CHERN FORMS OF POSITIVE VECTOR BUNDLES

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of the Ohio State University

By

Dincer GULER

*****

The Ohio State University 2006

Dissertation Committee: Approved by Prof. Fangyang Zheng, Advisor

Prof. Jeffery McNeal Advisor Prof. Andrzej Derdzinski Graduate Program in Mathematics

ABSTRACT

In this thesis we consider the problem posed by Griffiths, which asks to determine which polynomials of Chern forms will be always positive for any Griffiths positive vector bundle E. Following an idea used in the work of Yau and Zheng, we explore the induced metric on the projectivized bundle, the Grothendieck equation and the push forward of forms and we are able to prove that the signed Segre forms are always positive.

ii Dedicated to my wife and my parents

iii ACKNOWLEDGMENTS

I would like to thank my advisor, Professor Fangyang Zheng, for his patience, guidance and constant encouragement which made this work possible. I am grateful for the enlightening discussions and suggestions as well as the intellectual support he has provided.

I would like to thank Professor Jeffery McNeal for his enlightening discussions on

L2 methods in Complex Differential Geometry that has given me new insights in the theory.

I would like to thank Professor Andrzej Derdzinski for his invaluable time and professional insight.

I am thankful for the Department of Mathematics for their support expressed through several teaching and research fellowships that I enjoyed.

Last but not the least I wish to thank my family and my wife for their constant support and encouragement.

iv VITA

1975 ...... Born in Razgrad, Bulgaria

1998 ...... B.Sc. in Mathematics, Middle East Technical University, TURKEY

1998-Present ...... Graduate Teaching Associate, The Ohio State University

PUBLICATIONS

Guler, Dincer; Zheng, Fangyang On Ricci rank of Cartan-Hadamard manifolds, Internat. J. Math. 13 (2002), no. 6, 557–578

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Complex Differential Geometry

v TABLE OF CONTENTS

Abstract ...... ii

Dedication ...... ii

Acknowledgments ...... iv

Vita...... v

CHAPTER PAGE

1 Introduction ...... 1

2 Chern Forms, Segre Forms and Schur Polynomials ...... 8

2.1 Preliminaries ...... 8 2.2 Chern and Segre Forms ...... 10 2.3 Positive Polynomials ...... 13

3 Positive Differential Forms, Push Forward and Positive Vector Bundles 21

3.1 Positive Differential Forms ...... 21 3.2 Push Forward of Differential Forms ...... 24 3.3 Positive Vector Bundles ...... 26

4 Positivity of the Segre Forms ...... 33

4.1 Proof of the Main Theorem ...... 33 4.2 Semipositivity ...... 40 4.3 Some Remarks For Further Research ...... 43

Bibliography ...... 44

vi CHAPTER 1

INTRODUCTION

In complex geometry the concept of positivity has played a central role. Many peo- ple contributed to the development of it, most noticably Grauert, Hartshorne and

Griffiths.

In Griffiths paper [G] there were five notions of positivity, all closely related. In the end it seems that two of them become the most important. One is the ampleness of holomorphic vector bundles over complex projective varieties, coined by Hartshorne.

For a line bundle E over Xn, one of the many equivalent definitions of E being ample is that there exists an integer m and a holomorphic embedding f : X → PN of X

∗ ⊗m N N into a complex projective space such that f OP (1) = E , where OP (1) is the hyperplane line bundle. When the rank of E is greater than one, one may consider

−1 ∗ the projectivized bundle π : P(E) → X, where for any x ∈ X, π (x) = P(Ex) is the space of the hyperplanes in the fibre Ex. Following Grothendieck and Hartshorne,

∗ ∗ denote by OP(E)(1) the dual of the tautological line subbundle of π E , then E is defined to be ample (over X) if OP(E)(1) is ample over P(E). OP(E)(1) is also called the associate (hyperplane) line bundle over P(E).

