DOCSLIB.ORG

17 the Complex Projective Line

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

17 The complex projective line Now we will to study the simplest case of a complex projective space: the complex projective line. We will see that even this case has already very rich geometric interpretations. The close relation of complex arithmetic operation allows us to express geometric properties by nice algebraic structures. In par- ticular, this case will be the first example of a projective space in which we will be properly able to deal with circles. 1 17.1 CP Let us recall how we introduced the real projective line. We took the one- dimensional space R considered it as a euclidean line and added one point at infinity. Topologically we obtained a circle. The best way to express the elements of the real projective line algebraically was to introduce homogeneous coordinates. For this we associated to each number x R the vector (x, 1)T and identified non-zero scalar multiples. Finally we identi∈fied the vector (1, 0)T (and all its non-zero multiples) with the point at infinity. We ended up with the space R2 (0, 0)T RP1 = −{ }. R 0 −{ } We will do exactly the same for the complex numbers! To obtain the complex projective line we start with all the numbers in C. We associate every number z C with the vector (z,1)T and identify non-zero scalar multiples. By this we∈ associate all vectors of the form (a, b)T ; b = 0 to a number in C. What is left is the vector (1, 0)T and all its non-zero# multiples. They will represent a unique point at infinity. All in all we obtain the space CP1 defined by C2 (0, 0)T CP1 = −{ }. C 0 −{ } 268 17 The complex projective line This space is isomorphic to all all complex numbers together with one point at infinity. What is the dimension of this space? In a sense this depends on the point of few. From the perspective of real numbers already the complex plane C is a two-dimensional object, since it requires two real parameters to specify the objects of C. Hence one would say that also CP1 is a real-two-dimensional object since it differs from C just by one point. On the other hand from a complex perspective C is just a one-dimensional object. It contains just one (complex) parameter. As R is the real number line one could consider C as the complex number line. Thus C is complex-one-dimensional, and so is CP1. This is the reason why we call the space CP1 the complex projective line! There is another issue important to mention in this context. Identifying vectors that only differ by a non-zero multiple this time also includes mul- tiplication by complex numbers. Thus (1, 2)T , (3i, 6i)T , (2 + i, 4 + 2i)T all represent the same point. For every point represented by (a, b)T with b =0 we can reconstruct the corresponding number of C by multiplication with 1# /b. The dehomogenized number is then a/b C. We will also frequently identify ∈ CP1 with the space C := C . As in the case of real numbers we use the standard rules for arithmetic∪{ operations∞} with : ∞ ! 1/ = 0; 1/0= ; 1 + = ; + = . ∞ ∞ ∞ ∞ ∞ ∞ ∞ 17.2 Testing geometric properties We will know consider identify the finite part of the complex projective (i.e. C) with the Euclidean plane R2 and investigate how certain geometric properties can be expressed in terms of algebraic expressions in C. Most of these tests will however not correspond to projectively invariant properties. We will first express the property that two vectors associated to two com- plex numbers z1 and z2 point into the same (or in the opposite) direction. For this we simply have to calculate the quotient z1/z2. Using polar coordinates we get (if z = 0) 2 # iψ1 iψ2 i(ψ1−iψ2) z1/z2 =(r1e /r2e ) = (r1/r2)e . If the two vectors point in to the same direction we have ψ1 = ψ2 and the above expression is a positive real number. If they point in opposite directions we have ψ = π + ψ and (since e−iπ = 1) the above number is real and 1 2 − negative. In other word we could say that in the complex plane 0 z1 and z2 are collinear if the quotient z1/z2 is real (provided z2 = 0). Using complex conjugation we can even turn this into an equality, since# z is real if and only if z = z. We get z and z point into the same or opposite direction z /z = z /z . 1 2 ⇐⇒ 1 2 1 2 17.2 Testing geometric properties 269 C z2 B z1 A Fig. 17.1. Testing simple geometric properties. We can use a similar test to decide whether three arbitrary distinct num- bers correspond to collinear points. Let A, B and C be three points in the complex plane. We represent these points by their corresponding complex numbers. Then these three points are collinear if the quotient (B A)/(C A) is real (provided the denominator does not vanish). This can is an− immediate− consequence of our previous considerations since this quotient simply com- pares the directions of the vectors B A and C A. Analogously to the last statement we get: − − A, B, C are collinear (B A)/(C A)=(B A)/(C A). ⇐⇒ − − − − Unfortunately, the last two expressions do not fit into our concepts of projective invariants. The next one, however, will. We will describe whether four points lie commonly on a circle. For this we first need a well known theorem of elementary geometry: the peripheral angle theorem. This theorem states that if we have a circle and a secant from A to B on this circle. Then all points on the circle on one side of the secant “see” the secant under the same angle. The two (invariant) angles on the left and on the right side of the secant sum up to an angle of π. Figure 17.2 illustrates this theorem. We omit a proof here (it can be found in many textbooks of elementary geometry) but we formulate the theorem on the level of complex numbers. For this we denote by ∠A(B, C) the counterclockwise angle between the vectors B A and C A. We can summarize both cases of the peripheral angle theorem− by measuring− angle differences modulo multiples of π. In such a version the peripheral angle theorem reads: Theorem 17.1. Let A and B, C, D be four points on a circle embedded in the complex plane. Then the angle difference ∠C(A, B) ∠D(A, B) is a multiple of π − We postpone a proof of this version of the peripheral angle theorem until we have the possibility to prove it by a simple projective argument. If C and D 270 17 The complex projective line α α C α D α A B A π α − π α B − Fig. 17.2. The peripheral angle theorem. are on the same side of the secant through A and B both angles are equal and the difference is 0 π. If they are on opposite sides of the secant the difference is +π or π depending· on the orientation. We will− interpret this theorem in the light of complex numbers. The angle ∠C (A, B) can be calculated as the angle ψ1 of the following complex number C A − = r eiψ1 . C B 1 − Similarly the angle ∠D(A, B) can be calculated as angle of D A − = r eiψ2 . D B 2 − We can get the difference of the angles simply by dividing these two numbers. We get C A D A − − =(r /r )ei(ψ1−ψ2). C B D B 1 2 − − Since the angle difference is" a multiple of π by the peripheral angle theorem this number must be real. Now, the amazing fact is: the expression on the left is nothing else but a cross-ratio in the complex projective plane. Thus we can say that if the four points are on a circle, this cross-ratio (A, B; C, D) is a real number. The converse is also “almost” true we only have to include the special case that the circle may have infinite radius and degenerates to a line. It is easy to check that if A, B, C, D are collinear the cross-ratio is real, as well. In our considerations we have to consider as real number as well. The cross-ratio assumes this value if either C = B or∞D = A. All in all we obtain the following beautiful theorem that highlights the close relation of complex projective geometry and the geometry of circles. Theorem 17.2. Four points in CP1 are cocircular or collinear if and only if the cross-ratio (A, B; C, D) is in R . ∪{∞} 17.3 Projective transformations 271 17.3 Projective transformations A projective transformation in CP1 can (as in the real case) be expressed T 2 by a matrix multiplication. If the vector (z1,z2) C represents a point in CP1 by homogeneous coordinates. Then a projective∈ transformation can be expressed as τ: CP1 CP1 → z1 ab z1 z2 )→ cd z2 # $ # $# $ As usual the matrix must be non-degenerate. All entries can be complex. Ma- trices differing only by a non-zero scalar multiple represent the same projective transformation. Since the matrix has for complex parameters and scalar mul- tiples will be identified we have three complex degrees of freedom (or six real degrees of freedom, if one prefers).
Recommended publications
  • Differentiable Manifolds

