Riemann Surfaces Mock Exam
|
|
- Beverley Tucker
- 5 years ago
- Views:
Transcription
1 Riemann Surfaces Mock Exam Notes: 1. Write your name and student number **clearly** on each page of written solutions you hand in. 2. You are expected to give solutions in English or Dutch. 3. You are expected to explain your answers. 4. You are allowed to consult text books but not your own notes. 5. Advice: read all questions first, then start solving the ones you already know how to solve or have good idea on the steps to find a solution. After you have finished the ones you found easier, tackle the harder ones. Exercise 1. Let M be a complex manifold, i.e., M is endowed with a set of coordinates which take values in C n and such that the change of coordinates is holomorphic. Show that a M has a natural almost complex structure I. Show that this natural almost complex structure is (algebraically) integrable, i.e., the +i-eigenspace of I is an involutive subbundle of T C M. Solution. We define an almost complex structure on M as follows. For each p M, we take a holomorphic chart centered at p, ϕ : U M C n, ϕ(p) = 0. We define an automorphism I : T p M and the automorphism of T 0 C n given by multiplication by i. We must check that this is a well defined automorphism. If ψ : U C n is another holomorphic chart centered at p, then we must that that iψ ϕ 1 = ψ ϕ 1 i, but this follows from the fact that ϕ and ψ are holomorphic coordinates, i.e., ψ ϕ 1 is a holoomorphic map, a condition which is defined by the relation above. Since I in coordinate corresponds to multiplication by I, we see that the space Tp 1,0 M is the space generated by the vectors / z i p. Sine this is true for all p, we conclude that the +i-eigenbundle of I is the bundle spanned by / z i is local coordinates and to check integrability it is enough to show that the bracket of generators still lies in the space they generate. In this case [ ], = 1 [ i, i ] = 0 z i z j 4 x i y i x j y j where the last equality follows from the fact that the Lie bracket of coordinate vector fields vanish and the Lie bracket is C-linear. Exercise 2. Let Σ be a Riemann surface of genus g, p Σ and L p be the holomorphic line bundle associated to the divisor p, i.e., L p has a section with a single simple zero at p. Compute the dimensions of H 0 (Σ, L p ) and H 1 (Σ, L p );
2 H 0 (Σ, L p) and H 1 (Σ, L p). Solution. We start with the cohomology with coefficients in L p. Since L p has a holomorphic section s with a simple zero at p, its Euler characteristic is 1. To determine h 0 (Σ, L p ), we observe that any other holomorphic section of L p is given by fs where f is a meromorphic function on Σ which is either holormorphic of has at worse a single simple pole at p (and holomorphic on Σ\{p}). In the first case f is holomorphic in a compact surface hence it is constant. In the second case, f : Σ CP 1 and infinity in CP 1 is a regular value which is taken only once, that is, the degree of f is one and f is and isomorphism between Σ and CP 1. That is, if Σ is not CP 1, any other holomorphic section of L p is a constant multiple of s and hence h 0 (Σ, L p ) = 1 For Σ = CP 1, the space of meromorphic functions with a simple pole, say at z = 0, is given by the set of bi-holomorphisms of CP 1 which map zero to infinity. Since the biholomorphisms of CP 1 are given by Möbius transformations, the condition that zero is a pole means that we are considering Möbius transformations of the form z az + b. z with b 0. Notice however that b = 0 corresponds to the constant holomorphic map z a, which gives rise to the sections of L p which are constant multiples of s. Hence we conclude that H 0 (CP 1, L p ) is 2-dimensional. Finally, Riemann Roch tells us h 0 (Σ, L p ) h 1 (Σ, L p ) = d g + 1 h 1 (Σ, L p ) = g 2 + h 0 (Σ, L p ). So, if g > 0, we get h 1 (Σ, L p ) = g 1 and for g = 0, h 1 (Σ, L p ) = 0. Next we deal with L p. Since L p has a nowhere vanishing meromorphic section with a simple pole at p, we have that the degree of L p is 1. Hence L p has no holomorphic sections, as the formula χ E = #(zeros of s) #(poles of s) shows that a holomorphic line bundle can only have a holomorphic section s (i.e., s has no poles) if its Euler characteristic is positive. Therefore h 0 (Σ, L p) = 0 and Riemann Roch gives h 1 (Σ, L p) = g. Exercise 3. Let λ C, λ 0, 1 and let Σ CP 2 be the Riemann surface determined by the polynomial. and consider the map P (Z 0, Z 1, Z 2 ) = Z 2 0Z 2 2 Z 2 1(Z 1 Z 2 )(Z 1 λz 2 ) π : CP 2 \{[1, 0, 0]} CP 1 π([z 0, Z 1, Z 2 ]) = [Z 1, Z 2 ]. a) Find the singular points of Σ and resolve them to obtain a smooth Riemann surface Σ in an appropriate blow-up of CP 2, b) Show that the map π extends as a holomorphic map to the whole of Σ, c) Find the degree and the branch points of π and determine the Euler characteristic of Σ. Solution.
