# Chern classes à la Grothendieck

Size: px
Start display at page:

Transcription

1 Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces all important properties of Chern classes from a small number of axioms imposed on some given data. These axioms are in particular fulfilled if one inputs the category of smooth quasi-projective varieties with their intersection theory, thus obtaining the familiar theory of Chern classes from this more general setup. Contents 1 Projective bundles Vector bundles and locally free sheaves Projective bundles The splitting principle Input and axioms Input Axioms Some fundamental lemmas Chern classes 7 4 Appendix 1: Example of a projective bundle 9 5 Appendix 2: Example of a Chern class computation 11 Conventions We work in the setting of Grothendieck s article, i.e. X will always denote an algebraic variety: an integral, separated scheme of finite type over an algebraically closed field k. If one is so inclined, there is always Fulton s book [1] which treats a more general case. Disclaimer A big part of this note is shamelessly copied from either Gathmann [3] or Grothendieck [4]. This has not been proofread thoroughly, so if you find any mistakes, please tell me. 1

2 1 Projective bundles Grothendieck s approach is based on taking iterated projective bundles, so in this section we give their definitions and some basic properties. We start out by reviewing some of the theory of vector bundles. 1.1 Vector bundles and locally free sheaves Definition 1. Let X be a scheme. A sheaf of X -modules is called locally free of rank r if there is an open cover {U i } of X such that Ui = r X for all i. Definition 2. A vector bundle of rank r over a field k is a k-scheme F and a k- morphism π : F X, together with an open covering {U i } of X and isomorphisms (1) ψ i : π 1 (U i ) U i r k, such that the automorphism ψ i ψ 1 j of (U i U j ) r is linear in the coordinates of r. Proposition 3. There is a one-to-one correspondence between vector bundles of rank r on X and locally free sheaves of rank r on X. Proof. To a vector bundle F one associates the sheaf defined by (2) (U) = {k morphisms s : U Fsuch that π s = id U }, which is called the sheaf of sections. Conversely, let be a locally free sheaf. Take an open cover {U i } of X such that there are isomorphisms ψ i : Ui r U i. Now glue the schemes U i r k together along the isomorphism (3) (U i U j ) r k (U i U j ) r k : (p, x) (p, (ψ i ψ 1 j )(x)). Notice that linearity follows from the fact that ψ i ψ 1 j is a morphism of X - modules. The following lemmas are easy to prove and show that locally free sheaves on an arbitrary scheme are nice, i.e. all linear algebra constructions go through. The second lemma is used very, very often in the literature but is easily forgotten, at least by the author. Lemma 4. Locally free sheaves are closed under direct sums, tensor products, symmetric products, exterior products, duals and pullbacks. Lemma 5. Let 0 0 be an exact sequence of locally free sheaves of ranks f, g and h on a scheme X. Then g = f h. From now on we will use the terms vector bundle and locally free sheaf interchangeably. 2

3 1.2 Projective bundles Informally, for a given vector bundle F on X, the associated projective bundle (F) replaces each fibre F x, x X with its projectivization (F x ), so (F) x = (F x ). Let us make this precise. Definition 6. Let π : F X be a vector bundle of rank r on a scheme X. The projective bundle (F) is defined by glueing U i r 1 to U j r 1 along the isomorphisms (4) (U i U j ) r 1 (U i U j ) r 1 : (p, x) (p, ψ i,j x). One says that (F) is a projective bundle of rank r 1 on X. Notice that the corresponding projection morphism pr : (F) X is proper (since properness if local on the base ), which is not the case for vector bundles. We should remark that the general construction of (projective) bundles is as follows. Starting from a locally free sheaf, the associated vector bundle (repsectively projective bundle) is defined to be (5) Spec S( ), respectively Proj S( ), i.e. one takes the relative spec (proj) of the symmetric algebra associated to, see Hartshorne, Ex. II.5.18, II It is important to note that if X is a variety, then Proj S( ) is as well. The property of being (quasi-)projective also passes to projective bundles. This follows from a more general result on blowups, see Hartshorne, Prop. II This is not true for ordinary vector bundles Spec S( )! We now want to construct a canonical line bundle on (F), called the tautological subbundle. Let π : (F) X denote the projection map, and consider the pullback bundle π F on (F). The corresponding open covering of (F) is (U i r 1 ) and the bundle is made up of patches (U i r 1 ) r, which are glued along the isomorphisms (6) (U i U j ) r 1 ) r (U i U j ) r 1 ) r : (p, x, y) (p, ψ i,j x, ψ i,j y). Definition 7. The tautological subbundle L F on (F) is the rank 1 subbundle of π F given locally by equations (7) x i y j = x j y i, for i, j = 1,..., r. Geometrically, the fiber of L F over a point (p, x) (F) is the line in the fiber F p whose projectivization is x. The reason for taking duals will become clear later on. 1.3 The splitting principle One would like to have a nice composition series for any vector bundle F of rank r over X. By this we mean a filtration by subbundles (8) 0 = F 0 F 1... F r 1 F r = F, 3

