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Home , Ample line bundle, Canonical bundle, Dual bundle, Euler sequence, Line bundle, Projective bundle

VECTOR BUNDLES AND PROJECTIVE VARIETIES

by

NICHOLAS MARINO

Submitted in partial fulfillment of the requirements for the degree of

Master of Science

Department of , Applied Mathematics, and Statistics

CASE WESTERN RESERVE UNIVERSITY

January 2019 CASE WESTERN RESERVE UNIVERSITY

Department of Mathematics, Applied Mathematics, and

Statistics

We hereby approve the thesis of

Nicholas Marino

Candidate for the degree of Master of Science

Committee Chair

Nick Gurski

Committee Member

David Singer

Committee Member

Joel Langer

Date of Defense: 10 December, 2018

1 Contents

Abstract 3

1 Introduction 4

2 Basic Constructions 5 2.1 Elementary Definitions ...... 5 2.2 Bundles ...... 8 2.3 Divisors ...... 12 2.4 Differentials ...... 13 2.5 Chern Classes ...... 14

3 Moduli Spaces 17 3.1 Some Classifications ...... 17 3.2 Stable and Semi-stable Sheaves ...... 19 3.3 Representability ...... 21

4 Vector Bundles on Pn 26 4.1 Cohomological Tools ...... 26 4.2 Splitting on Higher Projective Spaces ...... 27 4.3 Stability ...... 36

5 Low-Dimensional Results 37 5.1 2-bundles and Surfaces ...... 37 5.2 Serre’s Construction and Hartshorne’s Conjecture ...... 39 5.3 The Horrocks-Mumford ...... 42

6 Ulrich Bundles 44

7 Conclusion 48

8 References 50

2 Vector Bundles and Projective Varieties

Abstract

by

NICHOLAS MARINO

Vector bundles play a prominent role in the study of projective algebraic varieties. Vector bundles can describe facets of the intrinsic geometry of a variety, as well as its relationship to other varieties, especially projective spaces. Here we outline the general theory of vector bundles and describe their classification and structure. We also consider some special bundles and general results in low dimensions, especially rank 2 bundles and surfaces, as well as bundles on projective spaces. Finally, we indicate some open problems and current areas of research.

3 1 Introduction

There are several aspects by which vector bundles are of use in algebraic geome- try. Historically, the most straightfoward is the geometric one: vector bundles are higher-dimensional varieties whose local structure is only truly interesting in a lower . In this way, they provide digestible examples of varieties in higher dimen- sion. On the other hand, from this point of view vector bundles are slightly artificial in : other classes of varieties with fibered structures aren’t so strin- gent in the restrictions they put on their fibrations, e.g. elliptic surfaces, which might have singular fibers. There are also, of course, algebraic parts of the theory. As the sheaves associated to free modules of finite rank, they allow for the use of linear al- gebra in a milieu of considerable sophistication (the smug theorist might say that the study of vector bundles began with the case where X = Spec k, for k any field). On the other hand, and perhaps more importantly, there are topological consider- ations. Given the development of abstract schemes and varieties obtained by gluing affine ones together in a particular manner, it is then unsurprising to find a sort of fiber bundle construction on them which glues together in the simplest possible way. From this point of view it is almost obvious, even inevitable, that the study of algebraic vector bundles would begin in ernest in that radical period of abstrac- tion between the ideas of Weil and Grothendieck. This study would be bolstered by the introduction of -theoretic methods into algebraic geometry. Indeed, though Serre’s seminal paper “Faisceaux Alg´ebriqueCoher´ents” [Ser55] was principally con- cerened with coherent sheaves, the locally free sheaves make up a prominent class of concrete coherent sheaves which are easy to describe, manipulate, and work with. Vector bundles also pose categorical problems for mathematicians. For instance,

4 the construction of moduli spaces of various classes of objects is of central concern to some, and the study of moduli spaces of vector bundles provides both a fertile testbed for this theory, as well as several open problems. Along the same line, understanding how the underlying geometry of a affects the range of possible vector bundles on it has occupied the work of many mathematicians over the last epoch, and continues to do so today.

2 Basic Constructions

2.1 Elementary Definitions

In what follows X is a projective scheme or variety, which is to be taken as meaning a complete, separated, Noetherian, integral scheme of finite type over an algebraically closed field k of characteristic 0. Sometimes X is not assumed to be integral, and may not be either reduced or irreducible, but in most of those cass will be in the context of X being a closed subscheme of another scheme, and the need for this generality will be apparent.

Definition. A linear fibration of rank r over X is a projective variety E together with a surjective π : E → X such that the fibers π−1(x) have the structure of a vector of dimension r for all x ∈ X. The trivial fibration is the product

π : X ×Ar → X, π the projection map. A over X is a linear fibration

π : E → X which is locally trivial, i.e., there exists an open cover {Ui} of X, and

−1 ∼ n −1 r r ϕi : π (Ui) −→ Ui ×A such that ϕi ◦ϕj : Ui ∩Uj ×A → Ui ∩Uj ×A is a map of the form (x, v) 7→ (x, g,i,j(x)(v)), where gi,j : Ui ∩ Uj → GL(r, k) is a

morphism of varieties. The gi,j are the transition functions of the bundle. These

also satisfy the conditions that gi,i = IdAr , and on the intersection Ui ∩ Uj ∩ Uk,

5 gi,k = gi,jgj,k. [Pot97] A morphism f : E → F of vector bundles π : E → X and π0 : F → X is a

0 −1 morphism of varieties such that π ◦ f = π, and the maps on the fibers fx : π (x) → π0−1(x) is a .

One could alternatively (and often more conveniently) treat vector bundles as follows:

Definition. Let X be a projective variety with structure sheaf OX .A vector bundle of rank r over X is a locally free OX - E of rank r; i.e., an OX -module E and an open cover {Ui} of X such that E(Ui) is a free OX (U)-module of rank r.

Given a vector bundle π : E → X, the sections Γ(U, E) of E over U ⊆ X

(i.e., regular maps s : U → E with π ◦ s = IdU ) form an OX (U)-module, given by

(s + t)(x) = s(x) + t(x) and (α · s)(x) = α(x)s(x) for s, t ∈ Γ(U, E) and α ∈ OX (U).

On trivializing open sets Ui, this module is free, so one can obtain a locally free sheaf of rank r by this method, often denoted O(E). On the other hand, given such a locally free sheaf E , for each x ∈ X let mx denote the maximal ideal of the local

Ox,X , and let Ex denote the Ox,X -module given by the stalk of E at x. Then Ex/mxEx ` is a (Ox,X /mx)-, and E = x∈X Ex/mxEx → X can be given the structure of a vector bundle on X such that O(E) ∼= E , and in fact these constructions give an equivalence of categories: the of vector bundles π : E → X over X and of vector bundles with the category of locally free OX -modules and morphisms of OX -modules. However, there are several natural constructions on vector spaces that admit gen- eralizations to vector bundles, which are easier to define for locally free sheaves than for locally trivial linear fibrations. If one wishes to give these definitions for such bun- dles π : E → X, this mostly involves performing the construction on the pointwise

6 fibers Ex and then showing these can be sewn together into a locally trivial bundle. To avoid this, one can simply consider the vector bundles as coherent sheaves, and use the extant definitions there. In light of this, one may define the (also called the Whitney sum) of vector bundles E and F , denoted E ⊕ F , to be their direct sum as OX -modules. Similarly one can define the product E ⊗ F , the E∨, the exterior powers ∧kE, and the symmetric powers SymkE.A sub-bundle F of a

locally free OX -module E is a locally free sub-module; if E ⊆ F is a sub-bundle, the quotient bundle E/F is well-defined. In the case of vector bundles π : E → X and π0 : F → X, and a map of vector bundles f : E → F , this map has well-defined

and image bundles exactly when the linear map fx is has constant rank over all x ∈ X.

Equivalently, this is the statement that a map of locally free OX -modules has a locally free kernel and cokernel exactly when the rank of this map is constant on the stalks.

In general, this need not be the case, but if E and F are locally-free OX -modules, the

kernel and cokernel of f are well-defined coherent sheaves, i.e., OX -modules F such

that X admits a covering by open affine subsets {Ui}, on which F |Ui is of the form ˜ Mi for a finitely generated Γ(U, OX )-module Mi (One might also want to consider the

category of quasi-coherent sheaves, which is the category of F as above, where

the Mi are not assumed to be finitely generated [Ser55]. This includes, for example, locally free sheaves of infinite rank. It was part of Grothendieck’s philosophy that one should strive to work in good categories with bad objects, rather than study the desired objects if they live in a bad category). It is also worth pointint out that several natural constructions on vector spaces which do not yield other vector spaces are also applicable to vector bundles. The most

important is the analogue of the construction V 7→ P(V ) of forming the associated to V . Given a vector bundle E , the associated fiberwise construction

7 is called the projective space bundle associated to E , and is denoted P(E ). One can define similar constructions for , flag varieties, etc. There are also properties defined similarly to those of modules: a vector bundle E is indecompos- able if it cannot be written as a direct sum of proper sub-bundles. It is simple if the only endomorphisms of E are homotheties, i.e., given by scalar multiplication.

When considering a variety over C as a complex , there is a question of algebraic and holomorphic vector bundles, where one might have to make the distinction between the two. However, when the variety in question is projective, the categories of these objects are equivalent by Serre’s GAGA [Ser56], and in this case it is not worth making the distinction. In this vain, there are several constructions originating in algebraic and which are here described by their algebraic analogues (Chern classes, the , etc.). The only tool which does not admit this type of analogy is the exponential sequence of sheaves on a : 0 → 2πiZ → O → O∗ → 0, where 2πiZ denotes the sheaf of locally constant functions with values of the form 2πiz, z ∈ Z, and O∗ denotes the sheaf of non-zero holomorphic functions.

