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

Size: px
Start display at page:

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

Transcription

1 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 somehow to find a metric on S, conformal to a given metric, that had constant curvature. Actually, that such a metric exists is guaranteed by the Uniformization Theorem as follows: Given any metric, there is a conformal (Riemann surface) structure, as we have already noted. By Uniformization, this structure is biholomorphic to the standard CP structure. So the pullback of the standard (round) Riemannian metric on S, which is associated to the usual complex structure on CP, solves our problem! But suppose we did not know about uniformization (we are not going to have it in higher dimensions!). How would we find the conformal constant curvature metric? (Finding it would of course PROVE the uniformization in this case.) What we need is a characterization of the constant curvature metric among all (conformal to the given metric) metrics. Here is how to do it: K d(area) d(area) Kd(area). But Kd(area) = 4π by the Gauss Bonnet Theorem. ( ) 4π So Kd(area) where area is in the given new, conformal metric. If we assume area that we have scaled by a constant so that the area is 4π, then we have Kd(area) 4π with equality if and only if K is (a constant multiple of). Actually, for this particular choice of area normalization, K is the constant, but K would have to be some constant, whichever area normalization we chose. By the Cauchy Schwarz inequality, ( ) ( ) Thus we have reduced finding a conformal constant curvature metric to a minimization problem. Similar considerations can be used to characterize the constant curvature metrics on other compact Riemann surfaces. This suggests that one really ought to look for "canonical metrics" that minimize curvature integrals of one sort or another. This idea can be carried quite far, as we shall see. It is, however, not entirely reasonable to expect to carry it out on compact Riemannian manifolds as a whole. Indeed, there are results of Nabutovsky and Weinberg that show that this cannot really be expected at all in complete generality.

2 What is reasonable to try is the following problem: Let M be a compact Kähler manifold with Kähler form ω. Then try to minimize the curvature square integral on M among all Kähler metrics with the same Kähler class as ω, that is with Kähler form that is in the cohomology class of ω. This is in fact exactly what we would be doing on compact Riemann surfaces if we looked for mimizing the integral of K within a conformal class of metrics with fixed area. The conformal class being fixed means we have a fixed complex structure and the family of Kähler metrics relative to it, and fixing the area corresponds precisely to fixing the cohomology class of the Kähler form(since H (M,R) is mapped isomorphically to R by evaluating classes on the oriented fundamental cycle for M, and this is equivalent for forms to integrating the Kähler form over M, which of course gives the area of M relative to the Kähler metric) Of course, the question arises: which curvature should we try to minimize the square integral of? As it happens, it does not matter! Calabi ("Extremal Kähler Metrics" in "Seminar on Differential Geometry", Yau(ed.), Princeton University Press, 98 ) showed that the differences among the integral of the square of the scalar curvature, (one half of) the integral of the norm of the square of the norm of the Ricci curvature, and (one quarter of) the integral of the square of the norm of the Riemann curvature tensor, all these differences are constants independent of the Kähler metric choice (within a fixed Kähler class). So there is really only one minimization question. This fact is not obvious, but it is not quite as surprising as might at first appear. Recall that if ω and ω are Kähler forms in the same cohomology class, then there is a global function such that the difference of the two corresponding metrics is the Levi form of the function. (This is just the so-called Hodge Lemma: If a real (,) form is d-exact, then it is f for some function f.) So all three of the apparently different minimization problems involve in fact one and the same function. Thus it is perhaps not so very surprising that the minimizations, which one thinks of as picking out a canonical one of the functions, should not depend on which minimization one chooses. This is of course not the proof! It just makes the observation plausible. In the cases (negative and zero) where the Calabi Conjecture is always true (Aubin Yau, Yau), the minimization (for all three minimization problems) is given by the Einstein- Kähler metric with the given polarization ( Kähler form). Thus, the important and by now familiar Einstein Kähler metrics are obtained by minimizing the square of the curvature, analogous to the previous discussion of Riemann surfaces. But one can extend this to a more general bundle situation. This is the essential idea of Yang Mills theory.

