# η = (e 1 (e 2 φ)) # = e 3

Size: px
Start display at page:

Download "η = (e 1 (e 2 φ)) # = e 3"

Transcription

1 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 holonomy group contained in the group G 2. The second is a question of rigidity of totally geodesic cycles in riemannian symmetric spaces. In the future, I propose to continue to work on these questions, and also to study the gauge theory on manifolds of exceptional holonomy. Minimal Surfaces in Manifolds of G 2 Holonomy One idiosycrasy in dimension 7 is the fact that one can define a constant 3-form on R 7 that determines the multiplication of the octonion algebra O = R 8. More precisely, one can define φ(x, Y, Z) = X, Y Z where X, Y and Z are purely imaginary octonions. φ is a antisymmetric and nondegenerate. Moreover, φ determines the metric. In the context of Riemannian geometry, this means that a manifold M that supports a form ϕ that is at each point isomorphic to φ canonically admits a metric g ϕ. If the form ϕ is parallel with respect to the metric g ϕ, (M, g ϕ ) is said to be a a G 2 -manifold because the Riemannian holonomy group is necessarily isomorphic to a subgroup of the exceptional Lie group G 2. By the classification theorem of Berger, there exist only seven possibilities for the holonomy group for a non-symmetric and irreducible Riemannian manifold. Among these sits the group G 2. It was only relatively recently that the first example of a manifold of holonomy precisely G 2 was discovered. Over the last fifteen years their importance has become more evident: they have appeared in M- theory in high energy physics and are closely related to the geometry of Calabi-Yau manifolds and of self-duality in 4-dimensions. We show that the submanifolds of a manifold of G 2 -holonomy can be studied using the techniques of holomorphic bundles and of pseudo-holomorphic curves. Using the techniques of calibrated geometry, the minimal submanifolds of dimensions 3 and 4 of a G 2 manifold have been much studied. For submanifolds of dimension 2, we show that strong analogies exist with the minimal surfaces of 4-dimensional spaces. Just as in that case, we see that X admits a twistor space Z X and the minimal submanifolds of X are intimately related to the complex curves in Z X. We consider a 2 dimensional submanifold Σ X, where X is a G 2 manifold. To such a submanifold there exists canonically a normal vector field η. For example in R 7, in which case φ takes the form φ = e e 1 (e 45 e 67 ) + e 2 (e 46 e 75 ) + e 3 e 47 e 56 ). In this case, if the tangent space to Σ, T Σ = span{e 1, e 2 } is generated by e 1 and e 2 the normal vector is given by η = (e 1 (e 2 φ)) # = e 3 where denotes the contraction of a vector in a form and # denotes the musical duality. In particular, this permits us to define a Gauss map that lifts the surface Σ to a surface Σ in the unit 1

2 tangent bundle Z X = U 1 (T X ). We can consider this bundle to be the twistor space to the manifold X. Additionally, we can show that U 1 (T X ) admits a distribution E by hyperplanes, to which the Gauss lift Σ is always tangent. E admits canonically a hermitian structure, defined by the form ϕ. We define the notion of an adapted submanifold Σ X for which we have the following result. Theorem 1. The Gauss lift Σ of an adapted surface in a G 2 -manifold is pseudo-holomorphic with respect to the hermitian structure on E if and only if Σ is a minimal submanifold. The condition that a submanifold be adapted in this case is that the second fundamental form, considered as a symmetric tensor with values in the normal bundle, takes values everywhere orthogonal to the vector field η. Theorem 2. A surface Σ X is adapted in this sense if and only if the Levi-Civita connection defines a holomorphic structure on the bundle W = η N Σ,X, which is the set of normal vectors orthogonal to η. We would like to generalise and extend these results to manifolds with Spin(7) holonomy. In this case, the twistor space admits an almost complex structure. We would also like to use symplectic geometry and the techniques there-in of pseudo-holomorphic curves (à la Gromov and Labourie) to study adapted minimal surfaces. This work is submitted for publication and is available at arxiv:math.dg/ Rigidity of Totally Geodesic Cycles A simply connected symmetric space of compact type decomposes into a product of irreducible factors. We can show that the factors of rank one, with one possible exceptional case, are rigid when considered among minimal submanifolds. More precisely, let X = X 1 X 2 be a Riemannian symmetric space of compact type where X 1 is irreducible and of rank one. We exclude the example of the Cayley plane OP 2. Let f be the standard immersion of X 1 into X as a totally geodesic factor. Theorem 3. There exists a C 3 neighbourhood U of f in the set of immersions such that every minimal immersion contained in U is equivalent to f. The notion of equivalence that we use here is by the action of ambiant isometries and reparametrisations of the submanifold. In this sense, the non-exceptional components of rank one of the symmetric space are rigid, and isolated from minimal submanifolds of another type. Here we note that, except for the Cayley plane, every compact symmetric space of rank one admits a riemannian submersion from a sphere with totally geodesic fibres. We can hence deduce the preceding result from the corresponding one for X 1 = S n. We describe the result in that case. In his article of 1968, Simons showed that the second fundamental form A of a minimal submanifold satisfies the non-linear second order equation 2 A = A A A A + R(A) + R, (1) 2

