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

Size: px
Start display at page:

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

Transcription

1 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 by coordinate systems [a sub-atlas ] in which the transition functions from one coordinate system to another are not arbitrary C mappings but rather belong to some specific set of mappings from (open subsets of) Euclidean space to itself. [Precisely what is needed here is dealt with by the concept of pseudo-group of transformations]. Examples are 1. projective structures transformations are (restrictions of ) projective transformations of RP n, namely linear fractional transformations 2. conformal structures: overlap maps are conformal 3. affine structures: overlap maps are affine transformations of Euclidean space 4. complex structures: overlap maps are holomorphic maps [considered as realc maps on an even dimensional Euclidean space] These situations are different from each other and higher dimensions are often different from dimension 2. Important differences: Holomorphic maps do not have to come from a map of the whole space whereas, e.g., projective maps extend by definition. In dimension 2, there are many [infinite dimensional, not described by a finite set of parameters] maps [holomorphic functions] whereas in higher dimensions, conformal maps are a more rigid[extend to whole space except for possible definitely determined singularities, finite number of parameters, e.g., Liouville s Theorem that conformal maps of R 3 are generated by inversions and isometries and dilations/contractions].

2 Dimension 2: Riemannian metric always exists [manifolds are being assumed paracompact] and by historical theorems [Korn, Lichtenstein, early 1900s, etc.] there is a conformal structure attached to any given Riemannian metric, namely, there exist local coordinates [historically, isothermal parameters ] in which the metric has the form (positive function) (coordinate Euclidean metric) ;and a cover by such coordinate systems gives a conformal structure, clearly. Assuming orientability, as is assumed from now on, this gives also a complex structure since conformal orientation-preserving maps are necessarily holomorphic in real dimension 2. Almost complex structure: On a manifold with complex structure, we get a splitting at each point of the complexified tangent space [real tangent space tensored with the complex numbers] into two disjoint, spanning subspaces, with each the conjugate of the other. These are the holomorphic and antiholomorphic tangent spaces. This is the same as the span of the coordinate z j operators, resp. z operators, if ( z,... 1 z n ) is a holomorphic local coordinate system.. But the splitting also corresponds to eigenspaces of J (the real representation of multiplication by i on R 2n, J on tangent spaces is invariant under holomorphic maps), eigenvalues are +i and i since J composed with J is multiplication by -1. Also we can recover J from the splitting [because of this eigenvalue observation]. So we can think about an almost complex structure, namely a suitable splitting or equivalently a J operator at each point with J composed with J being multiplication by -1. [ J squared= -1 for short]. In two dimensions, every almost complex structure arises from a complex structure [e.g., one can J -average some Riemannian metric to make it J- invariant, then find the associated complex structure via the isothermal parameters idea, and the complex structure obtained has the same J as the original one.] From the general viewpoint from higher dimensions, this happens because ( ) 2 is always zero in dimension 2 on account of dimension reasons, while ( ) 2 = 0 is the condition for being able to attach a complex structure to an almost complex one.

3 In more detail, note that is always defined as a operator, if just an almost complex structure is given: the almost complex structure gives forms (p,q) types;one gets from there. The vanishing of the Nijenhuis tensor [see article on ~greene website, for course no 234], is equivalent to ( ) 2 being zero. And by the fundamental Newlander Nirenberg Theorem, this implies the integrability of the complex structure. Here integrability means the existence of local coordinate systems in which the given J (or splitting of the complexified tangent space) is the same as that determined by the coordinate system considered as a map into complex Euclidean n-space. In general, integrability of this sort is obtained in the real analytic case from the Cartan- Kaehler theory[ a generalized version of the Cauchy-Kowaleska Theorem in which integrability conditions can be treated, but only in the real analytic case in general]. But in the case where real analyticity is not assumed, other methods are needed.[one approach to proving the Newlander-Nirenberg Theorem: work through solving without using local coordinates, just using the J and integrability conditions to conclude that local holomorphic relative to given J functions exist in abundance]. Deformation especially in dimension 2: Complex structures are described up to equivalence by 3g-3 complex parameters(known to Riemann). Here equivalence means that there is a structure preserving diffeomorphism from one to the other. In general, deformation of almost complex structure means tilting the holomorphic tangent space within the whole complexified tangent space. So one can think of this as a linear map (at each point) from the holomorphic tangent space into the conjugate (or antihlomorphic) tangent space. Tracing through identifications gives the infinitesimal deformation as represented by an element of the sheaf cohomology H 1 (M, sheaf of germs of holomorphic vector fields) [Kodaira-Spencer theory, cf Morrow and Kodaira s book]. In the Riemann surface case, this becomes(via Serre Duality) the space of quadratic differentials, i.e., holomorphic sections of the square of the canonical(line)

