Exotic nearly Kähler structures on S 6 and S 3 S 3
|
|
- Darleen Booker
- 5 years ago
- Views:
Transcription
1 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
2 G 2 cones and nearly Kähler 6 manifolds (M 6, g) smooth Riemannian manifold Riemannian cone over M: C(M) = R + M endowed with the (incomplete) Riemannian metric dr 2 + r 2 g Hol(C) G 2 parallel forms ϕ = r 2 dr ω + r 3 ReΩ, ϕ = r 3 dr ImΩ r 4 ω 2 dϕ = 0 = d ϕ the SU(3) structure (ω, Ω) on M satisfies { dω = 3ReΩ dimω = 2ω 2 (NK) A 6 manifold M endowed with an SU(3) structure satisfying (NK) is called a nearly Kähler (nk) 6 manifold. nk manifolds and real Killing spinors nk manifolds and Gray Hervella classes of almost Hermitian manifolds every nk 6 manifold M is Einstein with Scal = 30 = M is compact with π 1 (M) < = we will assume M simply connected
3 The four known examples 1951, Steenrod s book: S 6 ImO 1968, Gray Wolf s classification of 3 symmetric spaces: S 3 S 3 = SU(2) 3 / SU(2) CP 3 = Sp(2)/U(1) Sp(1) F 3 = SU(3)/T 2 3 symmetric space: G/K, where G is a compact semisimple Lie group and K is the fixed point set of an automorphism of G of order 3 CP 3 and F 3 are the twistor spaces of S 4 and CP 2 respectively Connection with G 2 holonomy noted only in the 1980 s, e.g. Bryant s 1987 first example of a full G 2 holonomy metric is C(F 3 ) G 2 cones as local models for isolated singularities of G 2 manifolds Infinitely many Calabi Yau, hyperkähler and Spin(7) cones 2005, Butruille: the four known examples are the only homogeneous nk 6 manifolds 2006, Bryant: local generality of nk structures on 6 manifolds the same as for Calabi Yau structures
4 Main Theorem and possible proof strategies Main Theorem (F. Haskins, 2015) There exists an inhomogeneous nearly Kähler structure on S 6 and S 3 S 3. Two natural strategies to find nk 6 manifolds: Symmetries: cohomogeneity one nk 6 manifolds Desingularisation of nk sine-cones Both play a role in the proof of the Main Theorem Simplest singular nk spaces: cross-sections of split G 2 cones R C for a Calabi Yau cone C. (N 5, g N ) smooth Sasaki Einstein, i.e. C(N) is a Calabi Yau (CY) cone The sine-cone over N: SC(N) = [0, π] N endowed with the Riemannian metric dr 2 + sin 2 r g N SC(N) is nk but has two isolated singularities each modelled on C(N) Try to desingularise SC(N) by gluing complete asymptotically conical (AC) CY 3 folds
5 The simplest nk sine-cone and desingularisations N = N 1,1 := SU(2) SU(2)/ U(1) S 2 S 3 C(N 1,1 ) is the conifold {z1 2 + z2 2 + z2 3 + z2 4 = 0} C4 Up to symmetries the conifold admits two AC CY desingularisations (Candelas de la Ossa, 1990): the small resolution: total space of O( 1) O( 1) P 1 tip of the cone replaced by a round totally geodesic holomorphic S 2 the smoothing: T S 3 tip of the cone replaced by a round totally geodesic special Lagrangian S 3 The conifold itself and its asymptotically conical CY desingularisations are cohomogeneity one: SU(2) SU(2) acts isometrically with generic orbit of codimension one Question: Can we desingularise SC(N 1,1 ) as a cohomogeneity one nk space?