    Differentiable Manifolds

    Prof. A. Cattaneo Institut f¨urMathematik FS 2018 Universit¨atZ¨urich Differentiable Manifolds Solutions to Exercise Sheet 1 p Exercise 1 (A non-differentiable manifold). Consider R with the atlas f(R; id); (R; x 7! sgn(x) x)g. Show R with this atlas is a topological manifold but not a differentiable manifold. p Solution: This follows from the fact that the transition function x 7! sgn(x) x is a homeomor- phism but not differentiable at 0. Exercise 2 (Stereographic projection). Let f : Sn − f(0; :::; 0; 1)g ! Rn be the stereographic projection from N = (0; :::; 0; 1). More precisely, f sends a point p on Sn different from N to the intersection f(p) of the line Np passing through N and p with the equatorial plane xn+1 = 0, as shown in figure 1. Figure 1: Stereographic projection of S2 (a) Find an explicit formula for the stereographic projection map f. (b) Find an explicit formula for the inverse stereographic projection map f −1 (c) If S = −N, U = Sn − N, V = Sn − S and g : Sn ! Rn is the stereographic projection from S, then show that (U; f) and (V; g) form a C1 atlas of Sn. Solution: We show each point separately. (a) Stereographic projection f : Sn − f(0; :::; 0; 1)g ! Rn is given by 1 f(x1; :::; xn+1) = (x1; :::; xn): 1 − xn+1 (b) The inverse stereographic projection f −1 is given by 1 f −1(y1; :::; yn) = (2y1; :::; 2yn; kyk2 − 1): kyk2 + 1 2 Pn i 2 Here kyk = i=1(y ) .
  • HE WANG Abstract. a Mini-Course on Rational Homotopy Theory