3 a) First we look for singular points of Σ, i.e., solution to the equations P (Z 0, Z 1, Z 2 ) = 0 and P Z i (Z 0, Z 1, Z 2 ) = 0 for i = 0, 1, 2. Z0Z Z1(Z 2 1 Z 2 )(Z 1 λz 2 ) = 0 2Z 0 Z2 2 = 0 2Z 1 (Z 1 Z 2 )(Z 1 λz 2 ) Z1(Z 2 1 λz 2 ) Z1(Z 2 1 Z 2 ) = 0 2Z0Z Z1(Z 2 1 iλz 2 ) + λz1(z 2 1 Z 2 ) = 0 The second equation implies that at least one of Z 0 and Z 2 must vanish at a singular point. If Z 2 0, then Z 0 = 0 and the first equation means that Z 1 /Z 2 is a root of the polynomial p(z) = z 2 (z 1)(z λ). Similarly, the third equation means that Z 1 /Z 2 is also a solution to p (z) = 0 Since the only double root of p is z = 0, we see that the singular point must be of the form (0, 0, Z 2 ) and that such a point is indeed a solution to all four equations. Hence [0, 0, 1] is a singular point of Σ. Finally, if Z 2 = 0, then the third equation implies Z 1 = 0 and one can verify that (Z 0, 0, 0) is also a solution to the system, hence [1, 0, 0] is also a singular point of Σ. To resolve the singularities of Σ we take charts for CP 2 centered at the singular points and blow them up. We start with [0, 0, 1]. For that point we can take affine the chart (z 0, z 1 ) [z 0, z 1, 1] In this chart, the equation for Σ becomes z 2 0 z 2 1(z 1 1)(z 1 λ) = 0. A neighborhood of the exceptional divisor in the blow-up is parametrized by two charts which relate to the coordinates (z 0, z 1 ) by the following relations u 1 = z 0 ; v 1 = z 1 /z 0 ; u 2 = z 0 /z 1 ; v 2 = z 1. This means that Π, the blow-up map, obtained by writing (z 0, z 1 ) as a function of (u i, v i ) is given by Π(u 1, v 1 ) = (u 1, u 1 v 1 ) Π(u 2, v 2 ) = (u 2 v 2, v 2 ). In the (u 1, v 1 )-coordinate system, the pre-image of Σ\{[0, 0, 1]} is given by (Σ\{[0, 0, 1]}) = {(u 1, v 1 ) : u 2 1 u 2 1v 2 1(u 1 1)(u 1 λ) = 0 & u 1 0} (Σ\{[0, 0, 1]}) = {(u 1, v 1 ) : 1 v 2 1(u 1 1)(u 1 λ) = 0 & u 1 0} And the proper transform of Σ in this chart is given by (Σ\{[0, 0, 1]}) = {(u 1, v 1 ) : 1 v 2 1(u 1 1)(u 1 λ) = 0} and this is a smooth Riemann surface in this chart, as it is given as the zero of a function f(u 1, v 1 ) = 1 v 2 1(u 1 1)(u 1 λ) for which f = 0, f/ u 1 = 0 and f/ v 1 = 0 do not have a simultaneous solution. In the (u 2, v 2 )-coordinate system, the pre-image of Σ\{[0, 0, 1]} is given by (Σ\{[0, 0, 1]}) = {(u 2, v 2 ) : u 2 2v 2 2 v 2 2(u 2 v 2 1)(u 2 v 2 λ) = 0 & v 2 0} (Σ\{[0, 0, 1]}) = {(u 2, v 2 ) : u 2 2 (u 2 v 2 1)(u 2 v 2 λ) = 0 & v 2 0} And the proper transform of Σ in this chart is given by (Σ\{[0, 0, 1]}) = {(u 2, v 2 ) : u 2 2 (u 2 v 2 1)(u 2 v 2 λ) = 0} and this is a smooth Riemann surface in this chart, as it is given as the zero of a function f(u 2, v 2 ) = u 2 2 (u 2 v 2 1)(u 2 v 2 λ) = 0 for which f = 0, f/ u 2 = 0 and f/ v 2 = 0 do not have a simultaneous solution.