6 Cohomogeneity one nk manifolds 2010, Podestà Spiro: possible group diagrams of complete cohomogeneity one nk 6 manifolds G K K + K M SU(2) SU(2) U(1) SU(2) SU(2) S 3 S 3 SU(2) SU(2) U(1) SU(2) U(1) SU(2) S 6 SU(2) SU(2) U(1) U(1) SU(2) SU(2) U(1) CP 3 SU(2) SU(2) U(1) U(1) SU(2) U(1) SU(2) S 2 S 4 SU(3) SU(2) SU(3) SU(3) S 6
7 Outline of the proof 1. Understand the local theory for cohomogeneity one nk 6 manifolds in a neighbourhood of the principal orbit 2. Understand local cohomogeneity one nk manifolds in a neighbourhood of a singular orbit 3. Match pairs of solutions extending smoothly over a singular orbit
8 Nearly hypo structures On (0, T ) N 1,1 we write ω = η dt + ω 1 Ω = (ω 2 + iω 3 ) (η + idt), where (η, ω 1, ω 2, ω 3 ) defines an invariant SU(2) structure on N 1,1 The (NK) equation decouples into static and evolution equations for a family of invariant SU(2) structures on N 1,1 The SU(2) structure induced on a hypersurface in a nk 6 manifold is called a nearly hypo structure. It satisfies dω 1 = 3η ω 2 d(η ω 3 ) = 2ω 1 ω 1 The evolution equations to obtain a nk manifold by flowing a nearly hypo structure are: t ω 1 = dη 3ω 3 t (η ω 2 ) = dω 3 t (η ω 3 ) = dω 2 + 4η ω 1 Proposition The space of invariant nearly hypo structures on N 1,1 is a smooth manifold diffeomorphic to an open connected subset of SO 0 (1, 2). Up to symmetries, there exists a 2 parameter family of local cohomogeneity one nk stuctures on (0, T ) N 1,1.
9 Solutions extending over a singular orbit Problem: recognise which local solutions extend to complete metrics Proceed in two steps; separate the two singular orbits that appear and study separately. 1. Understand which solutions extend over a given singular orbit. 2. Understand how to match a pair of solutions from the previous step. Podestà Spiro: up to symmetries possible singular orbits are SU(2) SU(2)/U(1) SU(2) S 2 SU(2) SU(2)/ SU(2) S 3 Proposition There exist two 1 parameter families {Ψ a } a>0 and {Ψ b } b>0 of solutions to the fundamental ODE system which extend smoothly over a singular orbit S 2 and S 3, respectively. As a, b 0, appropriately rescaled the local nk structures Ψ a and Ψ b converge to the CY structure on the small resolution and the smoothing, respectively.
10 Maximal volume orbits M complete = orbital volume function V has a unique maximum The space of invariant maximal volume orbits V: V R 2 V 1 on V and V = 1 precisely for the Sasaki Einstein structure on N 1,1 V {V C} is compact Proposition Every member of the families {Ψ a } a>0 and {Ψ b } b>0 has a unique maximal volume orbit. Proof uses a simple continuity argument in the parameter a or b Non-emptiness: 3 of 4 known homogeneous examples appear in the families {Ψ a } a>0 and {Ψ b } b>0 Openness follows from the nondegeneracy of critical points of V Closedness uses compactness of V {V C} plus standard ODE theory and basic comparison theory
11 Matching Strategy for finding complete nk metrics: Match pairs of solutions in the two families across their maximal volume orbits using discrete symmetries α, β continuous curves in V R 2 parametrising the maximal volume orbits of {Ψ a } a>0 and {Ψ b } b>0 (u, v) : V R 2 coordinates on V Discrete symmetries = reflections along the axes Matching means: curves α and β must intersect or self-intersect (after applying a discrete symmetry) or intersect either axis An intersection point of α with axis u = 0 gives S 2 S 4, with axis v = 0 gives CP 3 An intersection point of β with either axis gives S 3 S 3 An intersection point of α with β (after the action of a discrete symmetry) gives S 6 Question: How many intersection/self-intersection points do the curves α and β (and their images under the reflections) have?