2.2 Line Bundles

Definition. A on X is a locally free sheaf L of rank 1. The trivial line bundle is just OX , sometimes denoted 1.

Note that if E and F are vector bundles of rank r and s, respectively, then E ⊗ F is a vector bundle of rank rs. Then in the case where E and F are line bundles, coupled with the fact that E ⊗ OX = E , the set of classes of line bundles on X has the structure of an (abelian) monoid. This is in fact a group, by the following:

8 ∨ Proposition 2.1. Suppose L is a line bundle on X, and let L = H om(L , OX ) ∨ ∼ be the dual bundle. Then L ⊗ L = OX . (A sheaf L with this property is said to be invertible.)

∨ Proof. Consider the map L ⊗ L → OX given by f ⊗ s 7→ f(s). On any open set U trivializing L ∨ ⊗ L , this is (locally) an isomorphism, and since these cover X, it

is an isomorphism of OX -modules.

Definition. The group of isomorphism classes of line bundles on X is known as the Picard group of X, denoted Pic X.

The Picard group also appears as a group: If L ∈ Pic X, let ϕi :

−1 OUi → L |Ui be isomorphisms trivializing L on open sets Ui. Then for each i, j, ϕi ◦

ϕj : OUi∩Uj → OUi∩Uj is an . These transition maps then determine an ˇ 1 × ˇ element L ∈ H (A, OX ) in the first Cech cohomology group, where the given sheaf is the multiplicative sheaf of units of the structure sheaf, and the open cove A is given ˇ by the Ui. On varieties, Cech cohomology and the usual agree, so

1 × this determines an element of H (X, OX ), and two line bundles determine the same element of the cohomology group exactly when they are isomorphic. The Picard group can also be given an underlying scheme structure, which naturally makes it a group scheme. There are a few universally important constructions of line bundles, relating both to the intrinsic geometry of X and of vector bundles on X. As we are principally concerned with projective varieties, line bundles on Pn are also important.

n Consider X = Pk = Proj S for S = k[x0, . . . , xn]. Let S(n) denote S as an S- ˜ module with the grading shifted by n, and let OX (n) = S(n) be the OX -module associated to the S-module S(n).

Definition. OX (1) is called the twisting sheaf of Serre. For any OX -module F ,

9 let F (n) denote the nth twist F ⊗ OX (n).

Concretely, in this case, OX (1) can be thought of as the collection of homogeneous polynomials of degree 1 on Pn.

Proposition 2.2. Let X, OX (n) as above. [Har77]

1. OX (n) is a line bundle on X.

˜ ∼ ˜ ∼ 2. If M is a graded S-module, M(n) = M(n). In particular, OX (n) ⊗ OX (m) =

OX (n + m).

Proof. 1. Let f ∈ S1; then the open set D+(f) ⊂ X defined by the non-vanishing

of f is affine, isomorphic to Spec S(f) (where S(f) denotes the degree-0 subring

of the localization Sf ), and the restriction of OX (n) to this subset is isomorphic ˜ to S(n)(f). OX (n) is then free on D+(f) if and only if S(n)(f) is a free S(f)-

module, and in this case these have the same rank. S(f) is the group of elements

of degree 0 in Sf , and S(n)(f) is the group of elements of degree n; these are

n isomorphic by the map s 7→ f s, which is invertible because f is in Sf . The

sets D+(f) cover X, so this establishes OX (n) as locally free of rank 1.

˜ ∼ ˜ ˜ 2. If M and N are graded S-modules, then M ⊗S N = M ⊗OX N: this follows ∼ from the fact that for any f ∈ S1,(M ⊗S N)(f) = M(f) ⊗S(f) N(f), which one sees immediately.

n The sheaf OX (1) carries a lot of information about the geometry of P . Some of

this comes from the fact that OX (1) bears the following property:

Definition. An OX -module F is generated by global sections if there exist global sections {si} ⊆ Γ(X, F ) such that for every point x ∈ X, {si} generates Fx as an

Ox-module, where si denotes the image of si in Fx.

10 In the case of OPn (1), it is generated by the coordinates x0, . . . , xn. Suppose i : Y → Pn is an immersion of a variety Y in a projective space Pn. The

∗ twisting sheaf OPn (1) may be pulled back to Y by i to give an OY -module i (OPn (1)),

and this OY -module is also generated by global sections, namely by the pullbacks

n ∗ of the xi on P . In this case, i (O(1)) is also a line bundle. The converse of this situation also holds:

Theorem 2.3. 1. If i : X → Pn is a morphism of varieties, then i∗(O(1)) is a line bundle which is generated by global sections.

2. If L is a line bundle on X which is generated by finitely many global sections

n s0, . . . , sn ∈ Γ(X, L ), then there exists a unique morphism i : X → P such ∼ ∗ ∗ that L = i (O(1)), with si = i (xi). In such a case, say that L is very ample.

In the case of Pn, O(1) is very ample and corresponds to the identity map, and

O(d) is also very ample, and corresponds to the Veronese embedding (x0 : ··· : xn) 7→

(µ1(x0, . . . , xn): ··· : µm(x0, . . . , xn)), where µi ranges over all monomials of degree

d in the variables x0, . . . , xn. We thus have an (almost tautological) correspondence between very ample line bundles on X and morphisms X → Pn. Without knowing more about X, it may then seem like it is difficult to find such line bundles, since a priori giving such morphisms sounds difficult. However, very ample line bundles (which are of great general import) may be recovered from another class of line bundles defined in analogy to the following theorem:

Theorem 2.4. Let X be a projective scheme, O(1) a very on

X, and F a coherent OX -module. Then there exists an integer n0 such that for all n ≥ n0, F (n) is generated by finitely many global sections. [Har77]

11 Definition. Let X be a Noetherian scheme, L a line bundle on X. L is ample

when for every coherent OX -module F , there exists an integer n0 = n0(F ) such that

n for all n ≥ n0, F ⊗ L is generated by global sections.

We then have the following theorem:

Theorem 2.5. Let X be a scheme of finite type over k, and L and on X. Then L is ample if and only if L ⊗m is very ample for some m > 0.

Ample line bundles are not terribly difficult to find, though their existence is not obvious in all cases. They also do not determine unique maps to projective space:

OPn (1) is both ample and very ample, but each tensor power of it gives a different embedding into a different projective space. The existence of an ample line bundle on a variety V is so useful the following definition is necessary:

Definition. A polarized variety is a pair (V, L ) consisting of a variety V and an ample line bundle L on V .

Definition. Suppose E is a locally free OX -module of rank r. Then the exterior

k r r product ∧ E has rank k , so in particular, ∧ E has rank 1, i.e., is a line bundle. This line bundle is the of E, denoted det E. This is a useful invariant of a vector bundle, and it additionally comes with the identity det(E ⊕F ) = det(E)⊗ det(F ), since this is true locally by looking at tensor powers.

2.3 Divisors

Let D be a (for all generality, Cartier) divisor on an arbitrary scheme X, and let D

∗ be given by a collection {(Ui, fi)} of open sets Ui ⊆ X and elements fi ∈ Γ(Ui, K )

0 of the total of Γ(Ui, OX ), where D ∼ D are linearly equivalent if on

12 0 compatible open sets Ui, the defining sections of D and D are in the same equivalence

∗ ∗ class of Γ(Ui, K /OX ).

Definition. If D = {(Ui, fi)} is a Cartier divisor on X, define the line bundle OX (D)

∗ −1 by taking Γ(Ui, OX (D)) to be the OX -submodule of K generated by fi . This

∗ is well-defined, since on Ui ∩ Uj, fi/fj ∈ OX (Ui ∩ Uj) , so the Γ(Ui, OX (D)) glue accordingly.

The group structure of divisors is compatible with that of line bundles:

Proposition 2.6. The map D 7→ OX (D) gives an injective group homomorphism Cl(X) → Pic X. When X is a projective scheme over a field, this map is an isomor- phism.

In particular, on X = Pn, any divisor D of degree d is linearly equivalent to dH, where H is any hyperplane of Pn (all hyperplanes have degree 1), so ClX is isomorphic to Z, and generated by a hyperplane. Correspondingly, Pic X is generated by the twisting sheaf OX (1), which is associated to the linear of hyperplanes. Thus OX (1) is sometimes referred to as the hyperplane bundle. Its

∨ n+1 dual bundle, OX (−1) = OX (1) , is the bundle induced by the projection A \ O → Pn; this is referred to as the tautological line bundle. In the event that a Cartier divisor D on a scheme X is effective, i.e., is of the form

{(Ui, fi)} with fi ∈ Γ(Ui, OX ) for all i, then D has an associated closed subscheme

Y , whose ideal sheaf JY is locally generated by the fi. It is clear, then, that in fact ∼ JY = OX (−D) as a sheaf.