4 bundle (the bundle of (1,0) forms for a Riemann surface). [Note that {the square of }the canonical line bundle is a positive bundle precisely if the genus g is greater than one. It is negative and hence without nontrivial holomorphic sections if g=0. This corresponds to the fact that deformations of CP 1 are trivial{cpp1 has a unique complex structure up to equivalence}. Nontrivial deformations of higher genus surfaces are abundant, e,g., the 3g -3 parameters when g>1, the one {complex} parameter when g=1. The 3g-3 number arises as follows: by the Riemann Roch Theorem, the dimension of the space of holomorphic sections of a line bundle = degree of bundle g +1 + dim of space of holomorphic sections of ( canonical bundle tensored with the dual of the given bundle). Here degree of bundle= total order of a meromorphic section = (first) Chern class interpreted as integer. Since the degree of the canonical bundle is 2g-2 and hence of the square of the canonical bundle is 4g-4, we get dimension of space of sections of the squared canonical bundle = 4g-4 g+1 + dimension of space of sections of the dual of the canonical bundle. But the dual of the canonical bundle has degree 2-2g, which is <0 in case g>1, so the last dimension item is in fact 0. Thus the deformation space has dimension 3g-3 when g>1. The same argument, mutatis mutandis, explains the one(complex) parameter in the torus case, genus=1.] Existence of almost complex structures: For any n-manifold, the tangent bundle is obtained by pulling back the Grassmannian bundle of n-planes in a high dimensional Euclidean space via a map of the manifold into the Euclidean space. More explicitly, according to Whitney s embedding theorem, if M is a (smooth) manifold of dimension n then there is an embedding of M into a Euclidean space R N dimension N>>n. Consider the Grassmannian Gr(n, N) of n planes in N- dimensional Euclidean space, with its tautological n-bundle, wherein the fibre over a point, a point being an n-plane, is that n-plane itself. Then the embedding induces a map of M into the Grassmannian Gr(n,N) by sending each point p in M to the image under the differential of the embedding at p applied to the tangent space of M at p. In other words, we send each point p to the tangent space of M at p, considered to be a subspace of R N

5 when M is considered as being in R N via the embedding. It is easy to see that the pullback of the tautological bundle of Gr(n,N) by this mapping of M into Gr(n,N) is in fact the tangent bundle of M! This same construction can be extended to show that any n-plane bundle on M arises from pullback of the tautological bundle via a map into Gr(n,N), N>>n. Moreover, this map is unique up to homotopy, for a given bundle. This means that the pullback of the cohomology classes of Gr(n,N) to M via this map depends only on the bundle itself. These are the characteristic classes of the bundle, by definition. Since this whole construction is independent of N when N is large, one usually lets N go to infinity and looks only at the classifying space Gr( n, ). If the bundle has group G one writes B G. Gr( n, ) = BGl n or, (see Milnor, Characteristic Classes for details of all this). E.g., (, ) which is the same, B O( n). Now an almost complex structure on a manifold of real dimension 2n amounts to a reduction of the structure group from GL(2n,R) to Gl(n, C) or, equivalently from the homotopy viewpoint, from SO(2n, R) to U(n) [orientability is automatic if an almost complex structure is to exist so we take it for granted]. For this to happen, the classifying map for the tangent bundle (a real 2n plane bundle) has to factor through a map into the classifying space of U(n). Namely, the classifying space for U(n) has a natural projection map into the classifying space for SO(2n). This amounts to no more than observing that a complex subspace of complex dimension n of C N can be considered to be a real subspace of real dimension 2n in R 2N. The almost complex structure then gives a way to lift the classifying map into B SO(2n). Namely, there is a map into B U(n) which when followed by the projection is the original real classifying map. This can be translated into things about characteristic classes. The real characteristic classes, that is pullbacks from B SO (2 n ) of its cohomology classes, are generated by the Pontryagin classes p1, p 2, (integral classes) and the Stiefel-Whitney classes w 1, w 2,