4 such that F i /F i 1 is a line bundle on X. In general this is not possible, but it is possible after pulling back the bundle to another variety. This theorem is often referred to as the splitting principle. Theorem 8. Let F be a rank r vector bundle on X, then there exists a variety Y and a morphism f : Y X such that f F has a filtration by vector bundles (9) 0 = F 0 F 1 F r 1 F r = f F, such that rk F i = i. In fact, Y can be constructed as an iterated projective bundle. Proof. We prove the statement by induction. For rk F = 1, it is trivial, so suppose F has rank strictly greater than 1. Let Y := (F ) with associated morphism f : Y X. Remember that L (f ) F denotes the tautological line F bundle. By dualizing we get a surjective morphism such that the kernel is a vector bundle: (10) 0 F (f ) F L F 0. Now F has rank one less than the rank of F so by induction there is a variety Y and a morphism f : Y Y such that (f ) F has a filtration with subquotient line bundles. It now suffices to set f = f f such that f F has a filtration (11) 0 = F 0 F 1... F r 1 = (f ) F f F and we re done. From the proof it is clear that Y can be constructed as iterated projective bundle. 2 Input and axioms In this section we introduce the formal data Grothendieck needs to obtain a nice theory of Chern classes. Chern classes are invariants associated to a vector bundle E over a variety X. The idea, which Grothendieck attributes to Chern, is to use the multiplicative structure on the ring of classes of algebraic cycles on the projective bundle associated to a vector bundle E on X to obtain an explicit construction of the Chern classes associated to E. In the setting we are interested in, the Chern classes live in the Chow ring CH (X ) of X, but in fact, this is not the only possibility, and Grothendieck s framework is general enough to cover other interesting case, for which we refer to the article. 2.1 Input Let V denote some category of smooth algebraic varieties over k, where the morphisms are morphisms of algebraic varieties. This category has to satisfy: V 1 : If X V, and F is a vector bundle on X, then (F) V. Further, one needs the following as input: 4

5 1. A contravariant functor A : V GCR, where GCR denotes the category of graded commutative unital rings, i.e. one has x y = ( 1) deg x deg y y x; 2. A functorial homomorphism of abelian groups p X : Pic(X ) A 2 (X ), for X V; 3. Let i : Y X V be a closed algebraic subvariety of constant codimension p in X, such that Y is also in V. Then there is a group homomorphism (12) i : A(Y ) A(X ), increasing the degree by 2p. We need some more notation: for a morphism f : X Z in V, we will denote f := A(f ) : A(Z) A(X ). The unit of A(X ) will be denoted 1 X, and for i and Y as in 3. above, we define (13) p X (Y ) := i (1 Y ). Also, if F is a vector bundle on X, remember that L F denotes the tautological subbundle of (F). Then using 2. above, we define ξ F as follows (14) ξ F := p (F) (L F ) A 2 ( (F)). Also notice that A( (F)) can be considered as a left A(X )-module by applying the functor A to the projection morphism pr : (F) X. We will refer to the input items as I.1, I.2, I.3. Smooth projective varieties We briefly review this input for V be the category of smooth projective varieties over k. Details can be found in [1, 2]. The condition V 1 is satisfied by the remark in the section on projective bundles. The functor A = CH, which sends a variety to its Chow ring with doubled degree, i.e. we put CH i in degree 2i, since the Chow ring is commutative (for other natural occurring A this is not the case. For a morphism of nonsingular varieties f : X Y, one can define a pullback by α γ f (α [Y ]), where γ f denotes the graph of f. That this works and is well defined can be found in [2]. The functorial morphism p X is just the map sending a Cartier divisor to its associated Weil divisor: a Cartier divisor is represented by the data {(U i ), f i }, and one sends this to the Weil divisor V ord V U i (f )[V ], where V runs through the codimension 1 subvarieties of X. Notice that this is in fact an isomorphism for all smooth n-dimensional schemes, which uses a deep theorem from commutative algebra, the Auslander-Buchsbaum theorem, which says that regular local rings are unique factorization domains. Finally, for the third datum the induced pushforward map just sends the class of a subvariety [Z] of codimension l in Y to [Z]. So in X it is of codimension p + l, corresponding to a degree increase of 2p. 2.2 Axioms Given the input in the previous section, Grothendieck requires them to satisfy the following four axioms. 5