12 The curves α and β The homogeneous nk structures on CP 3, S 6 and S 3 S 3 give us 3 points on the curves α and β: CP 3, S 3 S 3 gives intersection points of α and β with the axes; S 6 gives an intersection point between α and (the reflection of) β. Study limits of α and β curves as the parameters a and b 0: Proposition As a, b 0 Ψ a and Ψ b converge to the sine-cone over the standard Sasaki Einstein structure on N 1,1. Proof uses: convergence after rescaling to AC CY structures the Böhm functional B for cohom 1 Einstein metrics scale-invariance of B and the fact that B = 20V 2 5 on a max vol orbit SE metric on N 1,1 is the absolute min of V on space V of max vol orbits Maximal volume orbit of SC(N 1,1 ) corresponds to origin in R 2 u,v so curves α and β come out of the origin
13 An inhomogeneous nk structure on S 3 S 3 Aim: find an intersection point of the curve β with the axis v = 0 C(b) = # zeroes of v before the max vol orbit of Ψ b (u, v) satisfies the system { u = 3 λ v v = 4λu, zeroes of v are non-degenerate = C(b) can only jump when zero of v occurs on max vol orbit b = 1 is standard S 3 S 3 and we check C(b) = 1 Show that C(b) 2 for all b sufficiently small. Convergence to sine-cone implies λ(t) sin t as b 0 Compare to a solution of the Legendre equation (sin t u ) + 12 sin t u = 0 Theorem There exists at leat one b (0, 1) such that v = 0 on the max vol orbit and thus at least one smooth inhomogeneous nk structure on S 3 S 3.
14 An inhomogeneous nk structure on S 6 Aim: force α and β to intersect in a second point besides the one giving the homogeneous nk structure on S 6 Idea: use the two solutions on S 3 S 3 to find a closed bounded region D in the plane encircling the origin; D is given by an arc of the curve β and its reflections along the axes The α curve starts at the origin (as a 0); want to show that α eventually leaves D Proposition The curve α exits any compact subset of R 2 as a. Idea of proof: based on explicit Taylor series for solution Ψ a consider a particular (non geometric) rescaling of Ψ a Show that solutions to the rescaled equations converge smoothly to some limiting object Scaling used shows V max ca 4
15 Summary Conjecture The Main Theorem yields all (inhomogeneous) cohomogeneity one nk structures on simply connected 6 manifolds α H S 6 std 0.4 CP 3 std w2 0.2 S 3 S 3 new S 6 new β H S 3 S 3 std
16 Deformation theory of nk manifolds Moroianu Nagy Semmelmann (2008): infinitesimal nk deformations modulo gauge identified with co-closed primitive (1, 1) forms of eigenvalue 12 for the Laplacian Moroianu Semmelmann (2010): all homogeneous nearly Kähler manifolds are rigid except for F 3 Theorem (F., 2016) All the infinitesimal deformations of F 3 are obstructed Proof uses Hitchin s volume functionals to reinterpret NK structures as the zeroes of a smooth map Φ : Ω 3 exact Ω 4 exact Ω 1 Ω 3 exact Ω 4 exact Dirac-type operators on nk manifolds are used to show that the linearisation of Φ together with a gauge fixing condition is an elliptic linear PDE with kernel and cokernel equal to Ω 1,1 0 (12) Non-vanishing obstructions to lift the infinitesimal deformations of F 3 to second order deformations (cf. Koiso (1982) for the Einstein case)
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 informationTwo 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 informationConstructing 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 informationManifolds 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 informationManifolds with holonomy. Sp(n)Sp(1) SC in SHGAP Simon Salamon Stony Brook, 9 Sep 2016
Manifolds with holonomy Sp(n)Sp(1) SC in SHGAP Simon Salamon Stony Brook, 9 Sep 2016 The list 1.1 SO(N) U( N 2 ) Sp( N 4 )Sp(1) SU( N 2 ) Sp( N 4 ) G 2 (N =7) Spin(7) (N =8) All act transitively on S N
More informationALF 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 informationALF 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 informationSome new torsional local models for heterotic strings
Some new torsional local models for heterotic strings Teng Fei Columbia University VT Workshop October 8, 2016 Teng Fei (Columbia University) Strominger system 10/08/2016 1 / 30 Overview 1 Background and
More informationA Joint Adventure in Sasakian and Kähler Geometry
A Joint Adventure in Sasakian and Kähler Geometry Charles Boyer and Christina Tønnesen-Friedman Geometry Seminar, University of Bath March, 2015 2 Kähler Geometry Let N be a smooth compact manifold of
More informationCompact 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 informationη = (e 1 (e 2 φ)) # = e 3