6 (mod 2 classes). The classes that come from B U( n) are the Chern classes (integral classes) c 1, c 2, For the existence of a lift as described, some relationships have to happen: for example, in complex dimension 2, it must be that w 2 = c 1 mod Z 2 while p 1 = 2c 2 c 2 1. In algebraic topology (obstruction theory), it is known how to compute what has to happen for a lift to exist (up to homotopy) of the type that will correspond to an almost complex structure. In this case, this is all determined by characteristic classes. This has been worked out explicitly by Wu up to real dimension 6(and in outline 8) and in principle could be carried out for higher dimensions. But this is hard to do in general form (similar to the situation with homotopy groups of spheres). And the problem of going from the almost complex structure (if there is one) to finding an integrable one is not solved in general. Only in dimension 2 is this automatic. Robert E. Greene

### Math 231b Lecture 16. G. Quick

Math 231b Lecture 16 G. Quick 16. Lecture 16: Chern classes for complex vector bundles 16.1. Orientations. From now on we will shift our focus to complex vector bundles. Much of the theory for real vector

### THE NEWLANDER-NIRENBERG THEOREM. GL. The frame bundle F GL is given by x M Fx

THE NEWLANDER-NIRENBERG THEOREM BEN MCMILLAN Abstract. For any kind of geometry on smooth manifolds (Riemannian, Complex, foliation,...) it is of fundamental importance to be able to determine when two

### 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

### 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.

### 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

### Atiyah-Singer Revisited

Atiyah-Singer Revisited Paul Baum Penn State Texas A&M Universty College Station, Texas, USA April 1, 2014 From E 1, E 2,..., E n obtain : 1) The Dirac operator of R n D = n j=1 E j x j 2) The Bott generator

### NOTES ON FIBER BUNDLES

NOTES ON FIBER BUNDLES DANNY CALEGARI Abstract. These are notes on fiber bundles and principal bundles, especially over CW complexes and spaces homotopy equivalent to them. They are meant to supplement

### 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

### Recall for an n n matrix A = (a ij ), its trace is defined by. a jj. It has properties: In particular, if B is non-singular n n matrix,

Chern characters Recall for an n n matrix A = (a ij ), its trace is defined by tr(a) = n a jj. j=1 It has properties: tr(a + B) = tr(a) + tr(b), tr(ab) = tr(ba). In particular, if B is non-singular n n

### 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

### Lecture 2. Smooth functions and maps

Lecture 2. Smooth functions and maps 2.1 Definition of smooth maps Given a differentiable manifold, all questions of differentiability are to be reduced to questions about functions between Euclidean spaces,

### Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

### Algebraic Curves and Riemann Surfaces

Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex

### LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be

### Cobordant differentiable manifolds

Variétés différentiables cobordant, Colloque Int. du C. N. R. S., v. LII, Géométrie différentielle, Strasbourg (1953), pp. 143-149. Cobordant differentiable manifolds By R. THOM (Strasbourg) Translated

### 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.

### 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)

### 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

### 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

### Constant Mean Curvature Tori in R 3 and S 3

Constant Mean Curvature Tori in R 3 and S 3 Emma Carberry and Martin Schmidt University of Sydney and University of Mannheim April 14, 2014 Compact constant mean curvature surfaces (soap bubbles) are critical

### Chern forms and the Fredholm determinant

CHAPTER 10 Chern forms and the Fredholm determinant Lecture 10: 20 October, 2005 I showed in the lecture before last that the topological group G = G (Y ;E) for any compact manifold of positive dimension,

### SOME EXERCISES IN CHARACTERISTIC CLASSES

SOME EXERCISES IN CHARACTERISTIC CLASSES 1. GAUSSIAN CURVATURE AND GAUSS-BONNET THEOREM Let S R 3 be a smooth surface with Riemannian metric g induced from R 3. Its Levi-Civita connection can be defined

### The Canonical Sheaf. Stefano Filipazzi. September 14, 2015

The Canonical Sheaf Stefano Filipazzi September 14, 015 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will go over

### We have the following immediate corollary. 1

1. Thom Spaces and Transversality Definition 1.1. Let π : E B be a real k vector bundle with a Euclidean metric and let E 1 be the set of elements of norm 1. The Thom space T (E) of E is the quotient E/E

### Hodge Theory of Maps

Hodge Theory of Maps Migliorini and de Cataldo June 24, 2010 1 Migliorini 1 - Hodge Theory of Maps The existence of a Kähler form give strong topological constraints via Hodge theory. Can we get similar

### Lecture 6: Classifying spaces

Lecture 6: Classifying spaces A vector bundle E M is a family of vector spaces parametrized by a smooth manifold M. We ask: Is there a universal such family? In other words, is there a vector bundle E