4 Notice that the blow-up process resolved the singularity of Σ at [0, 0, 1] and produced a suface which intersects the exceptional divisor in tow points, namely, in the (u 2, v 2 ) chart, these points are the solution to u 2 2 (u 2 v 2 1)(u 2 v 2 λ) = 0 & v 2 = 0. and hence u 2 = ±λ 1 2. Next we resolve the point [1, 0, 0]. Here we take affine coordinates so that Σ is given by the equation This time, the blow-up coordinates are given by Or, in terms of the blow-up map so In the (u 3, v 3 )-coordinate system, we have (z 1, z 2 ) [1, z 1, z 2 ] z 2 2 z 2 1(z 1 z 2 )(z 1 λz 2 ) = 0. u 3 = z 1, v 3 = z 2 /z 1 ; u 4 = z 1 /z 2, v 4 = z 2. Π 2 (u 3, v 3 ) = (u 3, u 3 v 3 ) Π 2 (u 4, v 4 ) = (u 4 v 4, v 4 ). 2 (Σ\{[1, 0, 0]}) = {(u 3, v 3 ) : u 2 3v 2 3 u 2 3(u 3 u 3 v 3 )(u 3 λu 3 v 3 ) = 0 & u 3 0} 2 (Σ\{[1, 0, 0]}) = {(u 3, v 3 ) : v 2 3 u 2 3(1 v 3 )(1 λv 3 ) = 0 & u 3 0} 2 (Σ\{[1, 0, 0]}) = {(u 3, v 3 ) : v 2 3 u 2 3(1 v 3 )(1 λv 3 ) = 0} Notice that in this case the proper transform of Σ is not yet smooth and (u 3, v 3 ) = (0, 0) is a singular point,so we have to blow his point up We have two new coordinate charts, (u 5, v 5 ) and (u 6, v 6 ) which relate to (u 3, v 3 ) by the blow-up map In the (u 5, v 5 )-coordinate system we have Π 3 (u 5, v 5 ) = (u 5, u 5 v 5 ) Π 3 (u 6, v 6 ) = (u 6 v 6, v 6 ). 3 2 (Σ\{[1, 0, 0]}) = {(u 5, v 5 ) : u 2 5v 2 5 u 2 5(1 u 5 v 5 )(1 λu 5 v 5 ) = 0 & u 5 0} 3 2 (Σ\{[1, 0, 0]}) = {(u 5, v 5 ) : v 2 5 (1 u 5 v 5 )(1 λu 5 v 5 ) = 0 & u 5 0}. And the proper transform is 3 2 (Σ\{[1, 0, 0]}) = {(u 5, v 5 ) : v 2 5 (1 u 5 v 5 )(1 λu 5 v 5 ) = 0}. which is now the equation of a smooth surface. The points on intersection of the resolution with the exceptional divisor correspond to v 5 = ±1. A similar computation in the chart (u 6, v 6 ) shows that (u 6, v 6 ) = (0, 0), the only point of the last exceptional divisor not included in the precious parametrization, is not part of the proper transform of Σ, so the proper transform of Σ is smooth in the charts considered so far. We have only missed the chart (u 4, v 4 ). In this chart, the proper transform of Σ is given by the equation and this is a smooth surface in this chart. 1 u 2 4v 2 4(u 4 1)(u 4 λ) = 0
5 So we conclude that after three blow-ups, we have resolved Σ into a smooth Riemann surface. b) Here we show that the map π extends to CP 2, the blow-up of CP 2 at the point [1, 0, 0] and hence it extends to Σ. To prove the claim, we simply write the map π in terms of the coordinates (u 3, v 3 ) and U 4, v 4 ) of the previous part of the exercise: π Π 2 (u 3, v 3 ) = π([1, u 3, u 3 v 3 ]) = [u 3, u 3 v 3 ] = [1, v 3 ]. π Π 2 (u 4, v 4 ) = π([1, u 4 v 4, v 4 ]) = [u 4 v 4, v 4 ] = [u 4, 1]. Showing that π Π 2 is well defined and holomorphic smooth on the blow up of CP 2 at the point [1, 0, 0]. Above we constructed a manifold CP 2, which is a blow-up of CP 2 at two points, so we have a blow-up map Π : CP 2 CP 2 since π Π 2 is smooth, we conclude that π Π 2 Π is smooth and Σ is a smooth submanifold of CP 2 hence the restriction of this smooth map to Σ is also smooth. c) In the (z 1, z 2 )-coordinates, we have that Σ is given by z 2 2 z 2 1(z 1 z 2 )(z 1 λz 2 ) = 0. For each point [a, b] CP 1, we have that the point (z 1, z 2 ) Σ maps to [a, b] if and only if z 1 = (a/b)z 2, so we see that generically, the points of Σ which map to a fixed point [a, b] CP 1 are solutions to a quadratic polynomial, and the degree of π is 2. Still in this chart, we see that if a/b = α is fixed, then the elements of Σ which project to [a, b] are given by z 2 2 α 2 z 4 2(α 1)(α λ) = 0 i.e. for α 0, 1λ there are always two solutions and for α = 0, 1 or λ the equation would force z 2 = 0 and hence z 1 = 0, which is a singular point of Σ. Similarly, if we fix b/a = β we obtain that the elements of Σ which project to [a, b] are given by β 2 z 2 1 z 4 1(1 β)(1 βλ) = 0. and β = 0 forces z 1 = 0 and hence z 2 = 0 This means that α = 0, 1, λ, are the only possible singular values of π. From b) we know that π Π 2 is well defined and that at the singular point (u 3, v 3 ) = (0, 0), π Π 2 (0, 0) = [1, 0]. Since the blow-up that resolves the singularity at (u 3, v 3 ) = (0, 0) introduced two points, we see that π 1 ([1, 0]) has two point in Σ, hence [1, 0] is not a branch point for π : Σ CP 1. Next, we must check the points which were not covered by the coordinates (z 1, z 2 ). All the remaining points are in the affine chart (z 0, z 1 ) [z 0, z 1, 1]. There Σ is given by z 2 0 z 2 1(z 1 1)(z 1 λ) and the map π is simply projection onto the z 1 -plane. From the above we see that z 1 = 0 is not a singular value, since the resolution of the singularity (z 0, z 1 ) = (0, 0) introduced two points both of which are mapped to [0, 1]. On the other hand, if z 1 = 1 or λ then the only value of z 0 that solves the euation above is z 0 = 0 and hence these are ramification points. So [1, 1] and [λ, 1] are the only ramification points of π and we can compute the Euler characteristic of Σ by the Riemann Hurwiz formula: So Σ is CP 1. χ Σ = dχ CP 1 b = = 2.
