kwong Fok National Center for Theoretical Sciences Math Division National Tsing Hua University (Joint work

More information

Cohomology of the Mumford Quotient

Cohomology of the Mumford Quotient Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten

More information

Constructing compact 8-manifolds with holonomy Spin(7)

Constructing compact 8-manifolds with holonomy Spin(7) Constructing compact 8-manifolds with holonomy Spin(7) Dominic Joyce, Oxford University Simons Collaboration meeting, Imperial College, June 2017. Based on Invent. math. 123 (1996), 507 552; J. Diff. Geom.

More information

arxiv:alg-geom/ v1 29 Jul 1993

arxiv:alg-geom/ v1 29 Jul 1993 Hyperkähler embeddings and holomorphic symplectic geometry. Mikhail Verbitsky, verbit@math.harvard.edu arxiv:alg-geom/9307009v1 29 Jul 1993 0. ntroduction. n this paper we are studying complex analytic

More information

A Convexity Theorem For Isoparametric Submanifolds

A Convexity Theorem For Isoparametric Submanifolds A Convexity Theorem For Isoparametric Submanifolds Marco Zambon January 18, 2001 1 Introduction. The main objective of this paper is to discuss a convexity theorem for a certain class of Riemannian manifolds,

More information

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.

B 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X. Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2

More information

Hyperbolic Geometry on Geometric Surfaces

Hyperbolic Geometry on Geometric Surfaces Mathematics Seminar, 15 September 2010 Outline Introduction Hyperbolic geometry Abstract surfaces The hemisphere model as a geometric surface The Poincaré disk model as a geometric surface Conclusion Introduction

More information

Geometria Simplettica e metriche Hermitiane speciali

Geometria Simplettica e metriche Hermitiane speciali Geometria Simplettica e metriche Hermitiane speciali Dipartimento di Matematica Universitá di Torino 1 Marzo 2013 1 Taming symplectic forms and SKT geometry Link with SKT metrics The pluriclosed flow Results

More information

Compact Riemannian manifolds with exceptional holonomy

Compact Riemannian manifolds with exceptional holonomy Unspecified Book Proceedings Series Compact Riemannian manifolds with exceptional holonomy Dominic Joyce published as pages 39 65 in C. LeBrun and M. Wang, editors, Essays on Einstein Manifolds, Surveys

More information

Elements of Topological M-Theory

Elements of Topological M-Theory Elements of Topological M-Theory (with R. Dijkgraaf, S. Gukov, C. Vafa) Andrew Neitzke March 2005 Preface The topological string on a Calabi-Yau threefold X is (loosely speaking) an integrable spine of

More information

Rigidity for Multi-Taub-NUT metrics

Rigidity for Multi-Taub-NUT metrics Rigidity for Multi-Taub-NUT metrics Vincent Minerbe To cite this version: Vincent Minerbe. Rigidity for Multi-Taub-NUT metrics. 2009. HAL Id: hal-00428917 https://hal.archives-ouvertes.fr/hal-00428917

More information

. TORIC KÄHLER METRICS: COHOMOGENEITY ONE EXAMPLES OF CONSTANT SCALAR CURVATURE IN ACTION-ANGLE COORDINATES

. TORIC KÄHLER METRICS: COHOMOGENEITY ONE EXAMPLES OF CONSTANT SCALAR CURVATURE IN ACTION-ANGLE COORDINATES . TORIC KÄHLER METRICS: COHOMOGENEITY ONE EXAMPLES OF CONSTANT SCALAR CURVATURE IN ACTION-ANGLE COORDINATES MIGUEL ABREU Abstract. In these notes, after an introduction to toric Kähler geometry, we present

More information

Homogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky

Homogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky Homogeneous para-kähler Einstein manifolds Dmitri V. Alekseevsky Hamburg,14-18 July 2008 1 The talk is based on a joint work with C.Medori and A.Tomassini (Parma) See ArXiv 0806.2272, where also a survey

More information

COMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD

COMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD COMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD MELINDA LANIUS 1. introduction Because Poisson cohomology is quite challenging to compute, there are only very select cases where the answer is

More information

Manifolds with exceptional holonomy

Manifolds with exceptional holonomy Manifolds with exceptional holonomy Simon Salamon Giornata INdAM, 10 June 2015, Bologna What is holonomy? 1.1 Holonomy is the subgroup generated by parallel transport of tangent vectors along all possible

More information

Donaldson Invariants and Moduli of Yang-Mills Instantons

Donaldson Invariants and Moduli of Yang-Mills Instantons Donaldson Invariants and Moduli of Yang-Mills Instantons Lincoln College Oxford University (slides posted at users.ox.ac.uk/ linc4221) The ASD Equation in Low Dimensions, 17 November 2017 Moduli and Invariants