### 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................................................

### Three Descriptions of the Cohomology of Bun G (X) (Lecture 4)

Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and

### Chern Classes and the Chern Character

Chern Classes and the Chern Character German Stefanich Chern Classes In this talk, all our topological spaces will be paracompact Hausdorff, and our vector bundles will be complex. Let Bun GLn(C) be the

### 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

### 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

### Math 550 / David Dumas / Fall Problems

Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;

### 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

### 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,

### Introduction to Teichmüller Spaces

Introduction to Teichmüller Spaces Jing Tao Notes by Serena Yuan 1. Riemann Surfaces Definition 1.1. A conformal structure is an atlas on a manifold such that the differentials of the transition maps lie

### 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

### Background on c-projective geometry

Second Kioloa Workshop on C-projective Geometry p. 1/26 Background on c-projective geometry Michael Eastwood [ following the work of others ] Australian National University Second Kioloa Workshop on C-projective

### 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

### Stable complex and Spin c -structures

APPENDIX D Stable complex and Spin c -structures In this book, G-manifolds are often equipped with a stable complex structure or a Spin c structure. Specifically, we use these structures to define quantization.

### 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.

### CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally

### 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

### k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim

### 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

### 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

### 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

### 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.

### Characteristic Classes, Chern Classes and Applications to Intersection Theory

Characteristic Classes, Chern Classes and Applications to Intersection Theory HUANG, Yifeng Aug. 19, 2014 Contents 1 Introduction 2 2 Cohomology 2 2.1 Preliminaries................................... 2

### Fiber bundles and characteristic classes

Fiber bundles and characteristic classes Bruno Stonek bruno@stonek.com August 30, 2015 Abstract This is a very quick introduction to the theory of fiber bundles and characteristic classes, with an emphasis

### QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 1, 2015 (Day 1)

QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 1, 2015 (Day 1) 1. (A) The integer 8871870642308873326043363 is the 13 th power of an integer n. Find n. Solution.

### Handlebody Decomposition of a Manifold

Handlebody Decomposition of a Manifold Mahuya Datta Statistics and Mathematics Unit Indian Statistical Institute, Kolkata mahuya@isical.ac.in January 12, 2012 contents Introduction What is a handlebody

### 1 Differentiable manifolds and smooth maps

1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set

### Mini-Course on Moduli Spaces

Mini-Course on Moduli Spaces Emily Clader June 2011 1 What is a Moduli Space? 1.1 What should a moduli space do? Suppose that we want to classify some kind of object, for example: Curves of genus g, One-dimensional

### 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

### Poles of Instantons and Jumping Lines of Algebraic Vector Bundles on P

No. 5] Proc. Japan Acad., 5, Ser. A (1979) 185 Poles of Instantons and Jumping Lines of Algebraic Vector Bundles on P By Motohico ]V[ULASE Research Institute for Mathematical Science.s, Kyoto University

### 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

### Dirac Operator. Göttingen Mathematical Institute. Paul Baum Penn State 6 February, 2017

Dirac Operator Göttingen Mathematical Institute Paul Baum Penn State 6 February, 2017 Five lectures: 1. Dirac operator 2. Atiyah-Singer revisited 3. What is K-homology? 4. The Riemann-Roch theorem 5. K-theory

### 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

### LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES

LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.

### Each is equal to CP 1 minus one point, which is the origin of the other: (C =) U 1 = CP 1 the line λ (1, 0) U 0

Algebraic Curves/Fall 2015 Aaron Bertram 1. Introduction. What is a complex curve? (Geometry) It s a Riemann surface, that is, a compact oriented twodimensional real manifold Σ with a complex structure.

### 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

### NONNEGATIVE CURVATURE AND COBORDISM TYPE. 1. Introduction

NONNEGATIVE CURVATURE AND COBORDISM TYPE ANAND DESSAI AND WILDERICH TUSCHMANN Abstract. We show that in each dimension n = 4k, k 2, there exist infinite sequences of closed simply connected Riemannian

### 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

### ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS

ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DG-MANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical

### Remarks on the Milnor number

José 1 1 Instituto de Matemáticas, Universidad Nacional Autónoma de México. Liverpool, U. K. March, 2016 In honour of Victor!! 1 The Milnor number Consider a holomorphic map-germ f : (C n+1, 0) (C, 0)

### Exercises on characteristic classes

Exercises on characteristic classes April 24, 2016 1. a) Compute the Stiefel-Whitney classes of the tangent bundle of RP n. (Use the method from class for the tangent Chern classes of complex projectives

### COMPLEX ALGEBRAIC SURFACES CLASS 9

COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution

### Patrick Iglesias-Zemmour

Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries

### 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

### Bredon, Introduction to compact transformation groups, Academic Press

1 Introduction Outline Section 3: Topology of 2-orbifolds: Compact group actions Compact group actions Orbit spaces. Tubes and slices. Path-lifting, covering homotopy Locally smooth actions Smooth actions

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

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

### 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.

### 1. Classifying Spaces. Classifying Spaces

Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.