3 if A is considered a section of a Riemannian vector bundle. Here A and A are quadratic expression in A and R(A) and R are respectively obtained by contraction of A into the ambient curvature and its covariant derivative R. We consider a closed n-dimensional submanifold M of S n X 2 where X 2 is a compact type symmetric space and we make a hypothesis on the size of the derivative of the projection to the second factor. Theorem 4. There exists C > 0 such that for all 0 < Λ < 1 and for all closed minimal submanifold M of dimension n for which π 2 Λ the term R(A) satisfies R(A), ( A 2ρ CΛ 2) A 2. Here, π 2 denotes the derivative of the map π 2 : M X 2, and ρ is the infimum of the Ricci curvature in directions tangent to S n X 2. This is to say that in particular ρ > 0. In particular, by integration by parts and using the equation 1, we obtain the inequality 0 A 2( A 2 ρ ) q M where q is a constant that depends only on the dimensions of the spaces. If M is a closed minimal submanifold in S n X 2 such that π 2 and A are sufficiently small, then the right hand side of this inequality cannot be positive. Therefore, Corollary 5. There exists Λ > 0 such that if M is a closed minimal submanifold that satisfies π 2 Λ and A 2 < ρ/q, then M is totally geodesic. Λ can be chosen sufficiently small so that this implies that M = S n {pt}. This implies the Theorem 3. The question for the other factors of rank one are resolved individually, for the cases of the real, complex and quaternionic projective spaces. If a closed minimal submanifold of KP n X 2 is sufficiently close to the projective factor, then it must be a totally geodesic factor. In this case though, the proximity is with respect to the scalar curvature of the intrinsic metric on the submanifold, rather than the size of the second fundamental form. This work is to appear in the Annales de l Institut Fourier, volume 61 of 2011, and is currently available at arxiv: Gauge Theory on Manifolds of G 2 Holonomy As mentioned above, the geometry of manifolds of G 2 holonomy shows many similarities with Riemannian geometry in 3 and 4 dimensions. Motivated by these analogies, we have commenced a programme of research on gauge theory on manifolds of G 2 holonomy. Gauge theory in higher dimensions is today an active field of research; important results have already been proven by Donaldson, Segal and Thomas, Tao and Tian, Lewis, Brendle and most recently Sá Earp. One problem that we propose to study is of the existence of instantons on G 2 manifolds. If (X, ϕ, g ϕ ) is a manifold of G 2 holonomy, one can see that the bundle of 2-forms Λ 2 TX decomposes into two factors Λ 2 T X = Λ 2 7 Λ

4 of dimensions 7 and 14 respectively. In particular, Λ 2 14 = ker{ ϕ : Λ2 Λ 6 } where ϕ is the Hodge dual of ϕ. A connection A on a vector bundle E X is a G 2 -instanton if the curvature of A satisfies ϕ F A = 0. This condition can be seen as the analogue of the equation F A = F A that defines anti-self-dual connections on a 4-dimensional manifold. The first construction of a complete and irreducible metric of holonomy G 2 was done by Bryant and Salamon. They considered the 7-dimensional manifold given as the total space of the spinor bundle on S 3 X = (F H)/SU(2), where F is the principal fibre bundle given by the spin structure on S 3. Bryant and Salamon showed that X admits an explicit and asymptotically conic metric with holonomy equal to G 2. Additionally, X tautologically admits a hermitian vector bundle of complex rank 2. In work in progress we have shown that X admits two forms A 1 and A 2 such that if A = fa 1 + ga 2 where f and g are functions of the radius of the fibre, the connection = d + A is a G 2 -instanton if and only if f and g satisfy a system of ordinary differential equations. This project is still in progress. Other interesting questions to consider include to determine the domain of definition of the connections that are obtained, and to determine the integrability of the curvature. Gauge Theory on the Kummer Surface Following the construction of Bryant and and Salamon, the first example of a compact Riemannian manifold woth G 2 holonomy was constructed by Joyce. He considered the orbifold X s = T 7 /Γ where Γ is a finite group of G 2 -automorphisms of the flat 7-torus. Γ was chosen so that a neighbourhood of the singular locus in the quotient was diffeomorphic to a set of copies of T 3 C 2 /Z 2. C 2 /Z 2 has as a resolution the space Y that supports the well known metric of Eguchi-Hanson. Using this metric, Joyce constructed an approximate G 2 metric ϕ ε on the desingularisation X of X s and using non-linear analysis was able to obtain a 3-form ϕ = ϕ ε + dη that determined a metric of holonomy G 2. We would like to, for ε 1, construct via a gluing theorem a connection A on a fibre bundle E X such that ϕ F A = 0. The construction of Joyce has been called a generalised Kummer construction because of its similarity with the K3 surface of Kummer M T 4 /Z 2. For all 0 < ε 1 we can graft the Eguchi-Hanson metric on Y to a flat metric to obtain a Kähler metric ω ε on M. Similarly, we can graft an instanton A on Y to a trivial connection to obtain a connection A ε on a bundle over M. We consider also an analogous question in gauge theory 4-dimensions. This is closely analogous to the 7-dimensional case and seems to exhibit many of the same properties as that more complicated 4