2.4 Differentials

1 Definition. On a non-singular variety X of dimension n, the sheaf ΩX of differential

n 1 n 1-forms on X is a vector bundle of rank n. Its determinant ∧ ΩX = ΩX is the

13 1 of X, denoted ωX . The TX is the dual of ΩX ,

1 H omOX (ΩX , OX ).

n ∼ In the case of P , we have ωPn = OPn (−n − 1). These vector bundles are enormously powerful invariants for varieties, and will appear again frequently. If Y ⊆ X is also non-singular, we have additional useful bundles:

Definition. Suppose Y is a non-singular subvariety of a non-singular variety X, with sheaf of ideals J . The sheaf J /J 2 is locally free of rank r = codim(Y,X), and is

2 called the conormal bundle of Y in X. It’s dual, NY/X = H omOY (J /J , OY ) is the of Y in X

These bundles fit together into an 0 → TY → TX ⊗OY → NY/X →

2 1 1 0, which is dual to the exact sequence 0 → J /J → ΩX ⊗ OY → ΩY → 0. From ∼ r this one can also derive the identity ωY = ωX ⊗ ∧ NY/X , when Y is non-singular of r in X, also non-singular; in particular, when Y has codimension 1 ∼ and can be considered as a divisor with associated line bundle L , we have ωY =

ωX ⊗ L ⊗ OY .

n 1 Additionally, in the case of , we have the 0 → Ω n → P P ⊕(n+1) ⊕(n+1) OPn (−1) → OPn → 0, equivalently given as 0 → OPn → OPn (1) →

TPn → 0 [OSS11].

2.5 Chern Classes

Chern classes are vector bundle invariants arising in algebraic geometry through in- tersection theory. Though they are difficult to give explicitly much of the time, their power lies in their relationships to each other. It is even difficult to describe their defi- nition concretely, but fortunately any definitions one could give are equivalent [Har77].

14 It is first necessary to describe the setting in which a lives:

Definition. Suppose X is a non-singular quasi-projective variety. Let Zk denote the cycles of codimension k in X, which is the free abelian group generated by the closed, irreducible subvarieties of X with codimension k. Suppose f : X → X0 is a map of varieties, with Y a closed, irreducible subvariety of X. If dim f(Y ) < dim Y , let f∗(Y ) = 0; if dim f(Y ) = dim Y , let f∗(Y ) = [K(Y ): K(f(Y ))] · f(Y ); extending by linearity, this gives a map from the cycles of X to those of X0. Two cycles Y,Y 0 of the same codimension are rationally equivalent if there exists a subvariety V ⊆ X, with normalization f : V˜ → V , and linearly equivalent

0 ˜ 0 0 r divisors D,D on V such that f∗(D) = Y , and f∗(D ) = Y . Let A (X) be the group of cycles of codimension r on X modulo rational equivalence.

In this formulation, there is a group homomorphism Pic X → A1(X), since lin-

0 ∼ early equivalent divisors are rationally equivalent cycles. We also have A (X) = Z (generated by itself), and Ak(X) = 0 when k > dim(X). In the case of projective

n ∼ n+1 space X = P , A(X) = Z[h]/(h ), where h is the rational equivalence class of a hyperplane; i.e., Ai(X) is generated by a single element h, which is a of codimension i.

Definition. An consists of, for all non-singular, quasi-projective X, a pairing (an intersection product) Ar(X) × As(X) → Ar+s(X), making the collection of all cycles on X a ring, denoted A(X) (the Chow ring). If X, and X0

0 0 0 are such varieties, let p1 : X × X → X, p2 : X × X → X be the projections, and

∗ 0 ∗ 0 −1 0 define f : A(X ) → A(X) by f (Y ) = p1∗(Γf .p2 (Y )), where Γf is the graph of f as a cycle on X × X0. Such a product must satisfy the following conditions:

• The map f ∗ : A(X0) → A(X) induced by a map f : X → X0 is a ring map,

15 such that if g : X0 → X00, f ∗ ◦ g∗ = (g ◦ f)∗.

0 0 • If f : X → X is proper, f∗ : A(X) → A(X ) is a map of graded abelian groups

0 00 (perhaps shifting degree). If g : X → X is another such map, g∗ ◦f∗ = (g◦f)∗.

∗ • Suppose f : X → Y is proper, x ∈ A(X), y ∈ A(Y ). Then f∗(x.f (y)) =

f∗(x).y.

• Suppose Y,Z are cycles on X, and let ∆ : X → X × X denote the diagonal map. Then Y.Z = ∆∗(Y × Z).

• If Y,Z ⊆ X are subvarieties such that every of Y ∩ Z has codimension codim Y + codim Z, then Y.Z = P i(Y,Z; W ) · W , where Wj j j

Wj are the irreducible components of Y ∩ Z, and i(Y,Z; Wj), the local inter-

multiplicity of Y and Z along Wj, is an integer which depends on

a neighborhood of the of Wj in X.

• If Y ⊆ X is a subvariety, and Z is an effective Cartier divisor such that Y ∩ Z satisfies the above condition, then Y.Z is equal to the Cartier divisor of Y ∩ Z on Y .

One may then define the Chern classes of a vector bundle in an equally opaque way:

Definition. Let E be a locally free sheaf of rank r on X. There exist, for i = 0, . . . , r,

i the Chern classes ci(E) ∈ A (X). For notational convenience, let c(E ) = c0(E ) +

Pr i ··· cr(E ) denote the total Chern class of E , and let ct(E ) = i=0 ci(E ) · t denote the formal Chern polynomial of E . The Chern classes are defined as follows:

• c0(E ) = 1 for all E . If i is greater than the rank of E , ci(E ) = 0.

16 • If L is a line bundle with corresponding divisor D (i.e., L = OX (D)), c(L ) = 1 + D.

∗ • If f : Y → X is a map of varieties, and E is locally free on X, then ci(f (E )) =

∗ f (ci(E )) for all i.

0 00 0 00 • If 0 → E → E → E → 0 is exact, then ct(E ) = ct(E ) · ct(E ).

• Suppose E splits, i.e., has a filtration of vector bundles E = E0 ⊇ E1 ⊇

· · · ⊇ Er = 0, such that for all i, Ei+1/Ei is a line bundle Li. Then ct(E) = Qr Qr i=1 ct(Li) = i=1(1 + Di · t)

In particular, for any E on X, there exists a variety Y , vector bundle E 0 on Y , and map f : Y → X such that f ∗ : A(X) → A(Y ) is injective, E 0 = f ∗(E ), and E 0 splits. This is the splitting principle [Har77].

There are other formulas for the computation of Chern classes, and some intersec- tion theory will be necessary for some of the theory. However, the most interesting one to remark on now would be the self-intersection formula: If i : Y → X is the inclusion of a non-singular subvariety Y of codimension r, and N = NY/X is

∗ the normal bundle on Y , then i ◦ i∗(1) = cr(N ). By properties of the intersection

product above, we then have i∗(cr(N )) = Y.Y , the self-intersection of Y on X.

3 Moduli Spaces

3.1 Some Classifications

Even imposing relatively simple constraints on the base space X can leave the question of classifying vector bundles on X rather unwieldy. When we limit the rank of the vector bundle, say to line bundles, and consider the Picard group, life is fairly simple.

17 But even moving up to rank 2 is difficult on most varieties, and there are still open problems around the existence of various species of rank 2 bundles. One attempt at rectifying this situation, in certain cases, is through the construction and analysis of moduli spaces of vector bundles; this relies on a notion of stability or semi-stability, and regularity, but allows one to examine the total space of vector bundles using the same tools as the vector bundles themselves. In any event, in a few cases there are well-known results, as in the following:

Theorem 3.1 (Grothendieck, 1957). Suppose E is a vector bundle on P1. Then there

L ai exist unique a , all but finitely many 0, such that = O 1 (i) . [Gro57] i E i∈Z P

That is, every vector bundle on the splits into a direct sum of line bundles. We have already seen that these line bundles must be isomorphic to the

OP1 (l) for varying l. This theorem was published by Grothendieck, but commentators are quick to point out the myriad of people before him to have “essentially known” this result.

From here, there are a couple possible choices for next steps: Pn for n > 1, or curves C of higher ? It turns out the next best classification is the latter, due to Atiyah. However, it is much more complicated than the rational case.

Theorem 3.2 (Atiyah). Let X be a non-singular over C. A vector bundle E over X has two principal invariants: its rank, and its degree, which can be

1 ∼ defined equivalently as its first chern class c1(E) ∈ A (X) = Z, or as the degree of its determinant line bundle det E, when viewed as a divisor; in either case, the degree d may be specified with an integer, and the rank r as a positive integer. We then have the following: [Ati57]

ind • Let VBr,d (X) denote the set of indecomposable vector bundles of rank r and degree d, i.e., those which cannot be written as a direct sum of lower-rank

18 bundles.

ind • VBr,d (X) is in bijection with the closed points of X.

ind ind • the map − ⊗ OX (d) gives a bijection between VBr,0 and VBr,d .

r ind ind • The map ∧ − is a map VBr,d (X) onto VB1,d (X), which is a bijection exactly when r, d are coprime. If not, say (r, d) = k, then for every line bundle L of degree d there are k2 vector bundles E which have rank r and det E = L .

3.2 Stable and Semi-stable Sheaves

In this section, let (X, O(1)) be a polarized projective variety, so that O(1) is an ample line bundle; however, we will first consider the situation where X is additionally a non-

singular curve. If L is a line bundle on X, it has a well-defined degree: L = OX (D) P P for a divisor D = niPi, and deg L = deg D = ni. If E is a locally free sheaf of rank r on X, its degree is defined by deg E = deg det E. The slope of E is then

defined to be µ(E) = deg(E)/r. An alternative definition of slope isµ ˆ(E) = αd−1/αd, which will appear later [HL10].

Definition. A vector bundle E on X is µ-stable (resp. µ-semistable) if for all sub-bundles F ⊂ E of lower rank than E, µ(F ) < µ(E) (resp. µ(F ) ≤ µ(E)).

This definition is also referred to as Mumford-Takemoto stability. It is conve- nient in the case of non-singular curves, but is less useful in higher dimensions. In order to give such a generalization, it is necessary to describe another popular notion of stability, which is equivalent in dimension 1: Giesecker stability [Pot97]. This requires a number of auxiliary definitions, which will also be useful later. In general,

these definitions make sense for an arbitrary F .