3 Namely, in the most general possible terms, suppose one has a vector bundle over a compact Riemannian manifold which has some real (non-torsion) characteristic class that is not zero. Then it is not possible for the bundle to have a metric that is arbitrarily uniformly close to 0, since there is a formula involving the curvature for the characteristic class in question. Like the Euler characteristic in our initial discussion, or the first Chern class of the canonical bundle in the higher dimensional Kähler case, this characteristic class is topological it does not depend on the metric chosen. (Of course, in the Riemann surface case, the first Chern class of the canonical bundle is the Euler characteristic, up to sign convention, so all dimensions are essentially the same situation.) So there is some definite restriction on how small the curvature can get. Clearly, under these circumstances, it is natural to look for minimization of the total curvature in some sense. The right sense usually turns out to be minimization of the integral of the pointwise squared norm of the curvature tensor in some form. (The square is motivated by our earlier example: it makes no real sense to minimize the integral of the curvature itself since that is, in our first example on the two-sphere, exactly the thing that is topologically fixed!) To make this a bit more explicit, suppose that E M is a vector bundle over a Riemannian manifold with a structure group G. (Typically, G will be an orthogonal or unitary group arising from a metric structure on the fibres of E). One can consider connections on E, that is operators that take local sections of E into E-valued one-forms on M, so that given a local section s and a tangent vector to M at a point p in M(around which the section is defined), one gets an element of the fibre of E at p. One thinks of this element of the fibre as the derivative of the section along the tangent vector at p. This is required to satisfy a Leibnitz relation: in terms of one-forms (fs) = f s + df s, or,if the vector is denoted by X, fs = f s + (Xf )s. X X Also, it is required that the connection is compatible with the group G in the following sense: Consider the parallel translation of elements of the fibre over a point p in M along a closed curve at a point p in M. This gives a linear map from the fibre at p to itself. This is required to be an element of the action by G on the fibre. This is required for all points p and closed curves at p. Attached to the situation of a G-connection on a bundle is a curvature idea exactly analogous to the usual Riemannian geometry idea of the curvature tensor as the commutator of covariant derivatives, namely R(X,Y) is an operator on local sections of V with values in local sections of V which is a -form (antisymmetric, bilinear in X and Y) on X and Y, defined by R (X,Y)s = ( s) ( s) s. X Y Y X [X,Y] 3

4 This operator can be considered as a two-form with values at point in M lying in the Lie algebra of the group G acting on the fibre at that point. For example, in the Riemannian metric case, this would correspond to the antisymmetry of the curvature in its last two indices. In other words, the curvature in this case is an antisymmetric linear functional on the tangent space of M with values in the endomorphisms of the fibre of the vector bundle. i.e., if E is the vector bundle then the curvature is a section of End E T M. And the endomorphism values belong to the Lie algebra of G acting on the fibres of E. To see how the Lie algebra arises, recall first the technique for proving directly that the Lie bracket is the commutator of flows: Let X and Y be (local) vector fields with flows f t and g t ( so that, e.g., f t (x), x in M fixed, t varies, is an integral curve of X). Then the difference between x and the composition (g -t o f -t o g t o f t ) (x) is order t as t goes to 0 ; here the difference is computed as coordinate vector subtraction in any coordinate system around x. This difference turns out to the coordinate form of t [X,Y] evaluated at x, up to higher order terms. This fact is established by a direct computation with Taylor expansions of the integral curves. In the present situation, something similar can be done. Namely, since R is a tensor in X and Y one can assume that the flows of X and Y commute. Consider parallel translation of an s in the fibre at x around the closed curve consisting of X-flow by t, followed by Y flow by t, then X flow by t and Y flow by t. Again, a direct calculation with Taylor series that this has lowest order term quadratic in t as t goes to 0. And it coefficient is R(X,Y) s, in the sense that the translation of s around this curve is s plus t R(X,Y)s + higher order terms. So parallel translation acts here on the fibre by identity plus the curvature item shown plus higher order. Now parallel translation around a closed curve gives an action on the fibre that is an element of G. It follows that the map that sends s to R(X,Y)s is an endomorphism of the fibre that must belong to the Lie algebra of G considered as mapped into the endomorphisms on the fibre at x via the action of G on that fibre. This rather abstract description is a generalization of a familiar fact in surface theory, that the parallel translation of vectors around a smoothly closed curve that bounds a (small) disc in a surface differs from π-rotation (identity mapping) by the integral of the Gauss curvature over the disc. This is a special case of the Gauss formula, since the integral of the geodesic curvature is the rotation angle of the curve's tangent vector relative to parallel translation. Overall, the curvature as Lie algebra valued for the vector bundle situation is a generalization of the familiar idea in Riemannian geometry that the curvature gives the infinitesimal generators of the (restricted) holonomy group as a sub-lie-algebra of GL(dimM, R) (or in the Riemannian case of SO(dim M, R)). 4