1 The other positivity notion for E is the so called Griffiths positive: If E admits a

Hermitian metric h whose curvature is Griffiths positive. To be precise, that means

1,0 that for any x ∈ X and any 0 6= v ∈ Ex and any 0 6= V ∈ Tx X, the directional curvature r ¯ X k ¯ Θvv¯(V, V ) = viv¯jhk¯jΘi (V, V ) i,j,k=1 P is positive, where r is the rank of E, v = viei, hi¯j = h(ei, ej), with {e1, ..., er}

k being a local frame of E near x and (Θj ) the curvature of E under this frame, which is an r × r matrix of (1, 1) forms. In other words, the curvature Θvv¯ is a positive

¯ (1, 1) form. Note that Θvv¯(V, V ) is also called in some literature Griffiths curvature or bisectional curvature .

For line bundles, these two positivity notions coincide. One side is clear, the

N standard metric on OP (1) has Griffiths positive curvature, so if E is ample, the

⊗m ∗ N pull back metric on E = f OP (1) naturally induces a Hermitian metric on E

(by taking the m-th root) which is still positively curved. So ample implies Griffiths positive for line bundles. The other direction is known as the Kodaira embedding theorem, a fundamental result in complex geometry.

For vector bundles of rank higher than one, Griffiths positivity implies ampleness.

This is because any Hermitian metric h on E naturally induces a Hermitian metric

ˆ ˆ h on OP(E)(1) (see section 3.3 for details), and h has Griffiths positive curvature if h does. The other direction is still an open question: Is any ample vector bundle

Griffiths positive?

The only known case is when the base manifold is a curve, in this case the answer 2 is yes, proved by Unemura in [U]. In general case E being ample means that the line

bundle OP(E)(1) admits a positively curved metric g. However g in general may not be the induced metric hˆ from a Hermitian metric h on E. Instead g only gives a so called Finsler metric on E. The complex Finsler geometry is still largely uncharted territory, and the answer to the above question remained illusive, despite the general belief that the answer should be an affirmative one.

One of the most fundamental properties associated to the positivity notion of E is the positivity of its characteristic classes/forms. Let ck(E) denote the k-th Chern class of E. It is a cohomology class in Hk,k(E)∩H2k(X, Z). When E is equipped with a Hermitian metric h, the curvature form gives rise to the Chern forms Ck(E), which are real global (k, k) forms on X representing the k-th cohomology classes ck(E).

Let P(c1, ..., cr) be the polynomial ring over Q of the Chern classes of E and denote by Pk the Q-vector space of weighted homogeneous polynomials of degree k, with weight assignement deg(ci) = i.

In [G] Griffiths raised the question ”What kind of polynomials in Pk will be always positive for any ample vector bundle E over X?”, where both the rank of E and the dimension of X are no less than k. Here a (k, k) cohomoly class is considered positive if its integral over any k dimensional subvariety of X is positive. In fact this question was raised at both the cohomology class level and the form level.

At the cohomology level, this question was answered by the celebrated work of

Fulton and Lazarsfeld in [FL], expanding the earlier result of Bloch-Gieseker [BG].

To describe their result let us first fix some notations. Let E be a holomorphic vector

3 bundle of rank r over a projective manifold X of dimension n. Denote by Λ(k) the set of all partitions of k by nonnegative integers. Thus an element λ ∈ Λ(k) is of the form λ = (λ1, ..., λk) where

X k ≥ λ1 ≥ λ1 ≥ ... ≥ λk ≥ 0, λj = k

Each λ ∈ Λ(k) gives rise to a Schur polynomial Pλ of degree k defined as the k × k determinant

c c . . . c λ1 λ1+1 λ1+k−1

c c . . . c λ2−1 λ2 λ2+k−2 Pλ = ......

c c . . . c λk−k+1 λk−k+2 λk where by convention c0 = 1 and ci = 0 if i does not belong to [0, k]. It turns out that {Pλ : λ ∈ Λ(k)} forms a basis of the vector space Pk, the weighted homogeneous polynomials of degree k of the Chern classes of E. So any polynomial of the Chern