19 Definition. If (X, O(1)) is a polarized projective variety, and F is a coherent sheaf

P i i on X, the of E is given by χ(F ) = i(−1) h (X, F ) (recall

i i h (X, F ) = dimk H (X, F )). The Hilbert polynomial of F is given by PF (m) = χ(F (m)) = χ(F ⊗ O(m)) = χ(F ⊗ O(1)⊗m).

If F is a coherent sheaf, let dim F denote the dimension of the of F as a closed subvariety of X (when F is a non-zero vector bundle, this is just dim X). Then the Hilbert polynomial of F can be written uniquely in the form

Pdim F i PF (m) = i=0 αi(F )m /i!, for rational numbers αi(F ). In particular, αdim(F )(F ) is a positive rational number; call this the multiplicity of F .

Definition. Let F be a coherent sheaf on X of rank r and dimension d. The degree of F is deg F = αd−1(F ) − r · αd−1(OX ), and define the slope of F to be µ(F ) = deg F /r.

µ-stability for such F may then be defined as above, in terms of slopes. However, it is also required that all torsion subsheaves of F are of codimension at least 2.

Continuing on, let pF (m) = PF (m)/αdim(F )(F ) be the reduced Hilbert poly- nomial of F . Additionally, if p, q are univariate polynomials, say f < g (resp. f ≤ g) if f(m) < g(m) (resp. f(m) ≤ g(m)) for all sufficiently large m.

Definition. Let F be a coherent sheaf of dimension d on X. F is stable (resp. semi-stable) in the sense of Giesecker when for all proper subsheaves F 0 ⊂ F , pF 0 < pF (resp. pF 0 ≤ pF ).

Without any assumptions on the dimension of X (which could even be a locally Noetherian scheme over k), we have the following:

Proposition 3.3. f is µ-stable ⇒ F is stable ⇒ F is semi-stable ⇒ F is µ- semistable. [Pot97]

20 We will see that semistability is an important construction later on, but it is also sufficiently general: any coherent sheaf admits a filtration by semistable ones.

Theorem 3.4 (Harder-Narasimhan Filtration). Let F be a non-trivial coherent sheaf of dimension d. There exists a filtration

0 = F1 ⊂ · · · ⊂ Fl = F

such that the factors Gi = Fi/Fi−1 are dimension d semistable sheaves. Their reduced

Hilbert polynomials pi = pGi satisfy p1 > ··· > pl.

Among semistable sheaves, another important filtration is the Jordan-H¨older filtration:

Theorem 3.5 (Jordan-H¨olderFiltration). Let F be a semistable coherent sheaf. F has a filtration 0 = E0 ⊂ E1 ⊂ · · · ⊂ El = F such that Ei/Ei−1 is stable for all i, and all of these factors have reduced hilbert polynomial equal to pF . This filtration may not be unique, but the sheaf ⊕iEi/Ei−1 is independent of the choice of Ei; it is called the Jordan-H¨oldergrading and denoted gr(F ).

Two semistable vector bundles are said to be S-equivalent if they have the same Jordan-H¨oldergrading.

3.3 Representability

A is a geometric object whose points parametrize some other family of objects. Classically, such spaces were found largely by experiment or intuition; finding a moduli space for a desired collection of objects (i.e., giving this collection a geometric structure, if it is a set) was very difficult. Indeed, the Picard group of a projective curve was not described as a variety until the 1940s, after which such

21 constructions became easier [FGI+05]. This is not to say that constructing moduli spaces is now easy, but there are a cornucopia of tools available to those today. In the case of vector bundles over a fixed polarized variety (X, O(1)), two tools in particular were of great import.

To any scheme X, one may associate to X the contravariant hX from schemes to sets sending a scheme Y to the set of all morphisms Y → X. The scheme

X is said to represent the functor hX , and given any other scheme Y with hY = hX , there is a unique isomorphism X ∼= Y by the Yoneda lemma. However, given an arbitrary such functor h, there is in general no way to know whether h is representable

by a scheme X, i.e., if an X exists with h = hX [HL10]. In particular, one would ideally like to represent the functor which associates to a projective variety X all of its vector bundles. This turns out to be impossible, in general; even restricting interest to vector bundles with the same invariant, such as rank, is not enough. This is because the set is in some sense too large to be represented. However, over different objects the sets of stable or semistable bundles/sheaves may be given a moduli space structure. The technique for the construction of such a moduli space first involves proving the existence of a more convenient functor: the Hilbert functor of a scheme, which associates to a scheme X the set of all closed subschemes Y ⊆ X which share a pre- scribed Hilbert polynomial. Grothendieck achieved this using his methods of theory, finding this functor’s representation, the . In the general case, he constructed the , which parametrizes the quotients of a given coherent sheaf with a prescribed Hilbert polynomial. Mumford then showed that one can embed the Hilbert scheme in a suitable projective space, and take a quotient of this embedding by the action of a group on the projective space to construct moduli spaces of many different types of objects; he did this by developing his Geometric

22 [HL10].

Definition (Castelnuovo-Mumford Regularity). A coherent sheaf F on a polarized projective scheme (X, O(1)) is m-regular if Hi(X, F (m − i)) = 0 for all i > 0.

Proposition 3.6. Suppose F is m-regular. [Pot97]

1. F is m0-regular for all m0 ≥ m.

2. F (m) is generated by global sections.

3. For all n ≥ 0, the natural maps Γ(X, F (m)) ⊗ Γ(X, O(n)) → Γ(X, F (m + n)) are surjective.

Definition. If F is a coherent sheaf on (X, O(1)), the regularity of F , reg(F ), is the least m such that F is m-regular.

Definition. A family of (isomorphism classes of) coherent sheaves is bounded if

there exists a scheme S over k of finite type, and a coherent OS×kX -sheaf F such that the family is contained in the the set {F |Spec (k(s))×X : s ∈ X is a closed point}.

Lemma 3.7. Let {Fi} be a family of isomorphism classes of sheaves. The following are equivalent: [HL10]

1. {Fi} is bounded.

2. The set of Hilbert polynomials {PFi } is finite, and there is a uniform bound

reg(Fi) ≤ ρ for all i.

3. The set of Hilbert polynomials {PFi } is finite, and there exists a coherent sheaf

F and surjective morphisms F → Fi.

Corollary 3.8. Suppose {Fi} is a bounded family of coherent sheaves on a polarized projective scheme (X, O(1)). Then there exists a uniform constant m such that Fi(m) is generated by global sections for all i.

23 Proposition 3.9. Let (X, O(1)) be a smooth, polarized projective curve of genus g. The set of isomorphism classes of vector bundles of fixed rank r and degree d is not bounded for r ≥ 2. [Pot97]

Proof. Let a ∈ X be a closed point, identified with its divisor class by abuse of notation. Then consider the subfamily of vector bundles Ek = O(−ka) ⊕ O((d +

r−2 1 1 k)a) ⊕ O . Then h (Ek) ≥ h (O(−ka)) = k + g − 1. The above lemma implies that in a bounded family, the possible values of h1 are finite, but in the given set these values are unbounded, so the family is too.

Theorem 3.10. If (X, O(1)) is as above, the set of isomorphism classes of semistable bundles of fixed rank r and degree d is bounded for all r and d.

Theorem 3.11 (Coarse Moduli Space of Semistable Sheaves). Let (X, O(1)) be a

polarized projective scheme over k. There exists a projective scheme MO(1)(P ) whose closed points are in bijection with S-equivalence classes of semistable sheaves with

s Hilbert polynomial P . There exists an open subset MO(1)(P ) ⊆ MO(1)(P ) whose closed points correspond to S-equivalence classes of stable bundles. [HL10]

Once the existence of the scheme Quot(H ,P ) parametrizing isomorphism classes of quotients of a coherent H which have Hilbert polynomial P has been established, the proof follows this outline: Since the set of semistable bundles F with Hilbert polynomial P is bounded, there exists some sufficiently large m such that each F (m) ⊕P (m) is generated by global sections. Then let H = OX ⊗ OX (−m); choosing the basis of Γ(X, F (m)) corresponds to selecting a surjective map ρ : H → F given by ⊕P (m) a map OX → Γ(X, F (m)) composed with the map Γ(X, F (m))⊗OX (−m) → F . This associates every semistable bundle to a closed point in the scheme Quot(H ,P ). Furthermore, there exists an action of GL(P (m)) on this scheme given by [ρ] · g =

[ρ◦g], ρ ∈ Quot(H ,P ), g ∈ GL(P (m)); there also exists an injective map Aut(F ) →

24 GL(V ) whose image is the stabilizer of [ρ : H → F ], and the quotient of Quot(H ,P )

by this action is well-defined, and gives the desired moduli space MO(1)(P ). A large amount of technical machinery goes into each part of this proof. The construction of the Quot scheme is interesting in its own right, and relies on a number of results on flat families that have been entirely omitted. That there exists a on Quot(H ,P ) whose quotient should give the desired moduli space also requires a number of results in Geometric Invariant Theory, as does the fact that this quotient can be taken at all. Semistability and S-equivalence were defined just so that this is possible. [HN75] From here, the theory of moduli spaces gets significantly more complicated. Over curves, the basic theory is sufficient to consider not only the moduli spaces of semistable sheaves, or stable vector bundles, but more exotic objects such as augmented or dec- orated sheaves (which carry additional data) or principal bundles (which are a gen- eralization of vector bundles whose transition maps may lie in more general groups). There is also significant research activity in a generalization of Brill-Noether theory to vector bundles, which further identifies, classifies, and understands vector bundles based on their number of global sections. [LBPR06] However, over higher-dimensional varieties, the picture is somewhat less clear. Even in the case of rank 2 bundles on surfaces, there are plenty of unanswered questions, including some as basic as “is the moduli space empty?”. [NJHO95] For sake of productivity we move away from moduli theory and toward understanding the structure of individual bundles, now on varieties which are in some sense “simple.”