    HE WANG Abstract. a Mini-Course on Rational Homotopy Theory

    RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition.
  • An Introduction to Complex Algebraic Geometry with Emphasis on The

    An Introduction to Complex Algebraic Geometry with Emphasis on The

    AN INTRODUCTION TO COMPLEX ALGEBRAIC GEOMETRY WITH EMPHASIS ON THE THEORY OF SURFACES By Chris Peters Mathematisch Instituut der Rijksuniversiteit Leiden and Institut Fourier Grenoble i Preface These notes are based on courses given in the fall of 1992 at the University of Leiden and in the spring of 1993 at the University of Grenoble. These courses were meant to elucidate the Mori point of view on classification theory of algebraic surfaces as briefly alluded to in [P]. The material presented here consists of a more or less self-contained advanced course in complex algebraic geometry presupposing only some familiarity with the theory of algebraic curves or Riemann surfaces. But the goal, as in the lectures, is to understand the Enriques classification of surfaces from the point of view of Mori-theory. In my opininion any serious student in algebraic geometry should be acquainted as soon as possible with the yoga of coherent sheaves and so, after recalling the basic concepts in algebraic geometry, I have treated sheaves and their cohomology theory. This part culminated in Serre’s theorems about coherent sheaves on projective space. Having mastered these tools, the student can really start with surface theory, in particular with intersection theory of divisors on surfaces. The treatment given is algebraic, but the relation with the topological intersection theory is commented on briefly. A fuller discussion can be found in Appendix 2. Intersection theory then is applied immediately to rational surfaces. A basic tool from the modern point of view is Mori’s rationality theorem. The treatment for surfaces is elementary and I borrowed it from [Wi].
  • Circles in a Complex Projective Space

    Circles in a Complex Projective Space

    Adachi, T., Maeda, S. and Udagawa, S. Osaka J. Math. 32 (1995), 709-719 CIRCLES IN A COMPLEX PROJECTIVE SPACE TOSHIAKI ADACHI, SADAHIRO MAEDA AND SEIICHI UDAGAWA (Received February 15, 1994) 0. Introduction The study of circles is one of the interesting objects in differential geometry. A curve γ(s) on a Riemannian manifold M parametrized by its arc length 5 is called a circle, if there exists a field of unit vectors Ys along the curve which satisfies, together with the unit tangent vectors Xs — Ϋ(s)9 the differential equations : FSXS = kYs and FsYs= — kXs, where k is a positive constant, which is called the curvature of the circle γ(s) and Vs denotes the covariant differentiation along γ(s) with respect to the Aiemannian connection V of M. For given a point lEJIί, orthonormal pair of vectors u, v^ TXM and for any given positive constant k, we have a unique circle γ(s) such that γ(0)=x, γ(0) = u and (Psγ(s))s=o=kv. It is known that in a complete Riemannian manifold every circle can be defined for -oo<5<oo (Cf. [6]). The study of global behaviours of circles is very interesting. However there are few results in this direction except for the global existence theorem. In general, a circle in a Riemannian manifold is not closed. Here we call a circle γ(s) closed if = there exists So with 7(so) = /(0), XSo Xo and YSo— Yo. Of course, any circles in Euclidean m-space Em are closed. And also any circles in Euclidean m-sphere Sm(c) are closed.
  • NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A

    NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A

    NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation.
  • Grassmannians Via Projection Operators and Some of Their Special Submanifolds ∗