6 Exercise 4. Let f : Σ 1 Σ 2 be a non constant holomorphic map between compact connected Riemann surfaces. Let α Ω 1,0 (Σ 2 ) be a meromorphic 1-form on Σ 2 whose set of zeros and poles is disjoint from the set of critical values of f. By counting the number of zeros and poles of f α, re-obtain the Riemann Hurwitz formula: χ Σ1 = deg(f) χ Σ2 b, where b is the total branch number of f. Solution. Under the hypotheses of the exercise, for each point p Σ 2 which is a zero or a pole of α, f 1 (p) has deg(f) points and at any such point q f 1 (p) f α is a zero or a pole or the same order, since once can find coordinates centered at p andq for which f is simply the identity map. Now, if q Σ 1 is a singular point for f with branch number k and p = f(q), then we can find coordinates centered at p and q such that f is given by w = f(z) = z k and since α does not vanish at p, it can be written as α = a(w)dw, with a(0) 0. Then f α = f (a(w)dw) = a(f(z))d(f w) = a(f(z))d(z k ) = kz k 1 a(f(z))dz. That is f α has a zero of order k 1 at q. Since this is true for all singular points q, we see that the number of zeros of f α is deg(f) times the number of zeros of α plus the total branch number of f, while the number of poles of f αis simply the number of poles of ]alpha times the degree of f. Since the Euler characteristic of a Riemann surface is given by the number of poles minus the number of zeros of any meromorphic 1-form, we obtain the result.
Part II. Riemann Surfaces. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised
More informationRiemann Surfaces and Algebraic Curves
Riemann Surfaces and Algebraic Curves JWR Tuesday December 11, 2001, 9:03 AM We describe the relation between algebraic curves and Riemann surfaces. An elementary reference for this material is [1]. 1
More informationThe 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
More informationRiemann surfaces. Paul Hacking and Giancarlo Urzua 1/28/10
Riemann surfaces Paul Hacking and Giancarlo Urzua 1/28/10 A Riemann surface (or smooth complex curve) is a complex manifold of dimension one. We will restrict to compact Riemann surfaces. It is a theorem
More informationFAMILIES OF ALGEBRAIC CURVES AS SURFACE BUNDLES OF RIEMANN SURFACES
FAMILIES OF ALGEBRAIC CURVES AS SURFACE BUNDLES OF RIEMANN SURFACES MARGARET NICHOLS 1. Introduction In this paper we study the complex structures which can occur on algebraic curves. The ideas discussed
More informationis holomorphic. In other words, a holomorphic function is a collection of compatible holomorphic functions on all charts.
RIEMANN SURFACES 2. Week 2. Basic definitions 2.1. Smooth manifolds. Complex manifolds. Let X be a topological space. A (real) chart of X is a pair (U, f : U R n ) where U is an open subset of X and f
More informationLet X be a topological space. We want it to look locally like C. So we make the following definition.
February 17, 2010 1 Riemann surfaces 1.1 Definitions and examples Let X be a topological space. We want it to look locally like C. So we make the following definition. Definition 1. A complex chart on
More information1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3
Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces
More informationAlgebraic 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
More informationEach 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.