25 4 Vector Bundles on Pn

4.1 Cohomological Tools

In many ways, for each n the projective space Pn has the simplest geometry among projective varieties of dimension n. Additionally, they are so well-studied that many problems which can be formulated for a general Pn leave available many tools with which to attack the problem. This does not make every question easy to answer, and there are many ’simple’ cases of conjectures for which results are not known, or about which little is known. One of the available tools which is effective across projective spaces is cohomology. Powerfully, we have the Bott formula [OSS11]:

 k+n−pk−1 q = 0; 0 ≤ p ≤ n; p < k  k p   1 k = 0; 0 ≤ p = q ≤ n q n p  h ( , Ω n (k)) = P P −k+p−k−1  q = n; 0 ≤ p ≤ n; k < p − n  −k n−p   0 otherwise

In fact, this is enough to provide a proof of Grothendieck’s theorem:

1 Theorem 4.1. If E is a vector bundle on P , it is of the form O(a1) ⊕ · · · ⊕ O(ar), with a1 ≥ · · · ≥ ar uniquely determined. [Gro57]

Proof. If E is of rank 1, i.e., a line bundle, the result follows from the fact that

Pic Pn = Z. Proceed by induction on the rank of E , supposing the result holds for bundles of rank ≤ r, and suppose E has rank r + 1. Then by Serre’s theorem B, for sufficiently large k, E(k) has non-zero global sections; let k0 be the least such integer

0 1 with this property (this is well-defined by Serre ), so that h (P , E (k0)) > 0,

26 0 1 but h (P , E (k)) = 0 for all k < k0.

0 1 Now let s ∈ H (P , E (k0)) be a non-zero global section. In fact, s must be

1 nowhere-zero: if sx = 0 for some x ∈ P , then s would be a non-zero section of

1 E (k0) ⊗ Jx. But Jx = O(−{x}) = O(−1), as {x} is a hyperplane of P , so s in this

case s is a non-zero global section of E (k0 ⊗ O(−1) = E (k0 − 1). But by definition of

k0, this is impossible, so s must be nowhere-zero. s s thus defines a trivial sub-line-bundle of E , and an exact sequence 0 → OP1 −→

E (k0) → F → 0, and F must therefore be locally free with rank strictly less than

that of E (k0). Then by inductive hypothesis, F = O(b1) ⊕ · · · ⊕ O(br) for some

bi. If it can be shown that the exact sequence above splits, E would be of the form L 1 O(−k ) ⊕ O(b − k ), completing the proof. This occurs when Ext 1 ( , O 1 ) = 0 i i 0 P F P 1 1 1 ∨ 1 1 0. Identically, Ext (F , O) = H (P , F ⊗ O) = ⊕iH (P , O(−bi)). By the Bott

formula, this is 0 precisely when bi < 2 for all i.

In fact, it must be the case that bi ≤ 0 for all i. Tensoring the above exact

sequence with O(−1), we have 0 → O(−1) → E (k0 − 1) → ⊕iO(bi − 1) → 0 exact. This has long exact sequence in cohomology 0 1 0 1 Lr 0 1 1 1 ··· H (P , O(−1)) H (P , E (k0 − 1)) i=1 H (P , O(bi − 1)) H (P , O(−1)) ···

Lr 0 1 ··· 0 0 i=1 H (P , O(bi − 1)) 0 ··· 0 1 Thus h (P , O(bi − 1)) = 0 for each i, so by the Bott formula, bi ≤ 0 for all i, and the original exact sequence considered must split.

4.2 Splitting on Higher Projective Spaces

Using an argument found in the above proof, it is easy to see that if 0 → O(a) →

E → O(b) → 0 is an exact sequence of OPn -modules, n ≥ 2, E a rank 2 vector bundle,

1 1 n ∨ then must split: this happens when Ext 2 (O(a), O(b)) = H ( , O(a) ⊗ O(b)) = E P P H1(Pn, O(−a + b)) = 0, which follows from the Bott formula. This also shows that

27 any 2-bundle on Pn which does not split cannot contain a proper sub-bundle. In general, there are many examples of vector bundles of rank r on Pn which do not split, though for particular choices of r and n, they may not, or be difficult to find. [OSS11] A useful criterion for splitting is the following result of Horrocks:

Theorem 4.2. A vector bundle E on Pn splits into a direct sum of line bundles if and only if Hi(Pn, E (k)) = 0 for i = 1, . . . , n − 1 and k ∈ Z.

Proof. If E is a direct sum of line bundles, the result follows from the Bott formula. Conversely, proceed by induction on n; the case n = 1 is the above theorem. Now suppose the result is true for all n0 < n, and supose E is a vector bundle on Pn satisfying the above condition. Taking the exact sequence 0 → OPn (−1) → OPn →

n−1 OPn−1 → 0 (for a fixed choice of hyperplane P ) and tensoring with E (k), we have a long exact sequence in cohomology

i n i n−1 i+1 n · · · → H (P , E (k)) → H (P , E |Pn−1 (k)) → H (P , E (k − 1)) → · · ·

i n−1 By assumption, the outer two groups are both 0, so H (P , E |Pn−1 (k)) = 0 for r i = 1, . . . , n − 2. Then by induction hypothesis, E |Pn−1 = ⊕i=1OPn−1 (ai). Now let r ∼ F = ⊕i=1OPn (ai). Clearly, F |Pn−1 = E |Pn−1 ; let ϕ : F |Pn−1 → E |Pn−1 be such an isomorphism.

Claim: ϕ lifts to a map Φ : F → E , i.e., there exists such a Φ which is equal to

n−1 ϕ on P . To see this, take the above exact sequence, tensor with the OPn -module H om(F , E ) = F ∨ ⊗ E , and take the long exact sequence in cohomology to obtain

0 n ∨ 0 n−1 ∨ 1 n ∨ · · · → H (P , F ⊗ E ) → H (P , (F ⊗ E ) |Pn−1 ) → H (P , F ⊗ E (−1)) → · · ·

28 Expanding, we have

1 n ∨ 1 n ∨ 1 n H (P , F ⊗ E (−1)) = H (P , (⊕iO(ai)) ⊗ E (−1)) = ⊕iH (P , E (−ai − 1)) = 0

where the last equality is by assumption on E . In other words, the map H0(Pn, F ∨ ⊗

0 n−1 ∨ ∨ E ) → H (P , (F ⊗ E ) |Pn−1 is surjective, so any global section of (F ⊗ E ) |Pn−1 = ∨ H omPn−1 (F |Pn−1 , E |Pn−1 ) is the image of a global section of F ⊗E = H omPn (F , E ). Taking Φ to be this map, the claim is proved.

Now, F and E have the same rank and determinant. Taking top exterior powers, we obtain an induced map det Φ : det F → det E , which is a global section of

∨ det F ⊗ det E = O(−c1(F )) ⊗ O(c1(E )) = O(−c1(F ) + c1(F )) = O = C

and hence is constant on Pn. Its restriction to Pn−1 (which is the same as the deter- minant of ϕ on Pn−1) is nowhere-zero, so in particular, det Φ must be nowhere-zero as well. Thus Φ is actually an isomorphism. This proves the claim, since it exhibits

E as a direct sum of line bundles.

An improvement can be made when the rank of the vector bundle is smaller than the dimension of the projective space:

Theorem 4.3. Let E be a vector bundle of rank r on Pn with r < n. Then E splits as a direct sum of line bundles if and only if Hi(Pn, E (k)) = 0 for i = 1, . . . , r − 1 and k ∈ Z. [KPR03]

Corollary 4.4. A vector bundle E on Pn splits if and only if its restriction to some P2 ⊆ Pn does. [OSS11]

Proof. Let n ≥ 3, Pn−1 ⊆ Pn a hyperplane. One direction is clear, so it suffices to show that if E |Pn−1 splits, then E does. By the above criterion, if E |Pn−1 splits, then

29 i n−1 H (P , E |Pn−1 (k)) = 0 for i = 1, . . . , n − 2, and k ∈ Z. The exact sequence

0 → E (k − 1) → E (k) → E |Pn−1 (k) → 0 has long exact sequence in cohomology

i−1 n−1 i n i n i n−1 · · · → H (P , E (k)|Pn−1 (k)) → H (P , E (k − 1)) → H (P , E (k)) → H (P , E (k)|Pn−1 ) → · · ·

i n ∼ i n whose outer groups are both 0, so we have H (P , E (k − 1)) = H (P , E (k)) for i = 2, . . . , n − 2, and all k. At i = n − 1, we have an injection Hn−1(Pn, E (k − 1)) → Hn−1(Pn, E (k)) for all k. By Serre’s theorem B, we must have Hi(Pn, E (k)) = 0 for i = 2, . . . , n − 1 and k ∈ Z. To see that this holds for i = 1, apply the same argument to E ∨; by , H1(Pn, E (k)) = Hn−1(Pn, E ∨(−k − n − 1)) = 0. Then by the above criterion, E splits into a direct sum of line bundles.

This also makes it easier to find bundles which do not split as a direct sum of line

i n p bundles. For exmaple, by the Bott formula, we have h (P , Ω (k)) = δi,p for all k ∈ Z, p so Ω n cannot split when 0 < p < n. However, a cohomological criterion can account P for this.

Theorem 4.5 (Horrocks). If E is a vector bundle on Pn with Hq(Pn, E (k)) = 0 for

p n p q 6= 0, n, p, and k ∈ , and h ( , (k)) = δ , then is a direct sum of Ω n and Z P E k,0 E P line bundles.