P classes in Pk can be expressed as P = aλPλ where each aλ ∈ Q. For k = 2 the two

2 Schur polynomials are c2 and c1 − c2. For k = 3 there are three Schur polynimials c3,

3 c1c2 − c3 and c1 − 2c1c2 + c3. For k = 4, there are 5 Schur polynomials:

2 2 2 4 2 2 c4, c1c3 − c4, c2 − c1c3, c1c2 − c1c3 − c2 + c4 , c1 − 3c1c2 + 2c1c3 + c2 − c4 corresponding to the partitions (4, 0, 0, 0), (3, 1, 0, 0), (2, 2, 0, 0), (2, 1, 1, 0) and (1, 1, 1, 1)

k respecyively. Note that for λ = (1, 1, ..., 1), pλ is just (−1) sk(E), where sk is the

Segre class of E determined by the inverse Chern classes, namely

(1 + s1(E) + s2(E) + ...)(1 + c1(E) + c2(E) + ...) = 1 4 for any vector bundle E.

Recall that a cohomology class α ∈ Hk,k(X) is said to be positive if

Z α > 0 Z for any k-dimensional subvariety Z of X.

1.0.1. Theorem([FL]): If E is ample then for any k ≤ min{n, r}, Pλ is positive for P each λ ∈ Λ(k). Conversely if P = aλPλ is such that P is positive for any ample vector bundle E over any projective manifold M with r, n ≥ k. Then all aλ ≥ 0 and at least one aλ > 0.

So nontrivial, nonnegative linear combinations of the Schur polynomials are all the weighted homogeneous polynomials of Chern classes that are positive for all ample bundles. For the special case of ck which is Pλ for λ = (k, 0, ...0), the positivity for ample E was established by Bloch and Gieseker earlier in [BG]. The works [BG] and

[FL] answers the Griffiths conjecture on the class level completely. On the form level, however, the question is still largely open.

1.0.2. Question(Griffiths): Let (E, h) be a Hermitian bundle of rank r over a projective manifold Xn whose curvature is Griffiths positive. If k ≤ min{n, r}, then

ˆ n is it true that every Schur form Pλ is always a positive (k, k) form on X ?

ˆ Here of course Pλ is defined by replacing the Chern classes ci(E) by the corre- sponding Chern forms Ci(E) in the definition of Pλ. For the second Chern form

C2(E) the question was answered affirmatively by Griffiths himself in [G]. 5 1.0.3. (Griffiths): Let (E, h) be any rank r Hermitian vector bundle with Griffiths positive curvature over a projective manifold Xn. If n, r ≥ 2, then the Chern form

n C2(E) is a positive (2, 2) form on X .

Note that in [G] this was proved for rank 2 vector bundles over surfaces, but the proof can be easily adopted to the above general case.

In this thesis we give a partial answer to the Griffiths question:

1.0.4. Main Theorem: Let (E, h) be a rank r Hermitian vecor bundle with positive

Griffiths curvature over a projective manifold Xn. Then for any k ≤ min{n, r} the signed Segre form

k ˆ (−1) Sk(E) = P(1,1,...,1)(E) is a positive (k, k) form on Xn.

2 In particular, C1 − C2 is a positive (2, 2) form for any Griffiths positive vector bundle over a projective surface. This together with the aforementioned result of

Griffiths answers the two dimensional case.

Note that the Griffiths’ proof of the positivity of C2 is purely algebraic and it seems to be difficult to generalize it to higher dimensions. In fact even in dimension three this becomes too complicated algebraically. Our approach relies more on the global geometry of the setting. It stems out of an idea used in the work of Yau and Zheng [YZ], and by exploring the induced metric on the projectivized bundle, the Grothendieck equation and the push forward of forms we were able to better exploit the positivity assumption on the curvature of (E, h) and establish the partial 6 answer to this long standing question of Griffiths. The thesis is organized as follows.

In chapter 2 we review the Chern forms and positive polynomials. In chapter 3 we review and establish some results in the theory of positive differential forms, push forward of differential forms and positive vector bundles. In chapter 4 we prove our main result and discuss the semipositive case.