More informationFrom the definition of a surface, each point has a neighbourhood U and a homeomorphism. U : ϕ U(U U ) ϕ U (U U )
3 Riemann surfaces 3.1 Definitions and examples From the definition of a surface, each point has a neighbourhood U and a homeomorphism ϕ U from U to an open set V in R 2. If two such neighbourhoods U,
More informationAN INTRODUCTION TO ARITHMETIC AND RIEMANN SURFACE. We describe points on the unit circle with coordinate satisfying
AN INTRODUCTION TO ARITHMETIC AND RIEMANN SURFACE 1. RATIONAL POINTS ON CIRCLE We start by asking us: How many integers x, y, z) can satisfy x 2 + y 2 = z 2? Can we describe all of them? First we can divide
More informationX G X by the rule x x g
18. Maps between Riemann surfaces: II Note that there is one further way we can reverse all of this. Suppose that X instead of Y is a Riemann surface. Can we put a Riemann surface structure on Y such that
More informationExplicit Examples of Strebel Differentials
Explicit Examples of Strebel Differentials arxiv:0910.475v [math.dg] 30 Oct 009 1 Introduction Philip Tynan November 14, 018 In this paper, we investigate Strebel differentials, which are a special class
More informationAbelian Varieties and Complex Tori: A Tale of Correspondence
Abelian Varieties and Complex Tori: A Tale of Correspondence Nate Bushek March 12, 2012 Introduction: This is an expository presentation on an area of personal interest, not expertise. I will use results
More informationIntroduction to. Riemann Surfaces. Lecture Notes. Armin Rainer. dim H 0 (X, L D ) dim H 0 (X, L 1 D ) = 1 g deg D
Introduction to Riemann Surfaces Lecture Notes Armin Rainer dim H 0 (X, L D ) dim H 0 (X, L 1 D ) = 1 g deg D June 17, 2018 PREFACE i Preface These are lecture notes for the course Riemann surfaces held
More informationNOTES ON DIVISORS AND RIEMANN-ROCH
NOTES ON DIVISORS AND RIEMANN-ROCH NILAY KUMAR Recall that due to the maximum principle, there are no nonconstant holomorphic functions on a compact complex manifold. The next best objects to study, as
More informationExercises for algebraic curves
Exercises for algebraic curves Christophe Ritzenthaler February 18, 2019 1 Exercise Lecture 1 1.1 Exercise Show that V = {(x, y) C 2 s.t. y = sin x} is not an algebraic set. Solutions. Let us assume that
More informationComplex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool
Complex Algebraic Geometry: Smooth Curves Aaron Bertram, 2010 12. First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool for classifying smooth projective curves, i.e. giving
More informationwhere m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism
8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the
More informationPROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013
PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013 1. Problems on moduli spaces The main text for this material is Harris & Morrison Moduli of curves. (There are djvu files
More informationGAUGED LINEAR SIGMA MODEL SPACES
GAUGED LINEAR SIGMA MODEL SPACES FELIPE CASTELLANO-MACIAS ADVISOR: FELIX JANDA Abstract. The gauged linear sigma model (GLSM) originated in physics but it has recently made it into mathematics as an enumerative
More informationmult V f, where the sum ranges over prime divisor V X. We say that two divisors D 1 and D 2 are linearly equivalent, denoted by sending
2. The canonical divisor In this section we will introduce one of the most important invariants in the birational classification of varieties. Definition 2.1. Let X be a normal quasi-projective variety
More informationFROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS
FROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS ZHIQIN LU. Introduction It is a pleasure to have the opportunity in the graduate colloquium to introduce my research field. I am a differential geometer.
More information1. Divisors on Riemann surfaces All the Riemann surfaces in this note are assumed to be connected and compact.