Of course, while the above corollary relates the splitting behavior of a bundle E on Pn to that of its restriction to a lower projective space, one can also consider

1 n restrictions to various lines P ⊆ P . In fat, for any line L, the restriction E |L must split as E|L = OL(a1(L))⊕· · ·⊕OL(ar(L)), with a1 ≥ · · · ≥ ar integers, and E of rank

30 r r. The association aE (L) = (a1(L), . . . , ar(L)) defines a aE : G(1, n) → Z , where G(1, n) denotes the of lines in Pn.

r Definition. For a given line L, the tuple aE (L) = (a1(L), . . . , ar(L)) ∈ Z is called the splitting type of E on L. Ordering such tuples lexicographically, the least such

aE (L) over all lines L is called the generic splitting type of E . Any bundle for

which the tuple aE is independent of L is called uniform.

An important tool for the analysis of uniform bundles and the restriction of bun-

dles to lines is the “standard construction,” which associates to E a commutative

diagram. [OSS11] This construction is performed analytically over C: Let Gn = G(1, n) denote the Grassmannian of lines in Pn; if L ⊆ Pn is a line, let

n n+1 ` ∈ G denote the corresponding point, and let E` denote the plane in C defined by `.

With this considered, note that Gn has a V = {(`, v) ∈ Gn ×

n+1 n n C : v ∈ E`}. The associated P(V ) = {(x, `) ∈ P × G : x ∈ L} is a flag variety; denote it Fn. the projection maps p : Fn → Pn, q : Fn → Gn form the standard diagram: q Fn Gn p

Pn

Here, the map q exhibits Fn as a P1-bundle on Gn, and p identifies Fn with the

n projective bundle associated to the tangent bundle TPn . Now let ` ∈ G , and let L˜ = q−1(`) = {(x, `): x ∈ L} ⊆ Fn; then p gives an isomorphism of L˜ to L. Let F(x) denote the fiber p−1(x) = {(x, `): x ∈ L} ⊆ Fn, and let G(x) = {` ∈ Gn : ∼ n−1 x ∈ L} = P . Then the restriction of q to F(x) is an isomorphism F(x) → G(x). Finally, let B(x) = {(y, `): x, y ∈ L} = q−1(G(x)) ⊆ Fn, and let f : B(x) → G(x) and σ : B(x) → Pn denote the restrictions of q and p, respectively. We can assemble

31 the following :

f B(x) G(x)

σ q Fn Gn p

Pn

One simple application of this construction is to identify when a bundle is trivial:

n n ⊕r Proposition 4.6. Suppose E is of rank r on P , x ∈ P , and suppose that E |L = OL for every line L through x. Then E is trivial.

Proof. Let s : G(x) → B(x) be given by s(`) = (x, `), so that s is a section with f ◦ s = 1, and consider the diagram

f B(x) s G(x)

σ q Fn Gn p

Pn

The inverse image σ∗E is trivial over every L˜ = f −1(`) for ` ∈ G(x), since we have ∗ ∗ ∼ an isomorphism σ E |L˜ = p E |L˜ = E |L, and E |L is trivial by assumption. Then if we

can exhibit an r-bundle F on G(x) such that σ∗E = f ∗F , we’re done: since σ ◦ s is constant, O⊕r ∼= s∗σ∗ ∼= s∗f ∗ ∼= (f ◦ s)∗ ∼= , so σ∗ ∼= f ∗ ∼= f ∗O⊕r ∼= G(x) E F F F E F G(x) ⊕r ⊕r ∼ ∗ ∼ ∼ O , and then E itself must be trivial, since σ∗O = σ∗σ E = E ⊗O n σ∗O n = E B(x) B(x) F P ∼ and σ∗OB(x) = OPn . Thus we need only exhibit such an F . ∗ Take F = f∗σ E . This is a vector bundle of rank r by a theorem on direct images.

∗ ∗ ∗ We therefore have a canonical sheaf map f f∗σ E → σ E , which is given on each

32 L˜ = F −1(`) by the evaluation map

∗ ∗ 0 ˜ ∗ ∗ f f∗σ E |L˜ = H (L, σ E |L˜) ⊗C OL˜ → σ E |L˜

∗ ∼ ∗ ∗ ∗ therefore this is an isomorphism σ E = f f∗σ E = f F .

Corollary 4.7. Suppose E is a vector bundle on Pn which is generated by global

sections, and such that c1(E ) = 0. Then E is trivial.

Proof. If E is generated by global sections, there exists an exact sequence 0 → K →

⊕N n O n → → 0 for some N. For some line L ⊆ , this restricts to 0 → K| → P E P L ⊕N OL → E |L → 0. If E |L = ⊕iOL(ai), then each OL(ai) must be generated by global P sections, i.e., ai ≥ 0. Since c1(E ) = 0, we must also have i ai = 0, so ai = 0 for all i, i.e., E is trivial on every line. Thus E is trivial on Pn.

For the special case of rank 2 bundles, the splitting type on certain lines can determine splitting on the whole space:

Proposition 4.8. Suppose E is a rank 2 bundle on Pn such that there exists a 2- dimensional family of lines in Pn for which the splitting type of E on any line in the family is the same. Then E splits on Pn. [Bal95]

While the definition of a uniform bundle is fairly straightforward, there is a related, simpler idea:

Definition. A vector bundle E on Pn is homogeneous if for every automorphism t n ∗ ∼ of P , we have an isomorphism t E = E .

Since any line in Pn can be taken to any other by a projective transformation, ho- mogeneous vector bundles must be uniform. Having computed some low-dimensional examples, it was conjectured that the converse holds, i.e., that all uniform bundles are

33 homogeneous. However, it was later shown that there exist uniform non-homogeneous

bundles on every Pn, n ≥ 2. It is still an open problem to determine, for a given n, the least rank r of a bundle which is uniform but not homogeneous. There is a partial result on the splitting of uniform bundles:

Theorem 4.9. Suppose E is a uniform vector bundle of rank r on Pn, r < n. Then E splits as a direct sum of line bundles.

Proof. Suppose the statement holds for all uniform bundles of rank less than r, and

suppose E is a uniform rank r bundle. Without loss of generality, E has splitting type

(a1, . . . , ar), such that ai = 0 for 1 ≤ i ≤ k, and ai+1 < 0. If i = r, then this follows

by the above proposition, so suppose i < r, and suppose 0 → F → E → F 0 → 0 is an exact sequence of uniform vector bundles. Then by induction hypothesis, F and

F 0 split. It then follows from the Bott formula that H1(Pn, F ⊗ F 0) = 0, in which case the above sequence splits (and hence so does E ). Thus the problem remains to construct such a sequence. Consider the standard diagram

q Fn Gn p

Pn

n ˜ −1 ∗ ∼ If ` ∈ G and L = q (`), we have p E |L˜ = E |L (where L corresponds to `. By assumption, r−i ∼ ⊕i M E |L = OL ⊕ OL(ai+j) j=1

0 −1 ∗ n ∗ with ai+j < 0, so in particular h (q (`), p E |q−1(`) = i for all ` ∈ G . Thus q∗p E is a vector bundle of rank k on Gn (as opposed to simply being a coherent sheaf). Let

∗ ∗ ∗ ∗ ∗ G = q q∗p E . Then the q q∗p E → p E exhibits G as a sub-bundle of

34 ∗ 0 ˜ ∗ ∗ p E . Considering the evaluation map G |L˜ = H (L, p E |L˜)⊗OL˜ → p E |L˜ allows us to

construct an exact sequence over Fn: 0 → G → p∗E → G 0 → 0, and when restricted to fibers L˜ we have the identifications 0 ˜ ∗ ∗ 0 0 H (L, p E |L˜ ⊗ OL˜ p E |L˜ G |L˜ 0

⊕k ⊕k Lr−i r−i 0 OL OL ⊕ j=1 OL(ai+j) ⊕j=1OL(ai+j) 0 which shows that G and G 0 have the same splitting type over every L˜ = q−1(`). Thus,

if we can show that there exist bundles G , G 0 on Pn such that G = p∗G , G 0 = p∗G 0, then the exact sequence

0 → p∗G → p∗E → p∗G 0 → 0

projects to an exact sequence

0 → G → E → G 0 → 0

on Pn. To show this, it suffices to show that G ,G 0 are trivial on any F(x). Since

0 → G → p∗E → G 0 → 0

is exact, so is its restriction to a fiber:

0 → | → O⊕r → 0| → 0 G F(x) F(x) G F(x)

By looking at chern classes, we must have c( | ) · c( 0| ) = c(Or ) = 1, so if G F(x) G F(x) F(x) 0 r < n, c(G |F(x)) = c(G |F(x)) = 1, so they both have trivial determinant. It is known that globally generated bundles with trivial determinant are trivial [OSS11], which completes the proof.

35 1 This bound is sharp: For any n, the Ω n is homogeneous (hence P uniform) but indecomposable. [OSS11]

4.3 Stability

For general r and n, there are not many results to categorize the splitting type of a semistable r-bundle on Pn; several results are known for rank-2 bundles, which are included below. On the other hand, there is one powerful general criterion:

Lemma 4.10. Suppose E is a vector bundle of rank r on Pn with generic splitting type (a1, . . . , ar). If for some s we have |as − as+1| ≥ 2, there exists a normal subsheaf of F ⊆ E (i.e., a coherent subsheaf of E such that the quotient E /F is a torsion-free coherent sheaf) such that for a generic line L in Pn, F splits as

s M F |L = OL(ai) i=1

Theorem 4.11. Suppose E is a semistable bundle of rank r on Pn with generic splitting type (a1, . . . , ar). Then |ai − ai+1| ≤ 1 for all i = 1, . . . , r − 1.

n Corollary 4.12. If E is uniform n-bundle on P of splitting type (a1, . . . , an), if E does not split, we must have |ai − ai+1| ≤ 1 for i = 1, . . . , n − 1.