7 CHAPTER 2

CHERN FORMS, SEGRE FORMS AND SCHUR

POLYNOMIALS

2.1 Preliminaries

In this section we will fix some definitions and notations for our later use. Let X be a compact Kahler manifold with a fixed Kahler metric ω. Ak(X) and Ap,q(X) will denote the sets of smooth k and (p, q) forms respectively. When these forms have values in a vector bundle E we use the notations Ak(X,E) and Ap,q(X,E).

Let E be a Hermitian vector bundle with a Hermitian metric h = (hjk¯). Then one can define a natural sesquilinear map

p q p+q A (X,E) × A (X,E) → A (X, C)

(ξ, η) 7→ {ξ, η} by combining the wedge product and the Hermitian metric in the following manner.

P P If ξ = ξj ⊗ ej and η = ηk ⊗ ek, then

r X {ξ, η} = ξjη¯khjk¯ j,k=1 8 The curvature matrix of a Hermitian vector bundle (E, h) will be denoted by

i i Θ = (Θj). Thus each Θj is a (1, 1) form which in local coordinates can be expressed as

j X i Θi = Rjαβ¯dzα ∧ dz¯β

Since the curvature is an End(E,E) valued (1, 1) form we can also define the curvature tensor as

˜ X ∗ Θ(E) = Rjkα¯ β¯dzα ∧ dz¯β ⊗ ej ⊗ ek

P i where Rjkα¯ β¯ = Rjαβ¯hik¯.

For the sake of completeness we include the proof ofthe following lemma which guarantees the existance of normal coordinates at a point p ∈ X.

2.1.1. Lemma: For every point p ∈ X and every coordinate system (z1, ..., zn) near p, with p being the origin, there exists a holomorphic frame {e1, ..., er} of E in a neighborhood of p such that

n X 3 hjk¯(z) = (ej(z), ek(z)) = δjk − Rjkα¯ β¯zαz¯β + O(|z |) α,β=1 where Rjkα,¯ β¯ are the components of the curvature of E at p

Proof. Let f = {f1, ..., fr} be a holomorphic frame of E. We can assume that f(p) is an orthonormal frame at p by taking linear combinations of fj’s with constant coefficients. Then around p we have

n X 0 2 (fj(z), fk(z)) = δjk + (cjkαzα + cjkαz¯α) + O(|z| ) α=1 9 0 with cjkα =c ¯jkα. First we set

X gj(z) = fj(z) − cjkαzαfk(z) α,k

Then there are coefficients ajkαβ, bjkαβ and djkαβ such that

2 (gj(z), gk(z)) = δjk + O(|z| )

X 3 = δjk + (ajkαβzαz¯β + bjkαβz¯αz¯β + djkαβzαzβ) + O(|z| ) α,β

The holomorphic frame e = {e1, ..., er} we are looking for is

X ej(z) = gj(z) − djkαβzαzβgk(z) α,β,k ¯ Since cjkαβ = bjkαβ we find that

X 3 (ej(z), ek(z)) = δjk + ajkαβzαz¯β + O(|z| ) α,β

It remains to show that −ajkαβ = Rjkαβ are the coeficients of the curvature tensor.

But this folllows from

0 X 2 ∂(ej, ek) = {D ej, ek} = ajkαβz¯αdzβ + O(|z| ) α,β

00 0 X Θ(E)ej = D (D ej) = ajkαβdz¯β ∧ dzα ⊗ ek + O(|z|) α,β,k

2.2 Chern and Segre Forms

We denote by P(E) the projectivized bundle of lines in E. Thus

∗ ∗ P(E) = ∪x∈X {E \ zero section}/C 10 The manifold P(E) has the tautological line bundle OP(E)(−1) which is a subbundle of π∗E∗ and satisfies the short exact sequence