Research Statement My research interests lie in differential geometry and geometric analysis. My work has concentrated according to two themes. The first is the study of submanifolds of spaces with riemannian
More informationSimon Salamon. Turin, 24 April 2004
G 2 METRICS AND M THEORY Simon Salamon Turin, 24 April 2004 I Weak holonomy and supergravity II S 1 actions and triality in six dimensions III G 2 and SU(3) structures from each other 1 PART I The exceptional
More informationHolonomy 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 informationDirac operators with torsion
Dirac operators with torsion Prof.Dr. habil. Ilka Agricola Philipps-Universität Marburg Golden Sands / Bulgaria, September 2011 1 Relations between different objects on a Riemannian manifold (M n, g):
More informationStable 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 informationComplete 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 informationAlgebraic 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 informationTWISTOR AND KILLING FORMS IN RIEMANNIAN GEOMETRY
TWISTOR AND KILLING FORMS IN RIEMANNIAN GEOMETRY Andrei Moroianu CNRS - Ecole Polytechnique Palaiseau Prague, September 1 st, 2004 joint work with Uwe Semmelmann Plan of the talk Algebraic preliminaries
More informationCALIBRATED 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 informationGeometric Structures in Mathematical Physics Non-existence of almost complex structures on quaternion-kähler manifolds of positive type
Geometric Structures in Mathematical Physics Non-existence of almost complex structures on quaternion-kähler manifolds of positive type Paul Gauduchon Golden Sands, Bulgaria September, 19 26, 2011 1 Joint
More informationarxiv:math.dg/ v3 29 Sep 2003
Lectures on special Lagrangian geometry arxiv:math.dg/0111111 v3 29 Sep 2003 1 Introduction Dominic Joyce Lincoln College, Oxford, OX1 3DR dominic.joyce@lincoln.ox.ac.uk Calabi Yau m-folds (M, J, ω, Ω)
More informationCritical metrics on connected sums of Einstein four-manifolds
Critical metrics on connected sums of Einstein four-manifolds University of Wisconsin, Madison March 22, 2013 Kyoto Einstein manifolds Einstein-Hilbert functional in dimension 4: R(g) = V ol(g) 1/2 R g
More informationHYPERKÄHLER 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 informationLinear 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 informationIntro to Geometry and Topology via G Physics and G 2 -manifolds. Bobby Samir Acharya. King s College London. and ICTP Trieste Ψ(1 γ 5 )Ψ
Intro to Geometry and Topology via G 2 10.07.2014 Physics and G 2 -manifolds Bobby Samir Acharya King s College London. µf µν = j ν dϕ = d ϕ = 0 and ICTP Trieste Ψ(1 γ 5 )Ψ The Rich Physics-Mathematics
More informationCOHOMOGENEITY ONE EINSTEIN-SASAKI 5-MANIFOLDS
COHOMOGENEITY ONE EINSTEIN-SASAKI 5-MANIFOLDS DIEGO CONTI Abstract. We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution
More informationEigenvalue estimates for Dirac operators with torsion II
Eigenvalue estimates for Dirac operators with torsion II Prof.Dr. habil. Ilka Agricola Philipps-Universität Marburg Metz, March 2012 1 Relations between different objects on a Riemannian manifold (M n,
More informationThe 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 informationHolonomy 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 informationThe exceptional holonomy groups and calibrated geometry
Proceedings of 12 th Gökova Geometry-Topology Conference pp. 110 139 Published online at GokovaGT.org The exceptional holonomy groups and calibrated geometry Dominic Joyce Dedicated to the memory of Raoul
More informationHyperkähler 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 informationMirror Symmetry: Introduction to the B Model
Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds
More informationHeterotic Torsional Backgrounds, from Supergravity to CFT
Heterotic Torsional Backgrounds, from Supergravity to CFT IAP, Université Pierre et Marie Curie Eurostrings@Madrid, June 2010 L.Carlevaro, D.I. and M. Petropoulos, arxiv:0812.3391 L.Carlevaro and D.I.,
More informationHermitian vs. Riemannian Geometry
Hermitian vs. Riemannian Geometry Gabe Khan 1 1 Department of Mathematics The Ohio State University GSCAGT, May 2016 Outline of the talk Complex curves Background definitions What happens if the metric
More informationTheorem 2. Let n 0 3 be a given integer. is rigid in the sense of Guillemin, so are all the spaces ḠR n,n, with n n 0.