5 Now suppose that M is a Riemannian manifold and E is given a (fibre) metric that is G- compatible. Then the differential forms with values in E and other natural bundles inherit metrics. In particular, there is a norm on tensors of the sort that the curvature tensor is. One defines the Yang Mills functional or "action" (terminology from physics) as R (or sometimes ½ this for physics reasons). M A Yang-Mills connection is by definition a connection that is a stationary (critical) point for this functional in the space of G-connections as described(this space makes sense: it is exactly that the first variation vanishes for all variations of the connection, as in classical calculus of variations). The Euler-Lagrange equation for this is computed to be that δ R = 0 where δ is the adjoint of the natural notion of covariant derivative operating on V valued forms. (This is defined exactly as in the coordinate invariant formula for d, namely d (X,..X 0 R) = ( ) Xα(X,..X {with X i 0 k) i omitted} + ( ) i + j α([x i,x J],X 0,..Xk) { with Xi and X j } i< j The Bianchi identity is that dr = 0 already. So the connection being Yang Mills is equivalent to R being harmonic in the sense that R belongs to both the kernel of d d and its adjoint δ. (Here one considers as a map into two forms with values in the Lie algebra of G, as above, so that the adjoint goes from two forms to one forms with vaues in the Lie algebra.) Thus Yang Mills theory is formally analogous to harmonic theory. (Recall that harmonic form theory is itself the solution to a minimization problem: the harmonic representative in a derham cohomology class is exactly the unique form in that class that has minimal L norm.) In the case of the tangent bundle of a manifold of dimension four, this assumes special aspects. Since two forms are mapped to themselves under the Hodge star operator, one can consider whether * R is a multiple of R (which multiple would have to be +- since * preserves length). This gives a decomposition of any R into a "self-dual" and "anti-self dual" part, namely the eigenspace decomposition for the action of *. One has a lower bound from characteristic class theory in terms of the first Pontryagin class of V for Yang Mills functional and one sees that this lower bound is achieved by a connection if and only if it is self-dual or anti-self-dual (which one depending on the sign of the Pontyagin class). In either case, self dual or anti-self-dual, R is automatically harmonic (as n Hodge theory of the ordinary sort). This is all an analogue of the Hodge theory of the middle dimension forms on an even dimensional (oriented) manifold, dimension say k: * of an harmonic k-form is again an harmonic k-form so that the space of harmonic k-forms decomposes into spaces of forms on which * acts as identity or as minus the identity. 5