    Grassmannians Via Projection Operators and Some of Their Special Submanifolds ∗

    GRASSMANNIANS VIA PROJECTION OPERATORS AND SOME OF THEIR SPECIAL SUBMANIFOLDS ∗ IVKO DIMITRIC´ Department of Mathematics, Penn State University Fayette, Uniontown, PA 15401-0519, USA E-mail: [email protected] We study submanifolds of a general Grassmann manifold which are of 1-type in a suitably defined Euclidean space of F−Hermitian matrices (such a submanifold is minimal in some hypersphere of that space and, apart from a translation, the im- mersion is built using eigenfunctions from a single eigenspace of the Laplacian). We show that a minimal 1-type hypersurface of a Grassmannian is mass-symmetric and has the type-eigenvalue which is a specific number in each dimension. We derive some conditions on the mean curvature of a 1-type hypersurface of a Grassmannian and show that such a hypersurface with a nonzero constant mean curvature has at most three constant principal curvatures. At the end, we give some estimates of the first nonzero eigenvalue of the Laplacian on a compact submanifold of a Grassmannian. 1. Introduction The Grassmann manifold of k dimensional subspaces of the vector space − Fm over a field F R, C, H can be conveniently defined as the base man- ∈{ } ifold of the principal fibre bundle of the corresponding Stiefel manifold of F unitary frames, where a frame is projected onto the F plane spanned − − by it. As observed already in 1927 by Cartan, all Grassmannians are sym- metric spaces and they have been studied in such context ever since, using the root systems and the Lie algebra techniques. However, the submani- fold geometry in Grassmannians of higher rank is much less developed than the corresponding geometry in rank-1 symmetric spaces, the notable excep- tions being the classification of the maximal totally geodesic submanifolds (the well known results of J.A.
  • CR Singular Immersions of Complex Projective Spaces

    CR Singular Immersions of Complex Projective Spaces

    Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 2, 451-477. CR Singular Immersions of Complex Projective Spaces Adam Coffman∗ Department of Mathematical Sciences Indiana University Purdue University Fort Wayne Fort Wayne, IN 46805-1499 e-mail: Coff[email protected] Abstract. Quadratically parametrized smooth maps from one complex projective space to another are constructed as projections of the Segre map of the complexifi- cation. A classification theorem relates equivalence classes of projections to congru- ence classes of matrix pencils. Maps from the 2-sphere to the complex projective plane, which generalize stereographic projection, and immersions of the complex projective plane in four and five complex dimensions, are considered in detail. Of particular interest are the CR singular points in the image. MSC 2000: 14E05, 14P05, 15A22, 32S20, 32V40 1. Introduction It was shown by [23] that the complex projective plane CP 2 can be embedded in R7. An example of such an embedding, where R7 is considered as a subspace of C4, and CP 2 has complex homogeneous coordinates [z1 : z2 : z3], was given by the following parametric map: 1 2 2 [z1 : z2 : z3] 7→ 2 2 2 (z2z¯3, z3z¯1, z1z¯2, |z1| − |z2| ). |z1| + |z2| + |z3| Another parametric map of a similar form embeds the complex projective line CP 1 in R3 ⊆ C2: 1 2 2 [z0 : z1] 7→ 2 2 (2¯z0z1, |z1| − |z0| ). |z0| + |z1| ∗The author’s research was supported in part by a 1999 IPFW Summer Faculty Research Grant. 0138-4821/93 $ 2.50 c 2002 Heldermann Verlag 452 Adam Coffman: CR Singular Immersions of Complex Projective Spaces This may look more familiar when restricted to an affine neighborhood, [z0 : z1] = (1, z) = (1, x + iy), so the set of complex numbers is mapped to the unit sphere: 2x 2y |z|2 − 1 z 7→ ( , , ), 1 + |z|2 1 + |z|2 1 + |z|2 and the “point at infinity”, [0 : 1], is mapped to the point (0, 0, 1) ∈ R3.
  • Projective Geometry Lecture Notes