This monograph is motivated by a fundamental rigidity problem in Riemannian geometry: determine whether the metric of a given Riemannian symmetric space of compact type can be characterized by means of
More informationMirror symmetry for G 2 manifolds
Mirror symmetry for G 2 manifolds based on [1602.03521] [1701.05202]+[1706.xxxxx] with Michele del Zotto (Stony Brook) 1 Strings, T-duality & Mirror Symmetry 2 Type II String Theories and T-duality Superstring
More informationHermitian Symmetric Spaces
Hermitian Symmetric Spaces Maria Beatrice Pozzetti Notes by Serena Yuan 1. Symmetric Spaces Definition 1.1. A Riemannian manifold X is a symmetric space if each point p X is the fixed point set of an involutive
More informationLagrangian Submanifolds with Constant Angle Functions in the Nearly Kähler S 3 S 3
Lagrangian Submanifolds with Constant Angle Functions in the Nearly Kähler S 3 S 3 Burcu Bektaş Istanbul Technical University, Istanbul, Turkey Joint work with Marilena Moruz (Université de Valenciennes,
More informationHeterotic Flux Compactifications
Heterotic Flux Compactifications Mario Garcia-Fernandez Instituto de Ciencias Matemáticas, Madrid String Pheno 2017 Virginia Tech, 7 July 2017 Based on arxiv:1611.08926, and joint work with Rubio, Tipler,
More informationM-Theory on Spin(7) Manifolds
HUTP-01/A041 ITEP-TH-47/01 DAMTP-2001-77 hep-th/0109025 arxiv:hep-th/0109025v2 12 Nov 2001 M-Theory on Spin(7) Manifolds Sergei Gukov Jefferson Physical Laboratory, Harvard University Cambridge, MA 02138,
More informationBobby Samir Acharya Kickoff Meeting for Simons Collaboration on Special Holonomy
Particle Physics and G 2 -manifolds Bobby Samir Acharya Kickoff Meeting for Simons Collaboration on Special Holonomy King s College London µf µν = j ν dϕ = d ϕ = 0 and ICTP Trieste Ψ(1 γ 5 )Ψ THANK YOU
More informationCALIBRATED FIBRATIONS ON NONCOMPACT MANIFOLDS VIA GROUP ACTIONS
DUKE MATHEMATICAL JOURNAL Vol. 110, No. 2, c 2001 CALIBRATED FIBRATIONS ON NONCOMPACT MANIFOLDS VIA GROUP ACTIONS EDWARD GOLDSTEIN Abstract In this paper we use Lie group actions on noncompact Riemannian
More informationRicci-flat metrics on complex cones
Ricci-flat metrics on complex cones Steklov Mathematical Institute, Moscow & Max-Planck-Institut für Gravitationsphysik (AEI), Potsdam-Golm, Germany Quarks 2014, Suzdal, 3 June 2014 arxiv:1405.2319 1/20
More informationModuli 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 informationConformal Killing forms on Riemannian manifolds
Conformal Killing forms on Riemannian manifolds Habilitationsschrift zur Feststellung der Lehrbefähigung für das Fachgebiet Mathematik in der Fakultät für Mathematik und Informatik der Ludwig-Maximilians-Universität
More informationRICCI SOLITONS ON COMPACT KAHLER SURFACES. Thomas Ivey
RICCI SOLITONS ON COMPACT KAHLER SURFACES Thomas Ivey Abstract. We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming
More informationReduction of Homogeneous Riemannian structures
Geometric Structures in Mathematical Physics, 2011 Reduction of Homogeneous Riemannian structures M. Castrillón López 1 Ignacio Luján 2 1 ICMAT (CSIC-UAM-UC3M-UCM) Universidad Complutense de Madrid 2 Universidad
More informationGeometry 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 informationRiemannian Curvature Functionals: Lecture III
Riemannian Curvature Functionals: Lecture III Jeff Viaclovsky Park City Mathematics Institute July 18, 2013 Lecture Outline Today we will discuss the following: Complete the local description of the moduli
More informationQuantising noncompact Spin c -manifolds