6 1. For X V, and F a vector bundle of rank r on X, the elements (15) 1 (F), ξ F, ξ 2 F,..., ξr 1 F form a basis of the A(X )-module A( (F)). 2. For X V, L a line bundle on X, and s a regular section of L transversal to the zero section, such that s 1 (0) V, one has (16) p X (s 1 (0)) = p X (L). 3. For Z i Y j X, all belonging to V, one has (17) (j i) = j i. 4. For Y i X, both belonging to V, one has (18) i (bi (a)) = i (b)a, for a A(X ), b A(Y ). We will refer to these as A.1 through A.4. Smooth projective varieties For smooth projective varieties, axiom 1 is quite hard, and can be found for example as Theorem 3.3 (b)in [1]. Axiom 2 follows immediately from the familiar correspondence between Cartier divisors and line bundles. Axiom 3 is immediate from the definition of the pushforward we gave in the Inputs section. Axiom 4 is known as the projection formula and can also be found in [1]. 2.3 Some fundamental lemmas The input and axioms can be split in two subsets, each with a specific purpose. The map i discussed in I.3, and the axioms pertaining to i, namely A.2, A.3 and A.4 have as main purpose the proof of the following technical lemma. Lemma 9. Let X V, F a vector bundle of rank r on X, and s a regular section of F. Further, let (19) F = F 0 F 1 F r 1 F r = 0 be a decreasing sequence of subbundles of F, such that rk(f i ) = r i. For each i = 1,..., r, define (20) Y i = {x X s(x) F i }, and suppose that Y i is a non-singular subvariety of X, which is contained in V. Now let s i be the section of (F i /F i+1 ) Yi induced by s, and suppose that all the s i are transversal to the zero section. If one finally defines (21) ξ i = p X (F i 1 /F i ), then one concludes that (22) p X (Y r ) = ξ i. 1 i r 6

7 Proof. We will prove that (23) p X (Y j ) = ξ j, 1 i j by induction on j. For j = 1 this is just axiom 2. Denote by Y j+1 respective inclusion. Now one finds i Y j u j X the (24) p Yj (Y j+1 ) = A.2 p Yj ([F j /F j + 1] Yj ) = p Yj (u j [F j/f j+1 ]) = I.2 u j p X (F j /F j+1 ) = u j (ξ j+1). Now apply u j to the equality and in the next equality A.3 on the LHS to get u j i (1 Yj+1 ) = u j u j ξ j+1 (25) p X (Y j=1 ) = u j (1 Yj u j ξ j+1) = A.4 u j (1 Yj ) ξ j+1 = p X (Y j )ξ j+1. Now it suffices to use the induction hypothesis. Perhaps more important still is the following corollary which immediately follows by combining the lemma with A.2. Corollary 10. Under the conditions of lemma 9, and assuming that s vanishes nowhere, one has (26) ξ i = 0. 1 i r Then there is one more lemma, which is crucial for the uniqueness property of Chern classes. Lemma 11. Let X V, and F a vector bundle of rank r on X. Let f : Y X be the morphism obtained by the splitting principle. Then the induced morphism (27) f : A(X ) A(Y ) is injective. Proof. By our construction of the Y as iterated projective bundle, we can assume to be working with (F), and here the statement follows immediately from V 1 and axiom 1, since it says ξ 0 F = 1 (F) is free over A(X ). An inductive argument then shows the claim. 3 Chern classes To introduce Chern classes and describe their characterising properties, we will need corollary 10, I.1, I.2 and A.1. 7

8 From axiom 1, we immediately find that there exist unique elements c i (F) A 2i (X ) for every natural number i 0 such that (28) r c i (F)(ξ F ) r i = 0 c 0 (F) = 1 c i (F) = 0 for i > r. i=0 Definition 12. The c i (F) defined above is called the i-th Chern class of F. The sum of all Chern classes is denoted (29) c(f) = c i (F), i and is called the total Chern class of F. The following theorem completely describes the Chern classes in terms of their properties, which are tailored to computability. Theorem 13. The Chern classes defined above satisfy the following 3 properties: 1. Functoriality: let f : X Y be a morphism in V, and F a vector bundle on Y, then (30) c(f F) = f (c(e)); 2. Normalization: let L be a line bundle on X V. Then (31) c(l) = 1 + p X (L); 3. Additivity: for X V, and 0 F F F 0 an exact sequence of vector bundles on X, one has (32) c(f) = c(f )c(f ). Moreover, these 3 properties uniquely characterize Chern classes given the input and axioms. Let us first show the uniqueness statement, since this basically tells one how to actually compute a Chern class from the input. Let f : Y X be the map from the splitting principle. Since Y is an iterated projective bundle, by V 1 we know that Y V. By lemma 11, we know the associated map f : A(X ) A(Y ) is injective. So if we know f (c(f)), then we know c(f). But now by functoriality, one has f (c(f)) = c(f F). We know from the splitting principle that f F has a filtration by line bundles, so from additivity one obtains (33) c(f) = r c(f i 1 /F i ) = i=1 r (1 + p X (F i 1 /F i )), i=1 where we used normalization in the last step. Functoriality is not that hard, but we skip the proof. Let us first show normallity. Since L is a line bundle, we have (L) = X and L L, the tautological line bundle, is just L. So now (34) ξ L = p (L) (L L ) = p X (L ) = p X (L). 8