    Projective Geometry Lecture Notes

    Projective Geometry Lecture Notes Thomas Baird March 26, 2014 Contents 1 Introduction 2 2 Vector Spaces and Projective Spaces 4 2.1 Vector spaces and their duals . 4 2.1.1 Fields . 4 2.1.2 Vector spaces and subspaces . 5 2.1.3 Matrices . 7 2.1.4 Dual vector spaces . 7 2.2 Projective spaces and homogeneous coordinates . 8 2.2.1 Visualizing projective space . 8 2.2.2 Homogeneous coordinates . 13 2.3 Linear subspaces . 13 2.3.1 Two points determine a line . 14 2.3.2 Two planar lines intersect at a point . 14 2.4 Projective transformations and the Erlangen Program . 15 2.4.1 Erlangen Program . 16 2.4.2 Projective versus linear . 17 2.4.3 Examples of projective transformations . 18 2.4.4 Direct sums . 19 2.4.5 General position . 20 2.4.6 The Cross-Ratio . 22 2.5 Classical Theorems . 23 2.5.1 Desargues' Theorem . 23 2.5.2 Pappus' Theorem . 24 2.6 Duality . 26 3 Quadrics and Conics 28 3.1 Affine algebraic sets . 28 3.2 Projective algebraic sets . 30 3.3 Bilinear and quadratic forms . 31 3.3.1 Quadratic forms . 33 3.3.2 Change of basis . 33 1 3.3.3 Digression on the Hessian . 36 3.4 Quadrics and conics . 37 3.5 Parametrization of the conic . 40 3.5.1 Rational parametrization of the circle . 42 3.6 Polars . 44 3.7 Linear subspaces of quadrics and ruled surfaces . 46 3.8 Pencils of quadrics and degeneration . 47 4 Exterior Algebras 52 4.1 Multilinear algebra .
  • Toric Polynomial Generators of Complex Cobordism

    Toric Polynomial Generators of Complex Cobordism

    TORIC POLYNOMIAL GENERATORS OF COMPLEX COBORDISM ANDREW WILFONG Abstract. Although it is well-known that the complex cobordism ring is a polynomial U ∼ ring Ω∗ = Z [α1; α2;:::], an explicit description for convenient generators α1; α2;::: has proven to be quite elusive. The focus of the following is to construct complex cobordism polynomial generators in many dimensions using smooth projective toric varieties. These generators are very convenient objects since they are smooth connected algebraic varieties with an underlying combinatorial structure that aids in various computations. By applying certain torus-equivariant blow-ups to a special class of smooth projective toric varieties, such generators can be constructed in every complex dimension that is odd or one less than a prime power. A large amount of evidence suggests that smooth projective toric varieties can serve as polynomial generators in the remaining dimensions as well. 1. Introduction U In 1960, Milnor and Novikov independently showed that the complex cobordism ring Ω∗ is isomorphic to the polynomial ring Z [α1; α2;:::], where αn has complex dimension n [14, 16]. The standard method for choosing generators αn involves taking products and disjoint unions i j of complex projective spaces and Milnor hypersurfaces Hi;j ⊂ CP × CP . This method provides a smooth algebraic not necessarily connected variety in each even real dimension U whose cobordism class can be chosen as a polynomial generator of Ω∗ . Replacing the disjoint unions with connected sums give other choices for polynomial generators. However, the operation of connected sum does not preserve algebraicity, so this operation results in a smooth connected not necessarily algebraic manifold as a complex cobordism generator in each dimension.
  • Complex Algebraic Geometry

    Complex Algebraic Geometry

    Complex Algebraic Geometry Jean Gallier∗ and Stephen S. Shatz∗∗ ∗Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] ∗∗Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] February 25, 2011 2 Contents 1 Complex Algebraic Varieties; Elementary Theory 7 1.1 What is Geometry & What is Complex Algebraic Geometry? . .......... 7 1.2 LocalStructureofComplexVarieties. ............ 14 1.3 LocalStructureofComplexVarieties,II . ............. 28 1.4 Elementary Global Theory of Varieties . ........... 42 2 Cohomologyof(Mostly)ConstantSheavesandHodgeTheory 73 2.1 RealandComplex .................................... ...... 73 2.2 Cohomology,deRham,Dolbeault. ......... 78 2.3 Hodge I, Analytic Preliminaries . ........ 89 2.4 Hodge II, Globalization & Proof of Hodge’s Theorem . ............ 107 2.5 HodgeIII,TheK¨ahlerCase . .......... 131 2.6 Hodge IV: Lefschetz Decomposition & the Hard Lefschetz Theorem............... 147 2.7 ExtensionsofResultstoVectorBundles . ............ 162 3 The Hirzebruch-Riemann-Roch Theorem 165 3.1 Line Bundles, Vector Bundles, Divisors . ........... 165 3.2 ChernClassesandSegreClasses . .......... 179 3.3 The L-GenusandtheToddGenus .............................. 215 3.4 CobordismandtheSignatureTheorem. ........... 227 3.5 The Hirzebruch–Riemann–Roch Theorem (HRR) . ............ 232 3 4 CONTENTS Preface This manuscript is based on lectures given by Steve Shatz for the course Math 622/623–Complex Algebraic Geometry, during Fall 2003 and Spring 2004. The process for producing this manuscript was the following: I (Jean Gallier) took notes and transcribed them in LATEX at the end of every week. A week later or so, Steve reviewed these notes and made changes and corrections. After the course was over, Steve wrote up additional material that I transcribed into LATEX. The following manuscript is thus unfinished and should be considered as work in progress.
  • 2 Lie Groups