6 Hermitian Yang Mills theory is another specific realization of this very general idea of minimizing in classes which have a topologically determined lower bound on some functional, usually the integral of a squared norm. Namely, let us start with a holomorphic vector bundle over a compact Kähler manifold (it will turn out eventually that the Kähler condition is not really needed. But we want to start slowly). We can put an Hermitian metric on the vector bundle, more or less arbitrarily. Call this h and let g be the Kähler metric on the manifold. Now it is a standard items that there is then a unique connection on the vector bundle with the properties:. the connection is compatible with the metric, i.e., parallel sections (along a curve) have constant length and. the connection is holomorphic, that is, the connection forms relative to a holomorphic local frame in the vector bundle are type (,0). This is the same as saying that the (0,) part of Ds, for some local section s, is s. This is an easy thing to check: Just take d h(s, s ), which must be h(d s, s ) + h( s, Ds ), and remember the conjugation in the second slot of h, then match up types of forms to get that the connection forms are the matrix of forms h h. (see page 47-48) From here on, we denote the endomorphism valued curvature tensor that arises as already described (from a given connection) by F. Now, since we have a Kähler metric g, we can define a trace on F by contracting with the metric tensor, namely, jk trf = g F β, α jk where Roman indices are for the manifold, Greek indices for the fibres of the bundle. By definition, the holomorphic vector bundle is Hermitian Yang Mills over the Kähler manifold if there is an Hermitian metric h for which trgf=μid for some constant μ. Note that this is an extension of the ideas of extremal Kähler metric (according to the Calabi ideas discussed above), the trace condition being in that case the Kähler-Einstein condition on the Ricci tensor(that it is a constant multiple of the metric). In turn, this implies that the Ricci curvature form (curvature in this Hermitian Yang-Mills sense) is harmonic since a constant multiple of the Kähler form is necessarily harmonic. 6

7 This harmonicity turns out to be equivalent to the metric being a critical point of a functional measuring in effect the integral squared norm of the curvature and thus it fits into the general picture already discussed. (In the Kähler Einstein situation of Yau, the Einstein Kähler metric critical point is, as noted, in fact an absolute minimum in the class of connection arising from Kähler metrics with the specified Kähler class.) In case the bundle E is the holomorphic tangent bundle, then one can check easily that the Kahler metric is Hermitian Yang Mills if and only if it satisfies the Einstein-Kähler condition that the Ricci tensor (interpreted as a (,) form) is a constant multiple of the Kähler form, or equivalently the ordinary Ricci tensor is a constant multiple of the metric tensor g. Note that for a Hermitian Yang Mills situation in general (E not necessarily the tangent bundle), the constant μ can be computed in terms of the first Chern form of E and the Kähler form of the manifold metric g. Namely the first Chern form i αβ j c(e) = k h F dz dz. From this, one obtains that αβ jk π n n αβ ω ω c(e) ω = μ h h μrank(e) αβ =. n! n! deg(e So μ ) = where deg(e) = c (E) ω vol(m) rank(e). At this point, one might want to refer back to the first paragraphs here to see how the general ideas persist: in general terms, the metric h is a stationary point for total squared curvature with the constant curvature being a topological item for the vector bundle, Kahler manifold combination. The basic result of Uhlenbeck Yau is this: A stable vector bundle over a compact Kähler manifold admits a unique Hermitian Yang Mills connection. Here stability is in the Mumford-Takemoto- Gieseker sense: the vector bundle E is stable if for each coherent reflexive subsheaf script F of the sheaf of germs of holomorphic sections of E, degree (F)/rank (F) < degree(e)/rank E. Here degree is defined as c(e) ω. Kobayashi had shown earlier that the existence of an Hermitian Yang Mills metric implies the stability of the bundle in this sense. The Uhlenbeck-Yau result (and its extension to the non- Kähler case by Li and Yau) is a converse to this. Robert E. Greene 7

### 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

### 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

### 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

### 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

### Topic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016

Topic: First Chern classes of Kähler manifolds itchell Faulk Last updated: April 23, 2016 We study the first Chern class of various Kähler manifolds. We only consider two sources of examples: Riemann surfaces

### MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES

MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.