9 Now writing out equation 28 for a line bundle we get ξ L +c 1 (L) = 0, so c 1 (L) = p X (L) and normality follows since c 0 (L) = 1. It is only additivity that requires real work. Let us give a sketch of a proof. Using functoriality and the splitting principle in a similar way as before, one can reduce to the question whether additivity holds when F and F have a complete filtration by subbundles. Then obviously F also has such a filtration and it will suffice to show that for every composition series of F, F and F thus obtained, the equation 33 holds. So in effect, it remains to show equation 33 for a vector bundle F that has a complete filtration by subbundles. Consider the following diagram (35) f F F (F) f X and let L F denote the tautological subbundle of (F). Let (F i) i be a complete filtration by subbundles of F. Defining F = L F f F, the filtration on F gives a filtration on this bundle with factors F i 1 /F i = L F (f F i 1 /f F i ), so in the Picard group of (F), we get the equality (36) F i 1 /F i = L F + (f F i 1 /f F i ). Applying the group morphism p (F), we get (37) p (F) (F i 1 /F i ) = ξ F + ξ i, where ξ i = p (F)(f F i 1 /f F i ), just like in lemma 9. Now from the inclusion L F f F, one obtains a non-vanishing section s of F that turns out to be transversal to the zero section (I m skipping the transversality proof). This allows one to apply corollary 10 to obtain (38) (ξ F + ξ i ) = 0. 1 i r This says that the c i (F), defined by equation 28, are elementary symmetric functions in the ξ i, which is exactly what equation 33 says (remember that this equation involves a pullback). Using the theory of symmetric functions, one can deduce formula for the Chern classes of tensor products, exterior products and duals. 4 Appendix 1: Example of a projective bundle Let X = 1 and let F be the rank 2 vector bundle X X ( 1) on X. Then (F) is a projective bundle of rank 1, so it is a scheme of dimension 2. Our claim is that (F) is isomorphic to the blow-up of 2 in a point p. At least intuitively, it is clear that the blow-up should be a 1 -bundle over 1, since one can project onto the exceptional divisor. So we are looking for a rank 2 bundle F 9

10 on the projective line. Since every vector bundle on 1 splits as a direct sum of line bundles, we know F is of the form (d ) (d ). Now tensoring a vector bundle with a line bundle multiplies the transition functions by a scalar, so it does not change the associated projective bundle, so we can assume that F = ( n), for some n 0. It is not too hard to see that this n gives rise to a curve of self-intersection n so in fact n = 1. Let us check this formally: (F) is obtained by glueing two copies U 1 and U 2 of 1 1 along the isomorphism (39) ( 1 \{0}) 1 ( 1 \{0}) 1 : (q, (x 1 : x 2 )) (1/q, (x 1 : qx 2 )). The changes in the affine coordinate q correspond to the glueing one uses to obtain 1. The first projective coordinate gets sent to itself, because of X and the second one to qx 2 because of X ( 1). The blow-up of 2 in p = (1 : 0 : 0) on the other hand is given by (40) Bl p ( 2 ) = {((x 0 : x 1 : x 2 ), (y 1 : y 2 )) x 1 y 2 = x 2 y 1 } 2 1. An explicit isomorphism between the two varieties is given by (41) U 1 = 1 1 Bl p ( 2 ) : (q, (x 1, x 2 )) ((x 1 : qx 2 : x 2 ), (q : 1)), U 2 = 1 1 Bl p ( 2 ) : (q, (x 1 : x 2 )) ((x 1 : x 2 : qx 2 ), (1 : q)), and note that these morphisms are compatible with the glueing isomorphism (39). Now let us compute the Chow groups of (F). The computation is based on the following proposition. Proposition 14. Let X be a scheme stratified by affine spaces, i.e. there is a filtration by closed subschemes (42) = X 1 X 0 X n = X, such that X k \X k 1 = k k, with k appearing a k times. Then A k (X ) = a k. The projective plane has a stratification 2 1 0, so identifying 0 with p, the projective bundle (F) has a stratification Denoting by q any point in (F), by L the strict transform of a line in 2 through p and by E the exceptional divisor, one finds (43) A 0 (X ) = [q] A 1 (X ) = [L] [E] A 2 (X ) = [ (F)]. In fact, one can explicitly check that there is no relation in A 1 (X ). Suppose that n[l]+m[e] = 0. Let π : (F) 2 be the projection to the base of the blow-up. This is proper and (44) 0 = π (0) = π (n[l] + m[e]) = n[m] + m 0 A 1 ( 2 ), where [M] is the class of a line in 2, so n = 0. Denote by f : (F) 1 the 1 - bundle map. Then (45) 0 = f (0) = f (n[l] + m[e]) = n 0 + m[ 1 ], 10