5 geometry. We would like to construct, again by a gluing argument, anti-self-dual connections on K3 surfaces that degenerate towards the quotient T 4 /Z 2. We search for a solution (f, α) to a non-linear differential equation such that 1. ω ε + i f is a Ricci-flat Kähler metric, 2. A ε + α is an anti-self-dual connection with respect to the metric ω ε + i f. Our method follows that of Donaldson, who chooses a good conformal model to construct a Ricciflat Kähler metric. It is also closely related to the original gluing results of Taubes. This work is in progress. References [BS] [D] [DT] [ES] R.L. Bryant, S. Salamon, On the construction of some metrics with exceptional holonomy, Duke Math. J. 58 (1989), S.K. Donaldson, Calabi-Yau metrics on Kummer surfaces as a model gluing problem, arxiv:math.dg/ v1. S.K. Donaldson, R.P. Thomas, Gauge theory in higher dimensions, The geometric universe, Oxford Univ. Press (1998) J. Eells, S, Salamon, Constructions twistorielles des applications harmoniques, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) [J] D.D. Joyce, Compact riemannian 7-manifolds with holonomy G 2. I.. J. Diff. Geom. 43 (1996) [Lab] F. Labourie, Immersions isométriques elliptiques et courbes pseudo-holomorphes. J. Diff. Geom., 30 (1989) [Law] H.B. Lawson Jr., Rigidity theorems in rank-1 symmetric spaces, J. Diff. Geom., 4 (1970) [N] H. Nakajima, Moduli spaces of anti-self-dual connections on ALE gravitational instantons, Invent. math., 102, (1990) [S] J. Simons, Minimal varieties in riemannian manifolds. Ann. of Math. 88 (1968) [T] C.H. Taubes, Self-dual connections on 4-manifolds with indefinite intersection matrix. J. Diff. Geom. 19 (1984)

### 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.

### 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

### 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)

### 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

### 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

### 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

### 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

### 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

### The 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

### 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

### 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

### 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

### 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

### On 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,

### 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

### Minimal surfaces in quaternionic symmetric spaces

From: "Geometry of low-dimensional manifolds: 1", C.U.P. (1990), pp. 231--235 Minimal surfaces in quaternionic symmetric spaces F.E. BURSTALL University of Bath We describe some birational correspondences

### 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

### 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.

### RICCI 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

### 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

### How 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

### Manifolds 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

### As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).

An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted

### 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

### How 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

### Geometric 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

### CALIBRATED 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

### Conjectures 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.

### Critical 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

### Theorem 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

### 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

### THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS

THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS AILANA FRASER AND JON WOLFSON Abstract. In this paper we study the topology of compact manifolds of positive isotropic

### 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

### Parallel 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

### 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

### 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

### Riemannian 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

### arxiv: v3 [hep-th] 22 Sep 2014

Prepared for submission to JHEP IPhT-t14/056 Spin(7-manifolds in compactifications to four dimensions arxiv:1405.3698v3 [hep-th] 22 Sep 2014 Mariana Graña, a C. S. Shahbazi a and Marco Zambon b a Institut

### 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)

### 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

### LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS

LAGRANGIAN HOMOLOGY CLASSES WITHOUT REGULAR MINIMIZERS JON WOLFSON Abstract. We show that there is an integral homology class in a Kähler-Einstein surface that can be represented by a lagrangian twosphere