    2 Lie Groups

    2 Lie Groups Contents 2.1 Algebraic Properties 25 2.2 Topological Properties 27 2.3 Unification of Algebra and Topology 29 2.4 Unexpected Simplification 31 2.5 Conclusion 31 2.6 Problems 32 Lie groups are beautiful, important, and useful because they have one foot in each of the two great divisions of mathematics — algebra and geometry. Their algebraic properties derive from the group axioms. Their geometric properties derive from the identification of group operations with points in a topological space. The rigidity of their structure comes from the continuity requirements of the group composition and inversion maps. In this chapter we present the axioms that define a Lie group. 2.1 Algebraic Properties The algebraic properties of a Lie group originate in the axioms for a group. Definition: A set gi,gj,gk,... (called group elements or group oper- ations) together with a combinatorial operation (called group multi- ◦ plication) form a group G if the following axioms are satisfied: (i) Closure: If g G, g G, then g g G. i ∈ j ∈ i ◦ j ∈ (ii) Associativity: g G, g G, g G, then i ∈ j ∈ k ∈ (g g ) g = g (g g ) i ◦ j ◦ k i ◦ j ◦ k 25 26 Lie Groups (iii) Identity: There is an operator e (the identity operation) with the property that for every group operation g G i ∈ g e = g = e g i ◦ i ◦ i −1 (iv) Inverse: Every group operation gi has an inverse (called gi ) with the property g g−1 = e = g−1 g i ◦ i i ◦ i Example: We consider the set of real 2 2 matrices SL(2; R): × α β A = det(A)= αδ βγ = +1 (2.1) γ δ − where α,β,γ,δ are real numbers.
  • On the Bordism Ring of Complex Projective Space Claude Schochet

    On the Bordism Ring of Complex Projective Space Claude Schochet

    proceedings of the american mathematical society Volume 37, Number 1, January 1973 ON THE BORDISM RING OF COMPLEX PROJECTIVE SPACE CLAUDE SCHOCHET Abstract. The bordism ring MUt{CPa>) is central to the theory of formal groups as applied by D. Quillen, J. F. Adams, and others recently to complex cobordism. In the present paper, rings Et(CPm) are considered, where E is an oriented ring spectrum, R=7rt(£), andpR=0 for a prime/». It is known that Et(CPcc) is freely generated as an .R-module by elements {ßT\r^0}. The ring structure, however, is not known. It is shown that the elements {/VI^O} form a simple system of generators for £t(CP°°) and that ßlr=s"rß„r mod(/?j, • • • , ßvr-i) for an element s e R (which corre- sponds to [CP"-1] when E=MUZV). This may lead to information concerning Et(K(Z, n)). 1. Introduction. Let E he an associative, commutative ring spectrum with unit, with R=n*E. Then E determines a generalized homology theory £* and a generalized cohomology theory E* (as in G. W. White- head [5]). Following J. F. Adams [1] (and using his notation throughout), assume that E is oriented in the following sense : There is given an element x e £*(CPX) such that £*(S2) is a free R- module on i*(x), where i:S2=CP1-+CP00 is the inclusion. (The Thom-Milnor spectrum MU which yields complex bordism theory and cobordism theory satisfies these hypotheses and is of seminal interest.) By a spectral sequence argument and general nonsense, Adams shows: (1.1) E*(CP ) is the graded ring of formal power series /?[[*]].