### 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 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

### Lecture 8: More characteristic classes and the Thom isomorphism

Lecture 8: More characteristic classes and the Thom isomorphism We begin this lecture by carrying out a few of the exercises in Lecture 1. We take advantage of the fact that the Chern classes are stable

### DONAGI-MARKMAN CUBIC FOR HITCHIN SYSTEMS arxiv:math/ v2 [math.ag] 24 Oct 2006

DONAGI-MARKMAN CUBIC FOR HITCHIN SYSTEMS arxiv:math/0607060v2 [math.ag] 24 Oct 2006 DAVID BALDUZZI Abstract. The Donagi-Markman cubic is the differential of the period map for algebraic completely integrable

### Algebraic geometry versus Kähler geometry

Algebraic geometry versus Kähler geometry Claire Voisin CNRS, Institut de mathématiques de Jussieu Contents 0 Introduction 1 1 Hodge theory 2 1.1 The Hodge decomposition............................. 2

### TRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE. Luis Guijarro and Gerard Walschap

TRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE Luis Guijarro and Gerard Walschap Abstract. In this note, we examine the relationship between the twisting of a vector bundle ξ over a manifold

### 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

### 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

### COHOMOLOGY OF ARITHMETIC GROUPS AND EISENSTEIN SERIES: AN INTRODUCTION, II

COHOMOLOGY OF ARITHMETIC GROUPS AND EISENSTEIN SERIES: AN INTRODUCTION, II LECTURES BY JOACHIM SCHWERMER, NOTES BY TONY FENG Contents 1. Review 1 2. Lifting differential forms from the boundary 2 3. Eisenstein

### Geodesic Equivalence in sub-riemannian Geometry

03/27/14 Equivalence in sub-riemannian Geometry Supervisor: Dr.Igor Zelenko Texas A&M University, Mathematics Some Preliminaries: Riemannian Metrics Let M be a n-dimensional surface in R N Some Preliminaries:

### Space of surjective morphisms between projective varieties

Space of surjective morphisms between projective varieties -Talk at AMC2005, Singapore- Jun-Muk Hwang Korea Institute for Advanced Study 207-43 Cheongryangri-dong Seoul, 130-722, Korea jmhwang@kias.re.kr

### Conformal field theory in the sense of Segal, modified for a supersymmetric context

Conformal field theory in the sense of Segal, modified for a supersymmetric context Paul S Green January 27, 2014 1 Introduction In these notes, we will review and propose some revisions to the definition

### LECTURE 2: SYMPLECTIC VECTOR BUNDLES

LECTURE 2: SYMPLECTIC VECTOR BUNDLES WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic Vector Spaces Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over

### Introduction (Lecture 1)

Introduction (Lecture 1) February 2, 2011 In this course, we will be concerned with variations on the following: Question 1. Let X be a CW complex. When does there exist a homotopy equivalence X M, where

### 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

### MATH 217C NOTES ARUN DEBRAY AUGUST 21, 2015

MATH 217C NOTES ARUN DEBRAY AUGUST 21, 2015 These notes were taken in Stanford s Math 217c class in Winter 2015, taught by Eleny Ionel. I live-texed them using vim, and as such there may be typos; please

### 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

### 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

### 30 Surfaces and nondegenerate symmetric bilinear forms

80 CHAPTER 3. COHOMOLOGY AND DUALITY This calculation is useful! Corollary 29.4. Let p, q > 0. Any map S p+q S p S q induces the zero map in H p+q ( ). Proof. Let f : S p+q S p S q be such a map. It induces

### The 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

### EQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms

EQUIVARIANT COHOMOLOGY MARTINA LANINI AND TINA KANSTRUP 1. Quick intro Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### 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

### ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES. 1. Introduction

ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES ANDREAS ČAP AND KARIN MELNICK Abstract. We use the general theory developed in our article [1] in the setting of parabolic

### 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