11 so m = 0. One can also show that for any line H in (F) not intersecting E, one has [H] = [L] + [E] A 1 ( (F)). Now Pic( (F)) = [H] [E], so to compute the intersection products it will thus suffice to compute H 2, H E and E 2. First of all, clearly H 2 = 1 and H E = 0. Now (46) E 2 = E (H L) = E H E L = 0 1 = 1. All in all, the Chow ring has the following presentation (47) A ( (F)) = [x, y, z]/(x 2 = z, x y = 0, y 2 = z), where deg(x) = deg(y) = 1 and deg(z) = 2. 5 Appendix 2: Example of a Chern class computation Consider X = 1 1. It is not hard to check that (48) CH (X ) = [x, y]/(x 2, y 2 ) and intuitively, this is clear since lines in the same ruling do not intersect and lines in a different ruling intersect in one point, corresponding to the polynomial x y. We consider the problem of determining the kernel of the morphism (49) Hom( (0, 1), (1, 1)) (0, 1) (1, 1) 0, which should be a vector bundle F of rank 1. There is very easy way of doing this, pointed out to me by Dennis Presotto, using the (told you I always forget) lemma 5, but at least the following illustrates how a Chern class computation can be carried out. We ll do it step by step, to clearly illustrate what is going on. By additivity, we know that (50) c(f) c( (1, 1)) = c( (0, 1) (0, 1)). Now using additivity and then normality, the RHS becomes (51) c( (0, 1) (0, 1)) = c( (0, 1)) c( (0, 1)) = (1 + y)(1 + y) = 1 + 2y. For the LHS we find using normality and the group operation in Pic(X ) that (52) c( (1, 1)) = 1 + p X ( (1, 1)) = 1 + p X ( (1, 0)) + p X ( (0, 1)) = 1 + x + y. Now it is easy to check that (53) c(f) = 1 x + y, and using the isomorphism between CH 1 (X ) and Pic(X ), we see that F = ( 1, 1). 11

12 References [1] W. Fulton: Intersection theory [2] W. Fulton: Introduction to intersection theory in algebraic geometry [3] A. Gathmann: Algebraic geometry, see ~gathmann/class/alggeom-2002/main.pdf [4] A. Grothendieck: La théorie des classes de Chern 12

### INTERSECTION THEORY CLASS 12

INTERSECTION THEORY CLASS 12 RAVI VAKIL CONTENTS 1. Rational equivalence on bundles 1 1.1. Intersecting with the zero-section of a vector bundle 2 2. Cones and Segre classes of subvarieties 3 2.1. Introduction

### INTERSECTION THEORY CLASS 7

INTERSECTION THEORY CLASS 7 RAVI VAKIL CONTENTS 1. Intersecting with a pseudodivisor 1 2. The first Chern class of a line bundle 3 3. Gysin pullback 4 4. Towards the proof of the big theorem 4 4.1. Crash

### INTERSECTION THEORY CLASS 6

INTERSECTION THEORY CLASS 6 RAVI VAKIL CONTENTS 1. Divisors 2 1.1. Crash course in Cartier divisors and invertible sheaves (aka line bundles) 3 1.2. Pseudo-divisors 3 2. Intersecting with divisors 4 2.1.

### INTERSECTION THEORY CLASS 19

INTERSECTION THEORY CLASS 19 RAVI VAKIL CONTENTS 1. Recap of Last day 1 1.1. New facts 2 2. Statement of the theorem 3 2.1. GRR for a special case of closed immersions f : X Y = P(N 1) 4 2.2. GRR for closed

### APPENDIX 3: AN OVERVIEW OF CHOW GROUPS

APPENDIX 3: AN OVERVIEW OF CHOW GROUPS We review in this appendix some basic definitions and results that we need about Chow groups. For details and proofs we refer to [Ful98]. In particular, we discuss

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

### n P say, then (X A Y ) P

COMMUTATIVE ALGEBRA 35 7.2. The Picard group of a ring. Definition. A line bundle over a ring A is a finitely generated projective A-module such that the rank function Spec A N is constant with value 1.