### COLLAPSING MANIFOLDS OBTAINED BY KUMMER-TYPE CONSTRUCTIONS

COLLAPSING MANIFOLDS OBTAINED BY KUMMER-TYPE CONSTRUCTIONS GABRIEL P. PATERNAIN AND JIMMY PETEAN Abstract. We construct F-structures on a Bott manifold and on some other manifolds obtained by Kummer-type

### 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

### 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

### 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

### Ω 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

### 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

### Periodic-end Dirac Operators and Positive Scalar Curvature. Daniel Ruberman Nikolai Saveliev

Periodic-end Dirac Operators and Positive Scalar Curvature Daniel Ruberman Nikolai Saveliev 1 Recall from basic differential geometry: (X, g) Riemannian manifold Riemannian curvature tensor tr scalar curvature

### From 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

### 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

### Poisson modules and generalized geometry

arxiv:0905.3227v1 [math.dg] 20 May 2009 Poisson modules and generalized geometry Nigel Hitchin May 20, 2009 Dedicated to Shing-Tung Yau on the occasion of his 60th birthday 1 Introduction Generalized complex

### HARMONIC MAPS INTO GRASSMANNIANS AND A GENERALIZATION OF DO CARMO-WALLACH THEOREM

Proceedings of the 16th OCU International Academic Symposium 2008 OCAMI Studies Volume 3 2009, pp.41 52 HARMONIC MAPS INTO GRASSMANNIANS AND A GENERALIZATION OF DO CARMO-WALLACH THEOREM YASUYUKI NAGATOMO

### 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

### Metrics and Holonomy

Metrics and Holonomy Jonathan Herman The goal of this paper is to understand the following definitions of Kähler and Calabi-Yau manifolds: Definition. A Riemannian manifold is Kähler if and only if it

### Heterotic 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.,

### Mathematical 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

### NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD

NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD INDRANIL BISWAS Abstract. Our aim is to review some recent results on holomorphic principal bundles over a compact Kähler manifold.

### Scalar curvature and the Thurston norm

Scalar curvature and the Thurston norm P. B. Kronheimer 1 andt.s.mrowka 2 Harvard University, CAMBRIDGE MA 02138 Massachusetts Institute of Technology, CAMBRIDGE MA 02139 1. Introduction Let Y be a closed,

### arxiv: v1 [math.dg] 2 Oct 2015

An estimate for the Singer invariant via the Jet Isomorphism Theorem Tillmann Jentsch October 5, 015 arxiv:1510.00631v1 [math.dg] Oct 015 Abstract Recently examples of Riemannian homogeneous spaces with

### September 27, :51 WSPC/INSTRUCTION FILE biswas-loftin. Hermitian Einstein connections on principal bundles over flat affine manifolds

International Journal of Mathematics c World Scientific Publishing Company Hermitian Einstein connections on principal bundles over flat affine manifolds Indranil Biswas School of Mathematics Tata Institute

### 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

### Pietro 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

### A 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

### 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).

### 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

### TWISTOR 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

### 1 v >, which will be G-invariant by construction.

1. Riemannian symmetric spaces Definition 1.1. A (globally, Riemannian) symmetric space is a Riemannian manifold (X, g) such that for all x X, there exists an isometry s x Iso(X, g) such that s x (x) =

### Quantising 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

### 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

### Let F be a foliation of dimension p and codimension q on a smooth manifold of dimension n.

Trends in Mathematics Information Center for Mathematical Sciences Volume 5, Number 2,December 2002, Pages 59 64 VARIATIONAL PROPERTIES OF HARMONIC RIEMANNIAN FOLIATIONS KYOUNG HEE HAN AND HOBUM KIM Abstract.

### HOLONOMY RIGIDITY FOR RICCI-FLAT METRICS

HOLONOMY RIGIDITY FOR RICCI-FLAT METRICS BERND AMMANN, KLAUS KRÖNCKE, HARTMUT WEISS, AND FREDERIK WITT Abstract. On a closed connected oriented manifold M we study the space M (M) of all Riemannian metrics

### On the Virtual Fundamental Class

On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview Donaldson-Thomas theory: counting invariants

### MAPPING CLASS ACTIONS ON MODULI SPACES. Int. J. Pure Appl. Math 9 (2003),

MAPPING CLASS ACTIONS ON MODULI SPACES RICHARD J. BROWN Abstract. It is known that the mapping class group of a compact surface S, MCG(S), acts ergodically with respect to symplectic measure on each symplectic

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### Positive isotropic curvature and self-duality in dimension 4

Positive isotropic curvature and self-duality in dimension 4 Thomas Richard and Harish Seshadri Abstract We study a positivity condition for the curvature of oriented Riemannian 4-manifolds: the half-p