Proof. Suppose we have as − as+1 ≥ 2 for some s. Then there must exist a uniform subbundle F ⊂ E with splitting type (a1, . . . , as), which would have a quotient of splitting type (as+1, . . . , an). But these split as direct sums of line bundles, so

0 → F → E → E /F → 0 would also split, and hence so would E . This, however, is a contradiction.

36 5 Low-Dimensional Results

5.1 2-bundles and Surfaces

It is a theme of algebraic geometry that in a given class of objects, those of dimension 1 are the best understood, having available tools which are out of reach in higher dimensions. At this level of generality, the dimension 2 case often more greatly resembles the general case (of fixed dimension), but is still more manageable than that general case. In the case of vector bundles, rank 2 is where some general theory begins to emerge. Below is a collection of results for vector bundles either of rank 2, or on varieties of dimension 2, or both. Of course, the simplest examples of rank 2 bundles are simply direct sums of line bundles, i.e., decomposable 2-bundles; not much need be said about them at this time. One might expect some new rank 2 bundles to appear as extensions

0 → L → E → L 0 → 0 of line bundles L , L 0. The space of such objects can be given some form.

0 0 0 Definition. Two extensions 0 → L1 → E → L2 → 0 and 0 → L1 → E → L2 → 0 are equivalent if there is an isomorphism α : E → E 0 that makes the diagram

0 L1 E L2 0 α

0 0 0 0 L1 E L2 0

0 commute. The extensions are weakly equivalent if the equalities Li = Li are

0 replaced by Li → Li .

Fix L1 and L2. The Li are line bundles, so in particular their only automorphisms

37 are induced by scalar multiplication, there is an induced action of k× × k×on the set of isomorphism classes of such extensions. When considering these classes, the scalar multiplications on E ∼= E 0 are automorphisms, so don’t affect the particular class. It

1 follows that such classes of weak equivalence classes are the quotient of Ext (L2, L1)

× 1 by an action of k , which describes the projective space P(Ext (L2, L1)). This is 1 Unless Ext (L2, L1) = 0, in which the extension must split and we have only the

2 direct sum L1 ⊕ L2 (in fact, this is the case with P ). [Fri98] The first results on these topics were essentially due to Schwarzenberger, [Sch61a] [Sch61b] who established a few constructions which unfortunately are not practically productive. For instance, any vector bundle E on a X may be constructed as an extension 0 → E 0 → E → E 00 → 0, where E 0 is a direct sum of line bundles, and E 00 is a 2-bundle. As another example, if X is a non-singular surface and E is a

2-bundle on X, E is of the form f∗F for a double-cover f : Y → X by a non-singular surface Y , with F a line bundle on Y . As a matter of existence, over C we have the following:

Theorem 5.1. Let X be a surface, r > 1, c1 a divisor on X, and c2 ∈ Z. Then there exists a vector bundle E of rank r on X such that c1(E ) = c1 and c2(E ) = c2.

On a surface, an important stability condition for bundles is the inequality of Bogmolov:

Theorem 5.2 (Bogmolov’s Inequality). Suppose (X, O(1)) is a smooth, polarized

2 projective surface, and E is a semistable vector bundle of rank r. Then (r−1)c1(E ) ≤

2rc2(E ). [Fri98]

The study of surfaces is dominated by the relationship between a surface and the curves embedded in them; equivalently, by the line bundles on them. Interestingly, in one case, the surface can be determined by a bundle on an underlying curve:

38 Definition. Let C be a nonsingular curve. A geometrically ruled surface over C

is a surface S, and a smooth morphism S → C whose fibers are all isomorphic to P1.

In fact, geometrically ruled surfaces are the minimal models of ruled surfaces. But the relevant idea is this: If π : E → C is a vector bundle over C, of rank r, E will

have dimension r + 1, and in this case, the fibers over C will all be isomorphic to Ar.

However, taking the associated projective bundle PC (E) gives a projective variety of dimension r, whose fibers over C are isomorphic to Pr−1. In the case of r = 2, this happens to cover an entire species of surface:

Theorem 5.3. Fix a nonsingular curve C. Every geometrically ruled surface over C is isomorphic (over C) to PC (E ) for a rank 2 vector bundle on C. The bundles PC (E )

0 and PC (E ) are C-isomorphic if and only if there exists a line bundle L ∈ Pic (C) such that E 0 = E ⊗ L . [Bea96]

5.2 Serre’s Construction and Hartshorne’s Conjecture

The association of line bundles and divisors means that some rank 1 bundles corre- spond to closed subvarieties of codimension 1. One might ask whether or not such a correspondence could be generalized to higher rank. While it is not as ubiquitous as the r = 1 case, Serre (and later others) gave a construction which associates to a

local Y ⊂ X on Pn a vector bundle on X of rank 2, equipped with a section s which vanishes on Y .

Theorem 5.4. Let n ≥ 3.

• Suppose E is a rank 2 vector bundle on Pn, and s ∈ H0(Pn, E ) a section whose ∼ zero set Y is of codimension 2, with ideal sheaf JY . Then E |Y = NY/Pn , i.e., E is an extension to all of Pn of the normal sheaf of Y in Pn.

39 • Suppose Y is a local complete intersection of codimension 2 in Pn with ideal

sheaf JY , and suppose that the determinant of the normal sheaf can be ex-

n tended to all of P , that is, det NY/Pn = OPn (k)|Y for some k ∈ Z. Then there exists a 2-bundle on Pn, equipped with a section s which has Y as its zero locus (scheme-theoretically). Furthermore, s induces an exact sequence

−·s 0 → OPn −−→ E → JY (k) → 0

and E has Chern classes c1(E) = k, c2(E) = deg Y . [Fri98]

It is more convenient to sketch the construction in each direction. Suppose E is a rank-2 bundle and s is a suitable global section. Then on a trivializing open affine

n set U ⊂ P , s = f1e1 + f2e2, where f1, f2 ∈ O(U) and e1, e2 ∈ E (U) are a basis for E over U. If Y is the zero set of s (as a closed subscheme), then JY is generated over

2 U by (f1, f2). It follows that Y is then a local complete intersection, and JY /JY is naturally a locally free OY -module of rank 2. By examining the Koszul complex

∨ ∼ 2 2 for (E , s), it can be determined that E = JY /JY , (recall that JY /JY is the conormal bundle) so E represents an extension of the normal bundle NY/Pn to all of Pn. On the other hand, if Y is a locally complete intersection of codimension 2, and det NY/Pn = OPn (k)|Y for some k, we should expect to find E as the dual of an extension

∨ 0 → OPn (−k) → E → JY → 0

1 n and such extensions are in correspondence with the group ExtPn (JY , OP (−k)). By 0 a spectral sequence argument, this group is isomorphic to H (Y, OY ), and we can choose an extension E corresponding to the global section 1. [Har74] This construction can be also be generalized to arbitrary smooth, projective vari-

40 eties; in those cases the construction is not as useful because of the difficulty finding

convenient subschemes Z Kleiman gave a generalization in one direction: If E is an

r-bundle on Pn for r < n, for sufficiently large k, E (k) admits a global section s whose zero set is a nonsingular subvariety Y of codimension r, such that Y is a complete

intersection exactly when E (k) splits as a direct sum of line bundles. [Kle69] Hartshorne gave a variant of Serre’s correspondence with a very similar argument:

Theorem 5.5. A nonsingular subvariety Y (of codimension 2 in Pn) occurs as the

n zero set of a section of a rank 2 bundle E on P if and only if its canonical bundle ωY

is equal to OY (k) for some k. In that case, Y is a complete intersection if and only

if E is a direct sum of line bundles. [Har74]

After considering the cases in low dimensions, he went on to pose the following problem:

Conjecture 5.6 (Hartshorne). Let n ≥ 7. Every vector bundle of rank 2 on Pn splits as a direct sum of line bundles.

In view of his modification to Serre’s correspondence, this equivalently asserts that

every nonsingular subvariety Y ⊂ Pn of codimension 2 is a complete intersection. This conjecture was consistent with the philosophy at the time that finding indecomposable (n − r)-bundles becomes increasingly difficult as n − r increases, to the point where it may be impossible. At present, this conjecture is still unresolved, despite continued evidence and heuristic justification in lower dimensions. [Fol05] [NJHO95] He also asked whether the generalization to higher-rank vector bundles above has a converse in any sense, and this is also still unknown in general.

41 5.3 The Horrocks-Mumford Bundle

The Horrocks-Mumford bundle is a particular 2-bundle over P4, which possesses sev- eral unique qualities. Other than similar bundles derived from it, it is essentially the only known stable 2-bundle on P4, and it is also indecomposable. Its construction touches on a few other important tools.

Definition. A monad over a projective variety X is a short chain complex of vector bundles

α β 0 → F −→ G −→ F 0 → 0 which is exact at F and F 0, and such that im (α) is a sub-bundle over G . In this case, E = ker(β)/im (α) is called the cohomology of the monad. Any monad of the above form has a display:

0 0

0 F A E 0

0 F α G B 0 β F 0 F 0

0 0

with exact columns and rows, where A = ker(β) and B = Coker(α).

It follows immediately from the display that c(F ) · c(F 0) · c(E ) = c(G ), and that the rank of G is the sum of the ranks of F , F 0, and E .