### Preliminary Exam Topics Sarah Mayes

Preliminary Exam Topics Sarah Mayes 1. Sheaves Definition of a sheaf Definition of stalks of a sheaf Definition and universal property of sheaf associated to a presheaf [Hartshorne, II.1.2] Definition

### Porteous s Formula for Maps between Coherent Sheaves

Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for

### Cycle groups for Artin stacks

Cycle groups for Artin stacks arxiv:math/9810166v1 [math.ag] 28 Oct 1998 Contents Andrew Kresch 1 28 October 1998 1 Introduction 2 2 Definition and first properties 3 2.1 The homology functor..........................

### 12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

### Algebraic Cobordism. 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006.

Algebraic Cobordism 2nd German-Chinese Conference on Complex Geometry East China Normal University Shanghai-September 11-16, 2006 Marc Levine Outline: Describe the setting of oriented cohomology over a

### Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle

Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,

### We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle

### MA 206 notes: introduction to resolution of singularities

MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k 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

### ALGEBRAIC GEOMETRY I, FALL 2016.

ALGEBRAIC GEOMETRY I, FALL 2016. DIVISORS. 1. Weil and Cartier divisors Let X be an algebraic variety. Define a Weil divisor on X as a formal (finite) linear combination of irreducible subvarieties of

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### Fourier Mukai transforms II Orlov s criterion

Fourier Mukai transforms II Orlov s criterion Gregor Bruns 07.01.2015 1 Orlov s criterion In this note we re going to rely heavily on the projection formula, discussed earlier in Rostislav s talk) and

### AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES

AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES MATTHEW H. BAKER AND JÁNOS A. CSIRIK This paper was written in conjunction with R. Hartshorne s Spring 1996 Algebraic Geometry course at

### 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

### Section Blowing Up

Section 2.7.1 - Blowing Up Daniel Murfet October 5, 2006 Now we come to the generalised notion of blowing up. In (I, 4) we defined the blowing up of a variety with respect to a point. Now we will define

### Oral 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

### AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES

AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are C-linear. 1.

### the complete linear series of D. Notice that D = PH 0 (X; O X (D)). Given any subvectorspace V H 0 (X; O X (D)) there is a rational map given by V : X

2. Preliminaries 2.1. Divisors and line bundles. Let X be an irreducible complex variety of dimension n. The group of k-cycles on X is Z k (X) = fz linear combinations of subvarieties of dimension kg:

### APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY

APPENDIX 1: REVIEW OF SINGULAR COHOMOLOGY In this appendix we begin with a brief review of some basic facts about singular homology and cohomology. For details and proofs, we refer to [Mun84]. We then

### Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

### STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY

STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY ALEXANDER MERKURJEV 1. Introduction Let p be a prime integer. For a pair of topological spaces A X we write H i (X, A; Z/pZ) for the i-th singular cohomology group

### 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

### HARTSHORNE EXERCISES

HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

### 3. Lecture 3. Y Z[1/p]Hom (Sch/k) (Y, X).

3. Lecture 3 3.1. Freely generate qfh-sheaves. We recall that if F is a homotopy invariant presheaf with transfers in the sense of the last lecture, then we have a well defined pairing F(X) H 0 (X/S) F(S)

### Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally

### LINE BUNDLES ON PROJECTIVE SPACE

LINE BUNDLES ON PROJECTIVE SPACE DANIEL LITT We wish to show that any line bundle over P n k is isomorphic to O(m) for some m; we give two proofs below, one following Hartshorne, and the other assuming

### INTERSECTION THEORY CLASS 16

INTERSECTION THEORY CLASS 16 RAVI VAKIL CONTENTS 1. Where we are 1 1.1. Deformation to the normal cone 1 1.2. Gysin pullback for local complete intersections 2 1.3. Intersection products on smooth varieties

### 3. The Sheaf of Regular Functions

24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice

### 1 Notations and Statement of the Main Results

An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main

### Topics in Algebraic Geometry

Topics in Algebraic Geometry Nikitas Nikandros, 3928675, Utrecht University n.nikandros@students.uu.nl March 2, 2016 1 Introduction and motivation In this talk i will give an incomplete and at sometimes

### which is a group homomorphism, such that if W V U, then

4. Sheaves Definition 4.1. Let X be a topological space. A presheaf of groups F on X is a a function which assigns to every open set U X a group F(U) and to every inclusion V U a restriction map, ρ UV

### LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL

LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for non-stable-rationality based on the decomposition of the diagonal in the Chow group. This criterion

### 10. Smooth Varieties. 82 Andreas Gathmann

82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves of ideals, and closed subschemes 1 2. Invertible sheaves (line bundles) and divisors 2 3. Some line bundles on projective

### De Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)

II. De Rham Cohomology There is an obvious similarity between the condition d o q 1 d q = 0 for the differentials in a singular chain complex and the condition d[q] o d[q 1] = 0 which is satisfied by the

### ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF

ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF MATTHEW H. BAKER AND JÁNOS A. CSIRIK Abstract. We give a new proof of the isomorphism between the dualizing sheaf and the canonical

### ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0.

ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. ANDREW SALCH During the last lecture, we found that it is natural (even just for doing undergraduatelevel complex analysis!)