### 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

### Lecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)

Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries

### 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

### Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current

Author, F., and S. Author. (2015) Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current, International Mathematics Research Notices, Vol. 2015, Article ID rnn999, 7 pages. doi:10.1093/imrn/rnn999

### Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18

Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18 Required problems (to be handed in): 2bc, 3, 5c, 5d(i). In doing any of these problems, you may assume the results

### 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

### Hermitian 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

### 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

### 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:

### William P. Thurston. The Geometry and Topology of Three-Manifolds

William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 00 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed

### 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

### 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

### 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.

### Introduction to Group Theory

Chapter 10 Introduction to Group Theory Since symmetries described by groups play such an important role in modern physics, we will take a little time to introduce the basic structure (as seen by a physicist)

### 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

### 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

### η = (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

### 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

### 1. Geometry of the unit tangent bundle

1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations

### 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.

### CHAPTER 1 PRELIMINARIES

CHAPTER 1 PRELIMINARIES 1.1 Introduction The aim of this chapter is to give basic concepts, preliminary notions and some results which we shall use in the subsequent chapters of the thesis. 1.2 Differentiable

### 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

### 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

### 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

### 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

### 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

### Introduction to Extremal metrics

Introduction to Extremal metrics Preliminary version Gábor Székelyhidi Contents 1 Kähler geometry 2 1.1 Complex manifolds........................ 3 1.2 Almost complex structures.................... 5 1.3

### Some brief notes on the Kodaira Vanishing Theorem

Some brief notes on the Kodaira Vanishing Theorem 1 Divisors and Line Bundles, according to Scott Nollet This is a huge topic, because there is a difference between looking at an abstract variety and local

### 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

### 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)

### Geometry of Twistor Spaces

Geometry of Twistor Spaces Claude LeBrun Simons Workshop Lecture, 7/30/04 Lecture Notes by Jill McGowan 1 Twistor Spaces Twistor spaces are certain complex 3-manifolds which are associated with special

### Vectors. January 13, 2013

Vectors January 13, 2013 The simplest tensors are scalars, which are the measurable quantities of a theory, left invariant by symmetry transformations. By far the most common non-scalars are the vectors,

### Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem

PETER B. GILKEY Department of Mathematics, University of Oregon Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem Second Edition CRC PRESS Boca Raton Ann Arbor London Tokyo Contents

### 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

### Connections for noncommutative tori

Levi-Civita connections for noncommutative tori reference: SIGMA 9 (2013), 071 NCG Festival, TAMU, 2014 In honor of Henri, a long-time friend Connections One of the most basic notions in differential geometry

### Riemannian Curvature Functionals: Lecture I

Riemannian Curvature Functionals: Lecture I Jeff Viaclovsky Park City athematics Institute July 16, 2013 Overview of lectures The goal of these lectures is to gain an understanding of critical points of

### Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

### CHARACTERISTIC CLASSES

1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact

### 5 Constructions of connections

[under construction] 5 Constructions of connections 5.1 Connections on manifolds and the Levi-Civita theorem We start with a bit of terminology. A connection on the tangent bundle T M M of a manifold M

### THE UNIFORMISATION THEOREM OF RIEMANN SURFACES

THE UNIFORISATION THEORE OF RIEANN SURFACES 1. What is the aim of this seminar? Recall that a compact oriented surface is a g -holed object. (Classification of surfaces.) It can be obtained through a 4g

### 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,

### Riemannian geometry of the twistor space of a symplectic manifold

Riemannian geometry of the twistor space of a symplectic manifold R. Albuquerque rpa@uevora.pt Departamento de Matemática, Universidade de Évora Évora, Portugal September 004 0.1 The metric In this short

### THE VORTEX EQUATION ON AFFINE MANIFOLDS. 1. Introduction

THE VORTEX EQUATION ON AFFINE MANIFOLDS INDRANIL BISWAS, JOHN LOFTIN, AND MATTHIAS STEMMLER Abstract. Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show

### THEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)

4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M

### Physics 557 Lecture 5

Physics 557 Lecture 5 Group heory: Since symmetries and the use of group theory is so much a part of recent progress in particle physics we will take a small detour to introduce the basic structure (as