5 Let V = C , with basis e1, . . . , e5 (and consider index arithmetic cyclically). Let

42 P4 = P(V ), and let

π 0 → OP4 (−1) → V ⊗ OP4 −→ TP4 (−1) → 0 be the Euler sequence, writing Q = T (−1). Taking 2nd exterior powers gives the exact sequence

2 0 → Q ⊗ O(−1) → ∧2V ⊗ O −−→∧∧ π 2Q → 0

The Horrocks-Mumford bundle can be constructed as the cohomology of a twisted monad of the form

β 0 → V ⊗ O −→α (∧2Q)⊕2 −→ V ∨ ⊗ ∧4Q → 0

+ − Defining α (ei) = ei+2 ∧ ei+3 and α (ei) = ei+1 ∧ ei+4 and extending by linearity, let

(α+,α−)⊗1 (∧2π)⊕2 α : V −−−−−−→ (∧2V )⊕2 −−−−→ (∧2Q)⊕2

Now let φ : ∧2Q → H om(∧2Q, ∧4Q) be the natural isomorphism, and define Φ :

2 ⊕2 2 ⊕2 4 0 1 (∧ Q) → H om((∧ Q) , ∧ Q) given by Φ = φ ⊗ ( −1 0 ). Finally, define

Φ α∨ β :(∧2Q)⊕2 −→ H om(∧2Q)⊕2, ∧4Q) −→ H om(V ⊗ O, ∧4Q)

Then a calculation shows that α, β make the above complex into a monad. Tensoring with O(2), we have the twisted monad

α(2) β(2) 0 → V ⊗ O(2) −−→∧2Q⊕2 ⊗ O(2) −−→ V ∨ ⊗ O(3) → 0 by the identity O(1) ∼= ∧4Q. Its cohomology, E , is the Horrocks-Mumford bundle.

43 [NJHO95] By a computation based on the monad’s display, it has Chern classes

2 n+1 ct(E ) = 1 + 5t + 10t ; this clearly can’t be factored in Z[t]/(t ), so it must be the case that E is indecomposable. [OSS11] Underlying the construction of this bundle is the Beilinson spectral sequence, a powerful homological tool in the study of bundles on Pn:

Theorem 5.7 (Beilinson). Suppose E is a vector bundle of rank r on Pn. There

pq exists a spectral sequence Er with E1-term

pq q n −p E = H ( , (p)) ⊗ Ω n (−p) 1 P E P

This converges to   E i = 0 Ei =  0 otherwise

pq n −p,p so that when p + q 6= 0, E∞ = 0, and ⊕p=0E∞ is the associated graded sheaf of a filtration on E . [OSS11]

The Beilinson spectral sequence is a very convenient tool, because it calls for computations involving locally free resolutions of sheaves; this is as close to linear algebra as one can get in this situation. Consequently, there has been some interest

in generalizing this sequence, which originally is only defined for Pn. [NJHO95] [Ful07]

6 Ulrich Bundles

Ulrich sheaves are a class of bundle of recent interest to algebraic geometers. Coming from obscurity in (where they were created as “Maximally- generated Cohen-Macaulay modules”), they have become a central object of study in the world of vector bundles due to their numerous useful properties and applications.

44 Mystically, they are both difficult to come across and expected to be ubiquitous. Here we sketch some of their basic properties.

Theorem 6.1. Suppose X ⊂ PN is a smooth subvariety of dimension n, and let E be a rank r vector bundle on X. The following are equivalent: [Bea18]

1. There exists a linear resolution

0 → Lc → Lc−1 → · · · → L0 → E → 0

N bi where c = codim(X, P ), and Li = OP(−i) for some bi.

2. Hi(X, E (−k)) = 0 for k = 1, . . . , n and all i.

n n 3. If π : X → P is a finite linear projection, π∗E is trivial on P .

Proof. First consider the case X = Pn; then we must show that any bundle with the appropriately vanishing cohomology is trivial. First note that since the cohomology

groups vanish beyond the dimension of X, we in fact have Hk(Pn, E (−k)) = 0 for all k > 0; thus E is 0-regular in the sense of Castelnuovo-Mumford, above. Thus E is

generated by global sections, and in particular, Hi(Pn, E ) = 0 for i > 0. Recall that the Hilbert polynomial of E may be given by the expression PE (t) = χ(E (t)); then in this case, the Hilbert polynomial vanishes for t = −1,..., −n. Its leading coefficient

0 must be r/n!, so PE (t) = r/n!(t + 1) ··· (t + n), and therefore h (E ) = r. This implies

r the existence of a surjective map O n → , which must be an isomorphism; thus P E E is trivial. Now, let X be an arbitrary nonsingular subvariety, of dimension n. The first condition implies the second by computation. If the third holds, then Hi(X, E (−k)) =

i n H (P , π∗E (−k)) = 0 at the relevant indices, so the second condition holds. Assuming

45 the second condition, by the same equation the relevant cohomology on Pn vanishes, which by the above means that π∗(E ) is trivial. Now, assume the second condition, and construct a sequence of coherent sheaves

N Ki on P , subject to the following conditions:

• K0 = E .

0 N • Ki+1(−1) is the kernel of the evaluation map H (P ,Ki) ⊗ OPN → Ki.

` N • H (P ,Ki(−j)) = 0 for 1 ≤ j ≤ n + i.

Suppose the Ki are defined for i = 0, . . . , p, and define Ki+1 by the second condition.

0 N Then we tautologically have an exact sequence 0 → Kp+1(−1) → H (P ,Kp) ⊗

OPN → Kp → 0. Then by the long exact sequence in cohomology we have

` N H (P ,Kp+1(−j)) = 0

q N q−1 N for j = 1, . . . , n + p + 1, and also H (P ,Kp+1(−q)) = H (P ,Kp(−(q − 1))) = 0 for all q. Then Kp+1 is 0-regular, and satisfies all the above conditions.

0 N Now define Li = H (P ,Ki) ⊗ OPN (−i). Then for each i we have a short exact sequence

0 → Ki+1(−i − 1) → Li → Ki(−i) → 0

which can be chained to give another exact sequence 0 → Kc(−c) → Lc−1 → · · · →

L0 → E → 0. By the third condition above, Kc satisfies the conditions of the theorem on PN , so is in fact trivial. Then, considering that each map in the above sequence is of the appropriate form, by induction each Li is of the correct form.

Definition. If a vector bundle E on a smooth projective variety X satisfies any of the above equivalent conditions, E is an Ulrich bundle.

46 Corollary 6.2. The following immediately follow from the definition: [Bea18]

• Ulrich bundles are semistable; if an Ulrich bundle is not stable, it is an extension of Ulrich bundles of smaller rank.

• All Ulrich bundles on a projective curve C ⊂ Pn are of the form E (1), where E is a vector bundle on C with vanishing cohomology.

• If E is an Ulrich bundle on X ⊂ Pn and Y is the intersection of X with a

hyperplane, E |Y is Ulrich.

In particular, when X is a of degree d in PN , the linear resolution is an exact sequence 0 → O (−1)rd −→OL rd → → 0, where L is a rd × rd matrix P P E of linear forms on PN . If X is defined by the equation F = 0, then it must also be the case that det L vanishes (precisely) along X, so we must have det L = F r (up to scalar multiple). This relates the existence of an Ulrich bundle to a positive answer to a classical problem: when can X be represented in this kind of determinental form?

Eisenbud conjectured in [ES03] that any smooth projective variety X ⊂ PN should carry an Ulrich bundle; however, the existence of Ulrich bundles is not known in general. Results are known for curves, complete intersections, linear determinental varieties, Grassmannians, several classes of surfaces, including ones of low degree in P3, and others, (all nonsingular). Unfortunately, much of the recent progress in this direction has been using quite specialized techniques, which do not admit easy generalization to wider classes of varieties. [Cos17] Another important application of Ulrich bundles is to the theory of Chow forms.

47 If X ⊂ Pn, consider the standard diagram

q Fn Gn p

Pn

n −1 Then the Chow divisor of X is the divisor on G given by DX = q(p (X)).

n The defining equation of DX in the Pl¨ucker coordinates of G is called the Chow form. Chow forms were of classical and early-modern interest, as they encode detailed information about X (in modern research they were made somewhat obsolete by the Hilbert scheme). As part of their foundational paper [ES03], Eisenbud and Schreyer use the existence of Ulrich bundles to compute Chow forms for their corresponding spaces, turning a very difficult problem into a practically solved one.

7 Conclusion

We have therefore seen the core components of the theory of vector bundles in al- gebraic geometry. Their uses are varied across the board; their structure is at once simple and complex; their classification provides the foundation of a rich theory of moduli spaces. At times, given how well we generally understand vector spaces on their own, the slow pace of progress can be frustrating. On the other hand, the study of vector bundles has been a leader in that part of algebraic geometry which focuses on objects of a more classical nature. There are yet ever more subject areas in the field of vector bundles. Over affine varieties (and schemes), vector bundles correspond exactly to projective modules; their study in that regard is one of the motivating connections of K-theory. They also play a role in the A1- theory founded by Morel and Voevodsky. There

48 are also related “problems in linear algebra:” constructions involving the exterior and Clifford algebras of a vector space which may be associated to vector bundles; these constructions have been inciting new approaches to some of the above problems for the last several decades [OSS11] [Har79]. Attached to the commutative algebra of Ulrich bundles are new perspectives on the cohomology of vector bundles: cones of Betti tables and supernatural cohomology, which seek to understand the structure underlying the cohomology of vector bundles. [ES09] There are connections to Yang- Mills theory in physics, which have generalized in a pure setting to the notion of a mathematical . In the analytic world, there problems concerning the existence of complex structures on topological vector bundles. [OSS11]

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53