### Construction of M B, M Dol, M DR

Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant

### DIVISORS ON NONSINGULAR CURVES

DIVISORS ON NONSINGULAR CURVES BRIAN OSSERMAN We now begin a closer study of the behavior of projective nonsingular curves, and morphisms between them, as well as to projective space. To this end, we introduce

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,

### Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism

Algebraic Cobordism Lecture 1: Complex cobordism and algebraic cobordism UWO January 25, 2005 Marc Levine Prelude: From homotopy theory to A 1 -homotopy theory A basic object in homotopy theory is a generalized

### Lecture 9 - Faithfully Flat Descent

Lecture 9 - Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of faithfully flat descent, which is the notion that to obtain an object on a scheme X,

### 6. Lecture cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in Suslin-Voevodsky s approach to not

6. Lecture 6 6.1. cdh and Nisnevich topologies. These are Grothendieck topologies which play an important role in Suslin-Voevodsky s approach to not only motivic cohomology, but also to Morel-Voevodsky

### 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

### Algebraic varieties and schemes over any scheme. Non singular varieties

Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two

### (1) is an invertible sheaf on X, which is generated by the global sections

7. Linear systems First a word about the base scheme. We would lie to wor in enough generality to cover the general case. On the other hand, it taes some wor to state properly the general results if one

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves

### LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT

LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral D-module, unless explicitly stated

### Geometry 9: Serre-Swan theorem

Geometry 9: Serre-Swan theorem Rules: You may choose to solve only hard exercises (marked with!, * and **) or ordinary ones (marked with! or unmarked), or both, if you want to have extra problems. To have

### 1 Existence of the Néron model

Néron models Setting: S a Dedekind domain, K its field of fractions, A/K an abelian variety. A model of A/S is a flat, separable S-scheme of finite type X with X K = A. The nicest possible model over S

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48 RAVI VAKIL CONTENTS 1. A little more about cubic plane curves 1 2. Line bundles of degree 4, and Poncelet s Porism 1 3. Fun counterexamples using elliptic curves

### BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

### Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

### WHY IS O(1) CALLED THE TWISTING SHEAF?: THE BUNDLES O(n) ON PROJECTIVE SPACE

WHY IS O(1) CALLED THE TWISTING SHEAF?: THE BUNDLES O(n) ON PROJECTIVE SPACE JONATHAN COX 1. Introduction Question: Why is O(1) called the twisting sheaf? For simplicity we specialize to P 1 C during most

### THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3

THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3 KAZUNORI NAKAMOTO AND TAKESHI TORII Abstract. There exist 26 equivalence classes of k-subalgebras of M 3 (k) for any algebraically closed field

### Systems of linear equations. We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K,

Systems of linear equations We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K, f 11 t 1 +... + f 1n t n = 0, f 21 t 1 +... + f 2n t n = 0,.

### Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 43 AND 44 RAVI VAKIL CONTENTS 1. Flat implies constant Euler characteristic 1 2. Proof of Important Theorem on constancy of Euler characteristic in flat families

### is a short exact sequence of locally free sheaves then

3. Chern classes We have already seen that the first chern class gives a powerful way to connect line bundles, sections of line bundles and divisors. We want to generalise this to higher rank. Given any

### PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES

PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES Angelo Vistoli Scuola Normale Superiore Bordeaux, June 23, 2010 Joint work with Niels Borne Université de Lille 1 Let X be an algebraic variety over C, x 0 X. What

### Coherent sheaves on elliptic curves.

Coherent sheaves on elliptic curves. Aleksei Pakharev April 5, 2017 Abstract We describe the abelian category of coherent sheaves on an elliptic curve, and construct an action of a central extension of

### AN INTRODUCTION TO AFFINE SCHEMES

AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,

LINKED ALTERNATING FORMS AND LINKED SYMPLECTIC GRASSMANNIANS BRIAN OSSERMAN AND MONTSERRAT TEIXIDOR I BIGAS Abstract. Motivated by applications to higher-rank Brill-Noether theory and the Bertram-Feinberg-Mukai

### Section Projective Morphisms

Section 2.7 - Projective Morphisms Daniel Murfet October 5, 2006 In this section we gather together several topics concerned with morphisms of a given scheme to projective space. We will show how a morphism

### Picard Groups of Affine Curves

Picard Groups of Affine Curves Victor I. Piercey University of Arizona Math 518 May 7, 2008 Abstract We will develop a purely algebraic definition for the Picard group of an affine variety. We will then