### 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

### NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES

NOTES ON FORMAL NEIGHBORHOODS AND JET BUNDLES SHILIN U ABSTRACT. The purpose of this note is to review the construction of smooth and holomorphic jet bundles and its relation to formal neighborhood of

### 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.

### An Introduction to Riemann-Finsler Geometry

D. Bao S.-S. Chern Z. Shen An Introduction to Riemann-Finsler Geometry With 20 Illustrations Springer Contents Preface Acknowledgments vn xiii PART ONE Finsler Manifolds and Their Curvature CHAPTER 1 Finsler

### 7 Curvature of a connection

[under construction] 7 Curvature of a connection 7.1 Theorema Egregium Consider the derivation equations for a hypersurface in R n+1. We are mostly interested in the case n = 2, but shall start from the

### RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY

RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY NICK MCCLEEREY 0. Complex Differential Forms Consider a complex manifold X n (of complex dimension n) 1, and consider its complexified tangent bundle T C

### Complex manifolds, Kahler metrics, differential and harmonic forms

Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on

### L6: Almost complex structures

L6: Almost complex structures To study general symplectic manifolds, rather than Kähler manifolds, it is helpful to extract the homotopy-theoretic essence of having a complex structure. An almost complex

### 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.

### 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

### A THEOREM ON COMPACT LOCALLY CONFORMAL KAHLER MANIFOLDS

proceedings of the american mathematical society Volume 75, Number 2, July 1979 A THEOREM ON COMPACT LOCALLY CONFORMAL KAHLER MANIFOLDS IZU VAISMAN Abstract. We prove that a compact locally conformai Kahler

### 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

### 1 Hermitian symmetric spaces: examples and basic properties

Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................

### Some 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

### LECTURE 8: THE SECTIONAL AND RICCI CURVATURES

LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ( V ) the set of 4-tensors that is anti-symmetric with respect to

### 4.7 The Levi-Civita connection and parallel transport

Classnotes: Geometry & Control of Dynamical Systems, M. Kawski. April 21, 2009 138 4.7 The Levi-Civita connection and parallel transport In the earlier investigation, characterizing the shortest curves

### Useful theorems in complex geometry

Useful theorems in complex geometry Diego Matessi April 30, 2003 Abstract This is a list of main theorems in complex geometry that I will use throughout the course on Calabi-Yau manifolds and Mirror Symmetry.

### Vanishing theorems and holomorphic forms

Vanishing theorems and holomorphic forms Mihnea Popa Northwestern AMS Meeting, Lansing March 14, 2015 Holomorphic one-forms and geometry X compact complex manifold, dim C X = n. Holomorphic one-forms and

### 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

### 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

### CANONICAL SASAKIAN METRICS

CANONICAL SASAKIAN METRICS CHARLES P. BOYER, KRZYSZTOF GALICKI, AND SANTIAGO R. SIMANCA Abstract. Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for

### Math 230a Final Exam Harvard University, Fall Instructor: Hiro Lee Tanaka

Math 230a Final Exam Harvard University, Fall 2014 Instructor: Hiro Lee Tanaka 0. Read me carefully. 0.1. Due Date. Per university policy, the official due date of this exam is Sunday, December 14th, 11:59

### Kähler manifolds and variations of Hodge structures

Kähler manifolds and variations of Hodge structures October 21, 2013 1 Some amazing facts about Kähler manifolds The best source for this is Claire Voisin s wonderful book Hodge Theory and Complex Algebraic

### THE MCKEAN-SINGER FORMULA VIA EQUIVARIANT QUANTIZATION

THE MCKEAN-SINGER FORMULA VIA EQUIVARIANT QUANTIZATION EUGENE RABINOVICH 1. Introduction The aim of this talk is to present a proof, using the language of factorization algebras, and in particular the

### A brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström

A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal

### Transparent connections

The abelian case A definition (M, g) is a closed Riemannian manifold, d = dim M. E M is a rank n complex vector bundle with a Hermitian metric (i.e. a U(n)-bundle). is a Hermitian (i.e. metric) connection