Quantising noncompact Spin c -manifolds Peter Hochs University of Adelaide Workshop on Positive Curvature and Index Theory National University of Singapore, 20 November 2014 Peter Hochs (UoA) Noncompact
More informationTorus actions and Ricci-flat metrics
Department of Mathematics, University of Aarhus November 2016 / Trondheim To Eldar Straume on his 70th birthday DFF - 6108-00358 Delzant HyperKähler G2 http://mscand.dk https://doi.org/10.7146/math.scand.a-12294
More informationGeometric inequalities for black holes
Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de Córdoba, CONICET, Argentina. 3 August, 2012 Einstein equations (vacuum) The spacetime is a four dimensional manifold M with
More informationGeometry of the Calabi-Yau Moduli
Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51
More informationCompact 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 informationMathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang
Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions
More informationHow to recognize a conformally Kähler metric
How to recognize a conformally Kähler metric Maciej Dunajski Department of Applied Mathematics and Theoretical Physics University of Cambridge MD, Paul Tod arxiv:0901.2261, Mathematical Proceedings of
More informationDifferential Geometry and its Applications
Differential eometry and its Applications 30 01 318 333 Contents lists available at SciVerse ScienceDirect Differential eometry and its Applications www.elsevier.com/locate/difgeo Soliton solutions for
More informationFrom holonomy reductions of Cartan geometries to geometric compactifications
From holonomy reductions of Cartan geometries to geometric compactifications 1 University of Vienna Faculty of Mathematics Berlin, November 11, 2016 1 supported by project P27072 N25 of the Austrian Science
More informationHow curvature shapes space
How curvature shapes space Richard Schoen University of California, Irvine - Hopf Lecture, ETH, Zürich - October 30, 2017 The lecture will have three parts: Part 1: Heinz Hopf and Riemannian geometry Part
More informationCalabi-Yau Fourfolds with non-trivial Three-Form Cohomology
Calabi-Yau Fourfolds with non-trivial Three-Form Cohomology Sebastian Greiner arxiv: 1512.04859, 1702.03217 (T. Grimm, SG) Max-Planck-Institut für Physik and ITP Utrecht String Pheno 2017 Sebastian Greiner
More informationOPEN PROBLEMS IN NON-NEGATIVE SECTIONAL CURVATURE
OPEN PROBLEMS IN NON-NEGATIVE SECTIONAL CURVATURE COMPILED BY M. KERIN Abstract. We compile a list of the open problems and questions which arose during the Workshop on Manifolds with Non-negative Sectional
More informationLECTURE 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 informationRigid Schubert classes in compact Hermitian symmetric spaces
Rigid Schubert classes in compact Hermitian symmetric spaces Colleen Robles joint work with Dennis The Texas A&M University April 10, 2011 Part A: Question of Borel & Haefliger 1. Compact Hermitian symmetric
More informationarxiv:math/ v1 [math.dg] 11 Oct 2004
DESINGULARIZATIONS OF CALABI YAU 3-FOLDS WITH A CONICAL SINGULARITY arxiv:math/0410260v1 [math.dg] 11 Oct 2004 YAT-MING CHAN Abstract We study Calabi Yau 3-folds M 0 with a conical singularity x modelled
More informationConification 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 informationA wall-crossing formula for 2d-4d DT invariants
A wall-crossing formula for 2d-4d DT invariants Andrew Neitzke, UT Austin (joint work with Davide Gaiotto, Greg Moore) Cetraro, July 2011 Preface In the last few years there has been a lot of progress