### 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,

### ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES

ARITHMETICALLY COHEN-MACAULAY BUNDLES ON HYPERSURFACES N. MOHAN KUMAR, A. P. RAO, AND G. V. RAVINDRA Abstract. We prove that any rank two arithmetically Cohen- Macaulay vector bundle on a general hypersurface

### IndCoh Seminar: Ind-coherent sheaves I

IndCoh Seminar: Ind-coherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means -category ). This section contains a discussion of

### 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

### Smooth morphisms. Peter Bruin 21 February 2007

Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,

### h M (T ). The natural isomorphism η : M h M determines an element U = η 1

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli

### R-Equivalence and Zero-Cycles on 3-Dimensional Tori

HU Yong R-Equivalence and Zero-Cycles on 3-Dimensional Tori Master thesis, defended on June 19, 2008 Thesis advisor: Dr. DORAY Franck Mathematisch Instituut, Universiteit Leiden Acknowledgements I would

### Cohomology and Base Change

Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)

### Serre s Problem on Projective Modules

Serre s Problem on Projective Modules Konrad Voelkel 6. February 2013 The main source for this talk was Lam s book Serre s problem on projective modules. It was Matthias Wendt s idea to take the cuspidal

### Motivic integration on Artin n-stacks

Motivic integration on Artin n-stacks Chetan Balwe Nov 13,2009 1 / 48 Prestacks (This treatment of stacks is due to B. Toën and G. Vezzosi.) Let S be a fixed base scheme. Let (Aff /S) be the category of

### 9. Birational Maps and Blowing Up

72 Andreas Gathmann 9. Birational Maps and Blowing Up In the course of this class we have already seen many examples of varieties that are almost the same in the sense that they contain isomorphic dense

### 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

### Segre classes of tautological bundles on Hilbert schemes of surfaces

Segre classes of tautological bundles on Hilbert schemes of surfaces Claire Voisin Abstract We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 18

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 18 CONTENTS 1. Invertible sheaves and divisors 1 2. Morphisms of schemes 6 3. Ringed spaces and their morphisms 6 4. Definition of morphisms of schemes 7 Last day:

### Intersection Theory course notes

Intersection Theory course notes Valentina Kiritchenko Fall 2013, Faculty of Mathematics, NRU HSE 1. Lectures 1-2: examples and tools 1.1. Motivation. Intersection theory had been developed in order to

### Curves on P 1 P 1. Peter Bruin 16 November 2005

Curves on P 1 P 1 Peter Bruin 16 November 2005 1. Introduction One of the exercises in last semester s Algebraic Geometry course went as follows: Exercise. Let be a field and Z = P 1 P 1. Show that the

### ALGEBRAIC CYCLES ON THE FANO VARIETY OF LINES OF A CUBIC FOURFOLD arxiv: v2 [math.ag] 18 Apr INTRODUCTION

ALGEBRAIC CYCLES ON THE FANO VARIETY OF LINES OF A CUBIC FOURFOLD arxiv:1609.05627v2 [math.ag] 18 Apr 2018 KALYAN BANERJEE ABSTRACT. In this text we prove that if a smooth cubic in P 5 has its Fano variety

### ON A THEOREM OF CAMPANA AND PĂUN

ON A THEOREM OF CAMPANA AND PĂUN CHRISTIAN SCHNELL Abstract. Let X be a smooth projective variety over the complex numbers, and X a reduced divisor with normal crossings. We present a slightly simplified

### Special cubic fourfolds

Special cubic fourfolds 1 Hodge diamonds Let X be a cubic fourfold, h H 2 (X, Z) be the (Poincaré dual to the) hyperplane class. We have h 4 = deg(x) = 3. By the Lefschetz hyperplane theorem, one knows

### Characteristic classes in the Chow ring

arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic

### PERVERSE SHEAVES ON A TRIANGULATED SPACE

PERVERSE SHEAVES ON A TRIANGULATED SPACE A. POLISHCHUK The goal of this note is to prove that the category of perverse sheaves constructible with respect to a triangulation is Koszul (i.e. equivalent to

### 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

### K-theory, Chow groups and Riemann-Roch

K-theory, Chow groups and Riemann-Roch N. Mohan Kumar October 26, 2004 We will work over a quasi-projective variety over a field, though many statements will work for arbitrary Noetherian schemes. The

### Geometric invariant theory

Geometric invariant theory Shuai Wang October 2016 1 Introduction Some very basic knowledge and examples about GIT(Geometric Invariant Theory) and maybe also Equivariant Intersection Theory. 2 Finite flat

### LINKED HOM SPACES BRIAN OSSERMAN

LINKED HOM SPACES BRIAN OSSERMAN Abstract. In this note, we describe a theory of linked Hom spaces which complements that of linked Grassmannians. Given two chains of vector bundles linked by maps in both

### ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need