1. Divisors on Riemann surfaces All the Riemann surfaces in this note are assumed to be connected and compact. Let X be a Riemann surface of genus g 0 and K(X) be the field of meromorphic functions on
More informationGEOMETRY OF SYMMETRIC POWERS OF COMPLEX DOMAINS. Christopher Grow
GEOMETRY OF SYMMETRIC POWERS OF COMPLEX DOMAINS Christopher Grow A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts Department of Mathematics Central Michigan
More informationMath 213br HW 3 solutions
Math 13br HW 3 solutions February 6, 014 Problem 1 Show that for each d 1, there exists a complex torus X = C/Λ and an analytic map f : X X of degree d. Let Λ be the lattice Z Z d. It is stable under multiplication
More informationConformal 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
More informationCOMPLEX ANALYSIS-II HOMEWORK
COMPLEX ANALYSIS-II HOMEWORK M. LYUBICH Homework (due by Thu Sep 7). Cross-ratios and symmetries of the four-punctured spheres The cross-ratio of four distinct (ordered) points (z,z 2,z 3,z 4 ) Ĉ4 on the
More informationThe Geometry of Cubic Maps
The Geometry of Cubic Maps John Milnor Stony Brook University (www.math.sunysb.edu) work with Araceli Bonifant and Jan Kiwi Conformal Dynamics and Hyperbolic Geometry CUNY Graduate Center, October 23,
More informationGeometry Qualifying Exam Notes
Geometry Qualifying Exam Notes F 1 F 1 x 1 x n Definition: The Jacobian matrix of a map f : N M is.. F m F m x 1 x n square matrix, its determinant is called the Jacobian determinant.. When this is a Definition:
More informationLECTURE 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
More informationAn invitation to log geometry p.1
An invitation to log geometry James M c Kernan UCSB An invitation to log geometry p.1 An easy integral Everyone knows how to evaluate the following integral: 1 0 1 1 x 2 dx. An invitation to log geometry
More informationClass Numbers, Continued Fractions, and the Hilbert Modular Group
Class Numbers, Continued Fractions, and the Hilbert Modular Group Jordan Schettler University of California, Santa Barbara 11/8/2013 Outline 1 Motivation 2 The Hilbert Modular Group 3 Resolution of the
More informationLECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in
More informationz, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1
3 3 THE RIEMANN SPHERE 31 Models for the Riemann Sphere One dimensional projective complex space P(C ) is the set of all one-dimensional subspaces of C If z = (z 1, z ) C \ 0 then we will denote by [z]
More informationThe Theorem of Gauß-Bonnet in Complex Analysis 1
The Theorem of Gauß-Bonnet in Complex Analysis 1 Otto Forster Abstract. The theorem of Gauß-Bonnet is interpreted within the framework of Complex Analysis of one and several variables. Geodesic triangles
More informationRIEMANN SURFACES. LECTURE NOTES. WINTER SEMESTER 2015/2016.
RIEMANN SURFACES. LECTURE NOTES. WINTER SEMESTER 2015/2016. A PRELIMINARY AND PROBABLY VERY RAW VERSION. OLEKSANDR IENA Contents Some prerequisites for the whole lecture course. 5 1. Lecture 1 5 1.1. Definition
More informationLECTURE 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.
More informationTopic: 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
More informationAlgebraic 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,
More informationPICARD S THEOREM STEFAN FRIEDL
PICARD S THEOREM STEFAN FRIEDL Abstract. We give a summary for the proof of Picard s Theorem. The proof is for the most part an excerpt of [F]. 1. Introduction Definition. Let U C be an open subset. A
More information1 Compact Riemann surfaces and algebraic curves
1 Compact Riemann surfaces and algebraic curves 1.1 Basic definitions 1.1.1 Riemann surfaces examples Definition 1.1 A topological surface X is a Hausdorff topological space provided with a collection
More informationComplex Analysis Problems
Complex Analysis Problems transcribed from the originals by William J. DeMeo October 2, 2008 Contents 99 November 2 2 2 200 November 26 4 3 2006 November 3 6 4 2007 April 6 7 5 2007 November 6 8 99 NOVEMBER
More informationAn introduction to holomorphic dynamics in one complex variable Informal notes Marco Abate
An introduction to holomorphic dynamics in one complex variable Informal notes Marco Abate Dipartimento di Matematica, Università di Pisa Largo Pontecorvo 5, 56127 Pisa E-mail: abate@dm.unipi.it November
More informationA TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY. We also have an isomorphism of holomorphic vector bundles
A TASTE OF TWO-DIMENSIONAL COMPLEX ALGEBRAIC GEOMETRY LIVIU I. NICOLAESCU ABSTRACT. These are notes for a talk at a topology seminar at ND.. GENERAL FACTS In the sequel, for simplicity we denote the complex
More information1. The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions.
Complex Analysis Qualifying Examination 1 The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions 2 ANALYTIC FUNCTIONS:
More informationMT845: ALGEBRAIC CURVES
MT845: ALGEBRAIC CURVES DAWEI CHEN Contents 1. Sheaves and cohomology 1 2. Vector bundles, line bundles and divisors 9 3. Preliminaries on curves 16 4. Geometry of Weierstrass points 27 5. Hilbert scheme
More informationRiemann surfaces. 3.1 Definitions
3 Riemann surfaces In this chapter we define and give the first properties of Riemann surfaces. These are the holomorphic counterpart of the (real) differential manifolds. We will see how the Fuchsian
More informationComplex 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
More informationThe Bierstone Lectures on Complex Analysis
The Bierstone Lectures on Complex Analysis Scribed by Oleg Ivrii during Spring 2007 Preface These notes of my lectures on Complex Analysis at the University of Toronto were written by Oleg Ivrii on his
More informationSolutions to practice problems for the final
Solutions to practice problems for the final Holomorphicity, Cauchy-Riemann equations, and Cauchy-Goursat theorem 1. (a) Show that there is a holomorphic function on Ω = {z z > 2} whose derivative is z
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 6
COMPLEX ALGEBRAIC SURFACES CLASS 6 RAVI VAKIL CONTENTS 1. The intersection form 1.1. The Neron-Severi group 3 1.. Aside: The Hodge diamond of a complex projective surface 3. Riemann-Roch for surfaces 4
More informationDefinition 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
More informationLecture 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
More informationInverse Galois Problem for C(t)
Inverse Galois Problem for C(t) Padmavathi Srinivasan PuMaGraSS March 2, 2012 Outline 1 The problem 2 Compact Riemann Surfaces 3 Covering Spaces 4 Connection to field theory 5 Proof of the Main theorem
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 4
COMPLEX ALGEBRAIC SURFACES CLASS 4 RAVI VAKIL CONTENTS 1. Serre duality and Riemann-Roch; back to curves 2 2. Applications of Riemann-Roch 2 2.1. Classification of genus 2 curves 3 2.2. A numerical criterion
More informationHolomorphic 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
More informationBoris Springborn Riemann Surfaces Mini Lecture Notes
Boris Springborn Riemann Surfaces Mini Lecture Notes Technische Universität München Winter Semester 2012/13 Version February 7, 2013 Disclaimer. This is not (and not meant to be) a textbook. It is more
More information5.3 The Upper Half Plane
Remark. Combining Schwarz Lemma with the map g α, we can obtain some inequalities of analytic maps f : D D. For example, if z D and w = f(z) D, then the composition h := g w f g z satisfies the condition
More informationRiemann s goal was to classify all complex holomorphic functions of one variable.