### RIEMANN S INEQUALITY AND RIEMANN-ROCH

RIEMANN S INEQUALITY AND RIEMANN-ROCH DONU ARAPURA Fix a compact connected Riemann surface X of genus g. Riemann s inequality gives a sufficient condition to construct meromorphic functions with prescribed

### Geometry for Physicists

Hung Nguyen-Schafer Jan-Philip Schmidt Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers 4 i Springer Contents 1 General Basis and Bra-Ket Notation 1 1.1 Introduction to

### 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

### FAKE PROJECTIVE SPACES AND FAKE TORI

FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.

### 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

### 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

### UNIVERSAL UNFOLDINGS OF LAURENT POLYNOMIALS AND TT STRUCTURES. Claude Sabbah

UNIVERSAL UNFOLDINGS OF LAURENT POLYNOMIALS AND TT STRUCTURES AUGSBURG, MAY 2007 Claude Sabbah Introduction Let f : (C ) n C be a Laurent polynomial, that I assume to be convenient and non-degenerate,

### Fisica Matematica. Stefano Ansoldi. Dipartimento di Matematica e Informatica. Università degli Studi di Udine. Corso di Laurea in Matematica

Fisica Matematica Stefano Ansoldi Dipartimento di Matematica e Informatica Università degli Studi di Udine Corso di Laurea in Matematica Anno Accademico 2003/2004 c 2004 Copyright by Stefano Ansoldi and

### Definition We say that a topological manifold X is C p if there is an atlas such that the transition functions are C p.

13. Riemann surfaces Definition 13.1. Let X be a topological space. We say that X is a topological manifold, if (1) X is Hausdorff, (2) X is 2nd countable (that is, there is a base for the topology which

### Origin, Development, and Dissemination of Differential Geometry in Mathema

Origin, Development, and Dissemination of Differential Geometry in Mathematical History The Borough of Manhattan Community College -The City University of New York Fall 2016 Meeting of the Americas Section

### Lecture VI: Projective varieties

Lecture VI: Projective varieties Jonathan Evans 28th October 2010 Jonathan Evans () Lecture VI: Projective varieties 28th October 2010 1 / 24 I will begin by proving the adjunction formula which we still

### 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

### 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

### MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY

MATH 263: PROBLEM SET 1: BUNDLES, SHEAVES AND HODGE THEORY 0.1. Vector Bundles and Connection 1-forms. Let E X be a complex vector bundle of rank r over a smooth manifold. Recall the following abstract

### 274 Curves on Surfaces, Lecture 4

274 Curves on Surfaces, Lecture 4 Dylan Thurston Notes by Qiaochu Yuan Fall 2012 4 Hyperbolic geometry Last time there was an exercise asking for braids giving the torsion elements in PSL 2 (Z). A 3-torsion

### ENOKI S INJECTIVITY THEOREM (PRIVATE NOTE) Contents 1. Preliminaries 1 2. Enoki s injectivity theorem 2 References 5

ENOKI S INJECTIVITY THEOREM (PRIVATE NOTE) OSAMU FUJINO Contents 1. Preliminaries 1 2. Enoki s injectivity theorem 2 References 5 1. Preliminaries Let us recall the basic notion of the complex geometry.

### MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus

MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let

### Affine Connections: Part 2

Affine Connections: Part 2 Manuscript for Machine Learning Reading Group Talk R. Simon Fong Abstract Note for online manuscript: This is the manuscript of a one hour introductory talk on (affine) connections.

### 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

### Geometry of the Moduli Space of Curves and Algebraic Manifolds

Geometry of the Moduli Space of Curves and Algebraic Manifolds Shing-Tung Yau Harvard University 60th Anniversary of the Institute of Mathematics Polish Academy of Sciences April 4, 2009 The first part

### Gravitation: Tensor Calculus

An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013