### 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

### 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

### Structure theorems for compact Kähler manifolds

Structure theorems for compact Kähler manifolds Jean-Pierre Demailly joint work with Frédéric Campana & Thomas Peternell Institut Fourier, Université de Grenoble I, France & Académie des Sciences de Paris

### Shifted Symplectic Derived Algebraic Geometry and generalizations of Donaldson Thomas Theory

Shifted Symplectic Derived Algebraic Geometry and generalizations of Donaldson Thomas Theory Lecture 2 of 3:. D-critical loci and perverse sheaves Dominic Joyce, Oxford University KIAS, Seoul, July 2018

### A Bird Eye s view: recent update to Extremal metrics

A Bird Eye s view: recent update to Extremal metrics Xiuxiong Chen Department of Mathematics University of Wisconsin at Madison January 21, 2009 A Bird Eye s view: recent update to Extremal metrics Xiuxiong

### Reduction 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

### ARITHMETICITY OF TOTALLY GEODESIC LIE FOLIATIONS WITH LOCALLY SYMMETRIC LEAVES

ASIAN J. MATH. c 2008 International Press Vol. 12, No. 3, pp. 289 298, September 2008 002 ARITHMETICITY OF TOTALLY GEODESIC LIE FOLIATIONS WITH LOCALLY SYMMETRIC LEAVES RAUL QUIROGA-BARRANCO Abstract.

### The Eells-Salamon twistor correspondence

The Eells-Salamon twistor correspondence Jonny Evans May 15, 2012 Jonny Evans () The Eells-Salamon twistor correspondence May 15, 2012 1 / 11 The Eells-Salamon twistor correspondence is a dictionary for

### Luis Guijarro, Lorenzo Sadun, and Gerard Walschap

PARALLEL CONNECTIONS OVER SYMMETRIC SPACES Luis Guijarro, Lorenzo Sadun, and Gerard Walschap Abstract. Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that

### 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

### Detecting 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

### Published as: J. Geom. Phys. 10 (1993)

HERMITIAN STRUCTURES ON HERMITIAN SYMMETRIC SPACES F. Burstall, O. Muškarov, G. Grantcharov and J. Rawnsley Published as: J. Geom. Phys. 10 (1993) 245-249 Abstract. We show that an inner symmetric space

### APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD. 1. Introduction

APPROXIMATE YANG MILLS HIGGS METRICS ON FLAT HIGGS BUNDLES OVER AN AFFINE MANIFOLD INDRANIL BISWAS, JOHN LOFTIN, AND MATTHIAS STEMMLER Abstract. Given a flat Higgs vector bundle (E,, ϕ) over a compact

### LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS

LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS Contents 1. Almost complex manifolds 1. Complex manifolds 5 3. Kähler manifolds 9 4. Dolbeault cohomology 11 1. Almost complex manifolds Almost complex structures.

### Lecture I: Overview and motivation

Lecture I: Overview and motivation Jonathan Evans 23rd September 2010 Jonathan Evans () Lecture I: Overview and motivation 23rd September 2010 1 / 31 Difficulty of exercises is denoted by card suits in

### SPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY

SPECTRAL ASYMMETRY AND RIEMANNIAN GEOMETRY M. F. ATIYAH, V. K. PATODI AND I. M. SINGER 1 Main Theorems If A is a positive self-adjoint elliptic (linear) differential operator on a compact manifold then

### CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =

CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.

### Darboux theorems for shifted symplectic derived schemes and stacks

Darboux theorems for shifted symplectic derived schemes and stacks Lecture 1 of 3 Dominic Joyce, Oxford University January 2014 Based on: arxiv:1305.6302 and arxiv:1312.0090. Joint work with Oren Ben-Bassat,

### Homological mirror symmetry via families of Lagrangians

Homological mirror symmetry via families of Lagrangians String-Math 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants

### 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

### Perspective on Geometric Analysis: Geometric Structures III

Perspective on Geometric Analysis: Geometric Structures III Shing-Tung Yau Harvard University Talk in UCLA 1 When the topological method of surgery and gluing fails, we have to find a method that does

### On Spectrum and Arithmetic

On Spectrum and Arithmetic C. S. Rajan School of Mathematics, Tata Institute of Fundamental Research, Mumbai rajan@math.tifr.res.in 11 August 2010 C. S. Rajan (TIFR) On Spectrum and Arithmetic 11 August

### Overview of classical mirror symmetry

Overview of classical mirror symmetry David Cox (notes by Paul Hacking) 9/8/09 () Physics (2) Quintic 3-fold (3) Math String theory is a N = 2 superconformal field theory (SCFT) which models elementary