More informationThe moduli space of Ricci-flat metrics
The moduli space of Ricci-flat metrics B. Ammann 1 K. Kröncke 2 H. Weiß 3 F. Witt 4 1 Universität Regensburg, Germany 2 Universität Hamburg, Germany 3 Universität Kiel, Germany 4 Universität Stuttgart,
More informationModuli of heterotic G2 compactifications
Moduli of heterotic G2 compactifications Magdalena Larfors Uppsala University Women at the Intersection of Mathematics and High Energy Physics MITP 7.3.2017 X. de la Ossa, ML, E. Svanes (1607.03473 & work
More informationGeometria 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 informationAFFINE 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 informationConjectures on counting associative 3-folds in G 2 -manifolds
in G 2 -manifolds Dominic Joyce, Oxford University Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics, First Annual Meeting, New York City, September 2017. Based on arxiv:1610.09836.
More informationPietro Fre' SISSA-Trieste. Paolo Soriani University degli Studi di Milano. From Calabi-Yau manifolds to topological field theories
From Calabi-Yau manifolds to topological field theories Pietro Fre' SISSA-Trieste Paolo Soriani University degli Studi di Milano World Scientific Singapore New Jersey London Hong Kong CONTENTS 1 AN INTRODUCTION
More informationRICCI CURVATURE: SOME RECENT PROGRESS A PROGRESS AND OPEN QUESTIONS
RICCI CURVATURE: SOME RECENT PROGRESS AND OPEN QUESTIONS Courant Institute September 9, 2016 1 / 27 Introduction. We will survey some recent (and less recent) progress on Ricci curvature and mention some
More informationDonaldson 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 informationQuantising proper actions on Spin c -manifolds
Quantising proper actions on Spin c -manifolds Peter Hochs University of Adelaide Differential geometry seminar Adelaide, 31 July 2015 Joint work with Mathai Varghese (symplectic case) Geometric quantization
More informationCohomogeneity one hypersurfaces in CP 2 and CH 2
Cohomogeneity one hypersurfaces in CP 2 and CH 2 Cristina Vidal Castiñeira Universidade de Santiago de Compostela 1st September 2015 XXIV International Fall Workshop on Geometry and Physics, Zaragoza 2015
More informationSUMMARY OF THE KÄHLER MASS PAPER
SUMMARY OF THE KÄHLER MASS PAPER HANS-OACHIM HEIN AND CLAUDE LEBRUN Let (M, g) be a complete n-dimensional Riemannian manifold. Roughly speaking, such a space is said to be ALE with l ends (l N) if there
More informationString Theory and Generalized Geometries
String Theory and Generalized Geometries Jan Louis Universität Hamburg Special Geometries in Mathematical Physics Kühlungsborn, March 2006 2 Introduction Close and fruitful interplay between String Theory
More informationOn the 5-dimensional Sasaki-Einstein manifold
Proceedings of The Fourteenth International Workshop on Diff. Geom. 14(2010) 171-175 On the 5-dimensional Sasaki-Einstein manifold Byung Hak Kim Department of Applied Mathematics, Kyung Hee University,
More informationCitation 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 informationGENERALIZED VECTOR CROSS PRODUCTS AND KILLING FORMS ON NEGATIVELY CURVED MANIFOLDS. 1. Introduction
GENERALIZED VECTOR CROSS PRODUCTS AND KILLING FORMS ON NEGATIVELY CURVED MANIFOLDS Abstract. Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we
More informationΩ Ω /ω. To these, one wants to add a fourth condition that arises from physics, what is known as the anomaly cancellation, namely that
String theory and balanced metrics One of the main motivations for considering balanced metrics, in addition to the considerations already mentioned, has to do with the theory of what are known as heterotic