More information

CALIBRATED GEOMETRY JOHANNES NORDSTRÖM

CALIBRATED GEOMETRY JOHANNES NORDSTRÖM CALIBRATED GEOMETRY JOHANNES NORDSTRÖM Summer school on Differential Geometry, Ewha University, 23-27 July 2012. Recommended reading: Salamon, Riemannian Geometry and Holonomy Groups [11] http://calvino.polito.it/~salamon/g/rghg.pdf

More information

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat

More information

arxiv:hep-th/ v2 14 Oct 1997

arxiv:hep-th/ v2 14 Oct 1997 T-duality and HKT manifolds arxiv:hep-th/9709048v2 14 Oct 1997 A. Opfermann Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK February

More information

Compact 8-manifolds with holonomy Spin(7)

Compact 8-manifolds with holonomy Spin(7) Compact 8-manifolds with holonomy Spin(7) D.D. Joyce The Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB. Published as Invent. math. 123 (1996), 507 552. 1 Introduction In Berger s classification

More information

Math 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim

Math 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim SOLUTIONS Dec 13, 218 Math 868 Final Exam In this exam, all manifolds, maps, vector fields, etc. are smooth. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each).

More information

Elementary Deformations HK/QK. Oscar MACIA. Introduction [0] Physics [8] Twist [20] Example [42]

Elementary Deformations HK/QK. Oscar MACIA. Introduction [0] Physics [8] Twist [20] Example [42] deformations and the HyperKaehler/Quaternionic Kaehler correspondence. (Joint work with Prof. A.F. SWANN) Dept. Geometry Topology University of Valencia (Spain) June 26, 2014 What do we have 1 2 3 4 Berger

More information

DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES. September 25, 2015

DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES. September 25, 2015 DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES MAGGIE MILLER September 25, 2015 1. 09/16/2015 1.1. Textbooks. Textbooks relevant to this class are Riemannian Geometry by do Carmo Riemannian Geometry

More information

Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1

Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1 Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines

More information

Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture)

Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) Building Geometric Structures: Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover

More information

Geometry and Dynamics of singular symplectic manifolds. Session 9: Some applications of the path method in b-symplectic geometry

Geometry and Dynamics of singular symplectic manifolds. Session 9: Some applications of the path method in b-symplectic geometry Geometry and Dynamics of singular symplectic manifolds Session 9: Some applications of the path method in b-symplectic geometry Eva Miranda (UPC-CEREMADE-IMCCE-IMJ) Fondation Sciences Mathématiques de

More information

An Invitation to Geometric Quantization

An Invitation to Geometric Quantization An Invitation to Geometric Quantization Alex Fok Department of Mathematics, Cornell University April 2012 What is quantization? Quantization is a process of associating a classical mechanical system to

More information

Conjectures in Kahler geometry

Conjectures in Kahler geometry Conjectures in Kahler geometry S.K. Donaldson Abstract. We state a general conjecture about the existence of Kahler metrics of constant scalar curvature, and discuss the background to the conjecture 1.

More information

Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields

Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields Chapter 16 Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields 16.1 Isometries and Local Isometries Recall that a local isometry between two Riemannian manifolds M

More information

Hard Lefschetz Theorem for Vaisman manifolds

Hard Lefschetz Theorem for Vaisman manifolds Hard Lefschetz Theorem for Vaisman manifolds Antonio De Nicola CMUC, University of Coimbra, Portugal joint work with B. Cappelletti-Montano (Univ. Cagliari), J.C. Marrero (Univ. La Laguna) and I. Yudin

More information

BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS

BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate

More information

Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields

Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields Chapter 15 Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields The goal of this chapter is to understand the behavior of isometries and local isometries, in particular

More information

Geometry of Ricci Solitons

Geometry of Ricci Solitons Geometry of Ricci Solitons H.-D. Cao, Lehigh University LMU, Munich November 25, 2008 1 Ricci Solitons A complete Riemannian (M n, g ij ) is a Ricci soliton if there exists a smooth function f on M such

More information

On the BCOV Conjecture

On the BCOV Conjecture Department of Mathematics University of California, Irvine December 14, 2007 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called

More information

ELEMENTARY DEFORMATIONS AND THE HYPERKAEHLER / QUATERNIONIC KAEHLER CORRESPONDENCE

ELEMENTARY DEFORMATIONS AND THE HYPERKAEHLER / QUATERNIONIC KAEHLER CORRESPONDENCE ELEMENTARY DEFORMATIONS AND THE HYPERKAEHLER / QUATERNIONIC KAEHLER CORRESPONDENCE Oscar Macia (U. Valencia) (in collaboration with Prof. A.F. Swann) Aarhus, DK March 2, 2017 1 Planning Introduction and

More information

Metrics of Special Curvature with Symmetry

Metrics of Special Curvature with Symmetry arxiv:math/0610738v1 [math.dg] 4 Oct 006 Metrics of Special Curvature with Symmetry Brandon Dammerman The Queen s College University of Oxford A thesis submitted for the degree of Doctor of Philosophy