Math 8320 Spring 2004, Riemann s view of plane curves Riemann s goal was to classify all complex holomorphic functions of one variable. 1) The fundamental equivalence relation on power series: Consider
More informationTotally Marked Rational Maps. John Milnor. Stony Brook University. ICERM, April 20, 2012 [ ANNOTATED VERSION]
Totally Marked Rational Maps John Milnor Stony Brook University ICERM, April 20, 2012 [ ANNOTATED VERSION] Rational maps of degree d 2. (Mostly d = 2.) Let K be an algebraically closed field of characteristic
More informationMinimal 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
More informationRiemann sphere and rational maps
Chapter 3 Riemann sphere and rational maps 3.1 Riemann sphere It is sometimes convenient, and fruitful, to work with holomorphic (or in general continuous) functions on a compact space. However, we wish
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about
More informationAN INTRODUCTION TO MODULI SPACES OF CURVES CONTENTS
AN INTRODUCTION TO MODULI SPACES OF CURVES MAARTEN HOEVE ABSTRACT. Notes for a talk in the seminar on modular forms and moduli spaces in Leiden on October 24, 2007. CONTENTS 1. Introduction 1 1.1. References
More informationOral exam practice problems: Algebraic Geometry
Oral exam practice problems: Algebraic Geometry Alberto García Raboso TP1. Let Q 1 and Q 2 be the quadric hypersurfaces in P n given by the equations f 1 x 2 0 + + x 2 n = 0 f 2 a 0 x 2 0 + + a n x 2 n
More information. Then g is holomorphic and bounded in U. So z 0 is a removable singularity of g. Since f(z) = w 0 + 1
Now we describe the behavior of f near an isolated singularity of each kind. We will always assume that z 0 is a singularity of f, and f is holomorphic on D(z 0, r) \ {z 0 }. Theorem 4.2.. z 0 is a removable
More informationALGEBRAIC GEOMETRY AND RIEMANN SURFACES
ALGEBRAIC GEOMETRY AND RIEMANN SURFACES DANIEL YING Abstract. In this thesis we will give a present survey of the various methods used in dealing with Riemann surfaces. Riemann surfaces are central in
More informationFAKE 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.
More informationCR SINGULAR IMAGES OF GENERIC SUBMANIFOLDS UNDER HOLOMORPHIC MAPS
CR SINGULAR IMAGES OF GENERIC SUBMANIFOLDS UNDER HOLOMORPHIC MAPS JIŘÍ LEBL, ANDRÉ MINOR, RAVI SHROFF, DUONG SON, AND YUAN ZHANG Abstract. The purpose of this paper is to organize some results on the local
More informationRIEMANN 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
More informationSome 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
More informationExotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group
Exotic Lefschetz Fibrations and Stein Fillings with Arbitrary Fundamental Group Anar Akhmedov University of Minnesota, Twin Cities February 19, 2015 Anar Akhmedov (University of Minnesota, Minneapolis)Exotic
More informationEXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION
EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION JAN CHRISTIAN ROHDE Introduction By string theoretical considerations one is interested in Calabi-Yau manifolds since Calabi-Yau 3-manifolds
More informationSummary 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
More informationRIEMANN SURFACES. max(0, deg x f)x.