More informationK-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 informationParallel and Killing Spinors on Spin c Manifolds. 1 Introduction. Andrei Moroianu 1
Parallel and Killing Spinors on Spin c Manifolds Andrei Moroianu Institut für reine Mathematik, Ziegelstr. 3a, 0099 Berlin, Germany E-mail: moroianu@mathematik.hu-berlin.de Abstract: We describe all simply
More informationEinstein Metrics, Minimizing Sequences, and the. Differential Topology of Four-Manifolds. Claude LeBrun SUNY Stony Brook
Einstein Metrics, Minimizing Sequences, and the Differential Topology of Four-Manifolds Claude LeBrun SUNY Stony Brook Definition A Riemannian metric is said to be Einstein if it has constant Ricci curvature
More informationDIFFERENTIAL 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 informationH-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 informationMoment 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 informationThe 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 informationTakao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,...
J. Korean Math. Soc. 40 (2003), No. 4, pp. 667 680 HOMOGENEOUS POLYNOMIAL HYPERSURFACE ISOLATED SINGULARITIES Takao Akahori Abstract. The mirror conjecture means originally the deep relation between complex
More informationDetecting submanifolds of minimum volume with calibrations
Detecting submanifolds of minimum volume with calibrations Marcos Salvai, CIEM - FaMAF, Córdoba, Argentina http://www.famaf.unc.edu.ar/ salvai/ EGEO 2016, Córdoba, August 2016 It is not easy to choose
More informationQuasi-local mass and isometric embedding
Quasi-local mass and isometric embedding Mu-Tao Wang, Columbia University September 23, 2015, IHP Recent Advances in Mathematical General Relativity Joint work with Po-Ning Chen and Shing-Tung Yau. The
More informationNew Aspects of Heterotic Geometry and Phenomenology
New Aspects of Heterotic Geometry and Phenomenology Lara B. Anderson Harvard University Work done in collaboration with: J. Gray, A. Lukas, and E. Palti: arxiv: 1106.4804, 1202.1757 J. Gray, A. Lukas and
More informationCANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES
CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES JULIUS ROSS This short survey aims to introduce some of the ideas and conjectures relating stability of projective varieties to the existence of
More informationHiggs 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 informationOn 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 informationThe topology of symplectic four-manifolds
The topology of symplectic four-manifolds Michael Usher January 12, 2007 Definition A symplectic manifold is a pair (M, ω) where 1 M is a smooth manifold of some even dimension 2n. 2 ω Ω 2 (M) is a two-form
More informationLuminy Lecture 2: Spectral rigidity of the ellipse joint work with Hamid Hezari. Luminy Lecture April 12, 2015
Luminy Lecture 2: Spectral rigidity of the ellipse joint work with Hamid Hezari Luminy Lecture April 12, 2015 Isospectral rigidity of the ellipse The purpose of this lecture is to prove that ellipses are
More informationTopics in Geometry: Mirror Symmetry
MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:
More informationGeneralized N = 1 orientifold compactifications
Generalized N = 1 orientifold compactifications Thomas W. Grimm University of Wisconsin, Madison based on: [hep-th/0602241] Iman Benmachiche, TWG [hep-th/0507153] TWG Madison, Wisconsin, November 2006
More information