More information

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory

More information

Quaternionic Complexes

Quaternionic Complexes Quaternionic Complexes Andreas Čap University of Vienna Berlin, March 2007 Andreas Čap (University of Vienna) Quaternionic Complexes Berlin, March 2007 1 / 19 based on the joint article math.dg/0508534

More information

Projective parabolic geometries

Projective parabolic geometries Projective parabolic geometries David M. J. Calderbank University of Bath ESI Wien, September 2012 Based partly on: Hamiltonian 2-forms in Kähler geometry, with Vestislav Apostolov (UQAM), Paul Gauduchon

More information

The structure of algebraic varieties

The structure of algebraic varieties The structure of algebraic varieties János Kollár Princeton University ICM, August, 2014, Seoul with the assistance of Jennifer M. Johnson and Sándor J. Kovács (Written comments added for clarity that

More information

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:

More information

Lecture Notes a posteriori for Math 201

Lecture Notes a posteriori for Math 201 Lecture Notes a posteriori for Math 201 Jeremy Kahn September 22, 2011 1 Tuesday, September 13 We defined the tangent space T p M of a manifold at a point p, and the tangent bundle T M. Zev Choroles gave

More information

Ricci-flat metrics on the complexification of a compact rank one symmetric space

Ricci-flat metrics on the complexification of a compact rank one symmetric space Ricci-flat metrics on the complexification of a compact rank one symmetric space Matthew B. Stenzel We construct a complete Ricci-flat Kähler metric on the complexification of a compact rank one symmetric

More information

zi z i, zi+1 z i,, zn z i. z j, zj+1 z j,, zj 1 z j,, zn

zi z i, zi+1 z i,, zn z i. z j, zj+1 z j,, zj 1 z j,, zn The Complex Projective Space Definition. Complex projective n-space, denoted by CP n, is defined to be the set of 1-dimensional complex-linear subspaces of C n+1, with the quotient topology inherited from

More information

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction

SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces

More information

Exotic nearly Kähler structures on S 6 and S 3 S 3

Exotic nearly Kähler structures on S 6 and S 3 S 3 Exotic nearly Kähler structures on S 6 and S 3 S 3 Lorenzo Foscolo Stony Brook University joint with Mark Haskins, Imperial College London Friday Lunch Seminar, MSRI, April 22 2016 G 2 cones and nearly

More information

Left-invariant metrics and submanifold geometry

Left-invariant metrics and submanifold geometry Left-invariant metrics and submanifold geometry TAMARU, Hiroshi ( ) Hiroshima University The 7th International Workshop on Differential Geometry Dedicated to Professor Tian Gang for his 60th birthday Karatsu,

More information

Changing sign solutions for the CR-Yamabe equation

Changing sign solutions for the CR-Yamabe equation Changing sign solutions for the CR-Yamabe equation Ali Maalaoui (1) & Vittorio Martino (2) Abstract In this paper we prove that the CR-Yamabe equation on the Heisenberg group has infinitely many changing

More information

Action-angle coordinates and geometric quantization. Eva Miranda. Barcelona (EPSEB,UPC) STS Integrable Systems

Action-angle coordinates and geometric quantization. Eva Miranda. Barcelona (EPSEB,UPC) STS Integrable Systems Action-angle coordinates and geometric quantization Eva Miranda Barcelona (EPSEB,UPC) STS Integrable Systems Eva Miranda (UPC) 6ecm July 3, 2012 1 / 30 Outline 1 Quantization: The general picture 2 Bohr-Sommerfeld

More information

Modern Geometric Structures and Fields

Modern Geometric Structures and Fields Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface

More information

L 2 Geometry of the Symplectomorphism Group

L 2 Geometry of the Symplectomorphism Group University of Notre Dame Workshop on Innite Dimensional Geometry, Vienna 2015 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of Outline 1 The Exponential Map on D s ω(m) 2 Existence

More information

Topological Solitons from Geometry

Topological Solitons from Geometry Topological Solitons from Geometry Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge Atiyah, Manton, Schroers. Geometric models of matter. arxiv:1111.2934.

More information

DIFFERENTIAL FORMS AND COHOMOLOGY

DIFFERENTIAL FORMS AND COHOMOLOGY DIFFERENIAL FORMS AND COHOMOLOGY ONY PERKINS Goals 1. Differential forms We want to be able to integrate (holomorphic functions) on manifolds. Obtain a version of Stokes heorem - a generalization of the

More information

Contents. Preface...VII. Introduction... 1

Contents. Preface...VII. Introduction... 1 Preface...VII Introduction... 1 I Preliminaries... 7 1 LieGroupsandLieAlgebras... 7 1.1 Lie Groups and an Infinite-Dimensional Setting....... 7 1.2 TheLieAlgebraofaLieGroup... 9 1.3 The Exponential Map..............................

More information