RIEMANN SURFACES 10. Weeks 11 12: Riemann-Roch theorem and applications 10.1. Divisors. The notion of a divisor looks very simple. Let X be a compact Riemann surface. A divisor is an expression a x x x
More informationK-stability and Kähler metrics, I
K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates
More informationMODULI SPACES AND INVARIANT THEORY 95. X s X
MODULI SPACES AND INVARIANT THEORY 95 6.7. Hypersurfaces. We will discuss some examples when stability is easy to verify. Let G be a reductive group acting on the affine variety X with the quotient π :
More informationMath 797W Homework 4
Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition
More informationPMATH 950 (Riemann Surfaces) Notes
PMATH 950 (Riemann Surfaces) Notes Patrick Naylor February 28, 2018 These are notes for PMATH 950, taught by Ruxandra Moraru in the Winter 2018 term at the University of Waterloo. These notes are not guaranteed
More information14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski
14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski topology are very large, it is natural to view this as
More information0. Introduction 1 0. INTRODUCTION
0. Introduction 1 0. INTRODUCTION In a very rough sketch we explain what algebraic geometry is about and what it can be used for. We stress the many correlations with other fields of research, such as
More informationRecall 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
More informationCHAPTER 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
More informationAlgebraic 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
More informationM g = {isom. classes of smooth projective curves of genus g} and
24 JENIA TEVELEV 2. Elliptic curves: j-invariant (Jan 31, Feb 4,7,9,11,14) After the projective line P 1, the easiest algebraic curve to understand is an elliptic curve (Riemann surface of genus 1). Let
More informationThe topology of symplectic four-manifolds
The topology of symplectic four-manifolds Michael Usher January 12, 2007 Definition A symplectic manifold is a pair (M, ω) where 1 M is a smooth manifold of some even dimension 2n. 2 ω Ω 2 (M) is a two-form
More informationRIEMANN MAPPING THEOREM
RIEMANN MAPPING THEOREM VED V. DATAR Recall that two domains are called conformally equivalent if there exists a holomorphic bijection from one to the other. This automatically implies that there is an
More informationQUALIFYING 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.
More informationMath 213br HW 12 solutions
Math 213br HW 12 solutions May 5 2014 Throughout X is a compact Riemann surface. Problem 1 Consider the Fermat quartic defined by X 4 + Y 4 + Z 4 = 0. It can be built from 12 regular Euclidean octagons
More informationBroken pencils and four-manifold invariants. Tim Perutz (Cambridge)
Broken pencils and four-manifold invariants Tim Perutz (Cambridge) Aim This talk is about a project to construct and study a symplectic substitute for gauge theory in 2, 3 and 4 dimensions. The 3- and
More informationDISTINGUISHING EMBEDDED CURVES IN RATIONAL COMPLEX SURFACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 16, Number 1, January 1998, Pages 305 310 S 000-9939(98)04001-5 DISTINGUISHING EMBEDDED CURVES IN RATIONAL COMPLEX SURFACES TERRY FULLER (Communicated
More informationF (z) =f(z). f(z) = a n (z z 0 ) n. F (z) = a n (z z 0 ) n
6 Chapter 2. CAUCHY S THEOREM AND ITS APPLICATIONS Theorem 5.6 (Schwarz reflection principle) Suppose that f is a holomorphic function in Ω + that extends continuously to I and such that f is real-valued
More informationQualifying Exam Complex Analysis (Math 530) January 2019
Qualifying Exam Complex Analysis (Math 53) January 219 1. Let D be a domain. A function f : D C is antiholomorphic if for every z D the limit f(z + h) f(z) lim h h exists. Write f(z) = f(x + iy) = u(x,
More informationMorse Theory and Applications to Equivariant Topology
Morse Theory and Applications to Equivariant Topology Morse Theory: the classical approach Briefly, Morse theory is ubiquitous and indomitable (Bott). It embodies a far reaching idea: the geometry and
More informationComplex Analysis Important Concepts
Complex Analysis Important Concepts Travis Askham April 1, 2012 Contents 1 Complex Differentiation 2 1.1 Definition and Characterization.............................. 2 1.2 Examples..........................................
More informationA NEW FAMILY OF SYMPLECTIC FOURFOLDS
A NEW FAMILY OF SYMPLECTIC FOURFOLDS OLIVIER DEBARRE This is joint work with Claire Voisin. 1. Irreducible symplectic varieties It follows from work of Beauville and Bogomolov that any smooth complex compact
More informationΩ Ω /ω. To these, one wants to add a fourth condition that arises from physics, what is known as the anomaly cancellation, namely that
String theory and balanced metrics One of the main motivations for considering balanced metrics, in addition to the considerations already mentioned, has to do with the theory of what are known as heterotic
More informationQualifying Examination HARVARD UNIVERSITY Department of Mathematics Tuesday, January 19, 2016 (Day 1)
Qualifying Examination HARVARD UNIVERSITY Department of Mathematics Tuesday, January 19, 2016 (Day 1) PROBLEM 1 (DG) Let S denote the surface in R 3 where the coordinates (x, y, z) obey x 2 + y 2 = 1 +
More information