∗ ∗ 0 → OP(E)(−1) → π E → OP(E)(−1) ⊗ TP(E)/X → 0

Twisting the above short exact sequence with OP(E)(1) we obtain

∗ ∗ 0 → OP(E) → π E ⊗ OP(E)(1) → TP(E)/X → 0

∗ ∗ Then by the Withney formula we get, c(OP(E)).c(TP(E)/X ) = c(π E ⊗ OP(E)(1)), where c stands for the total Chern class. But since c(OP(E)) = 1 and TP(E)/X has rank

∗ ∗ r−1 we obtain cr(π E ⊗OP(E)(1)) = 0. To simplify the notation let ϕ = c1(OP(E)(1))

∗ andc ˆi = π (ci(E)).

∗ ∗ Let a1, ..., ar be the Chern roots of the bundle π E . i.e. let

∗ ∗ c(π E ) = (1 + a1)...(1 + ar)

Since the Chern roots of a tensor product F ⊗ G of two vector bundles F and G are

∗ ∗ the sums of Chern roots of F and G, the Chern roots of π E ⊗ OP(E)(1) would be a1 + ϕ, ..., ar + ϕ so we get

∗ ∗ 0 = cr(π E ⊗ OP(E)(1)) = (a1 + ϕ)(a2 + ϕ)...(ar + ϕ) =

r r−1 r r−1 r−2 r ϕ + ϕ (a1 + ... + ar) + ... + a1...ar = ϕ − ϕ cˆ1 + ϕ cˆ2 − ... + (−1) cˆr where

∗ i X cˆi = ci(π E) = (−1) ak1 ...aki

1≤k1<...

11 We now recall the Chern forms of a vector bundle E. They will be denoted by

Ck(E) and are defined as the coefficients of the polynomial

n−1 n det(P + tI) = fn(P ) + fn−1(P )t + ... + f1(P )t + t

i i where P = 2π Θ(E). That is Ck(E) is the (k, k) form fk( 2π Θ(E)). The Chern forms depend on the Hermitian connection of the Hermitian metric, however the cohomology classes they represent, which are called Chern classes, do not depend on the connection or metric. From now on we will denote the Chern forms with capital letters and the corresponding classes by lowcase letters. Thus ck(E) = cl[Ck(E)].

From simple linear algebra we obtain that C1(E) is the trace, Cn(E) the determinant

i and Ck(E) is the sum of the determinants of all k ×k diagonal submatrices of 2π Θ(E) formed by the entries on the rows and columns selected by all subsets I ⊂ {1, ..., r} of cardinality k. To be more specific we have

i X C (E) = Θj 1 2π j

1 X C (E) = − (Θj ∧ Θk − Θk ∧ Θj ) 2 8π2 j k j k

The Segre class of a vector bundle E is defined as the inverse of the total Chern

P k class. In other words if we write c(t) = ckt for the Chern polynomial and with

P k s(t) = skt then it holds c(t) · s(t) = 1. This means

sk(E) + c1(E)sk−1(E) + ··· + ck−1(E)s1(E) + s0(E) = 0

for each k = 1, 2, ..., r. The first few Segre classes are s0(E) = 1, s1(E) = −c1(E),

12 2 and s2(E) = c1(E) −c2(E). If the Chern roots of E are a1,...,ar, then the Segre class sk is the class

k X j1 jr (−1) a1 ...ar j1+...+jr=k As in the case of Chern forms, we will denote the Segre forms with capital letters

Sk where Sk(E) is defined by Sk(E) = P (C1(E), ..., Ck(E)) where P is the degree k weighted homogeneous polynomial that gives sk = P (c1(E), ..., ck(E)). Clearly,

Sk(E) satisfies

Sk(E) + C1(E)Sk−1(E) + ··· + Ck−1(E)S1(E) + Ck(E) = 0

for each k. Recall that we denoted ϕ = c1(OP(E)(1)) and to be consistent with the notation from now on we will denote Φ = C1(OP(E)(1)).

2.3 Positive Polynomials

Let X be a projective manifold of dimension n and let E be a holomorphic vector bundle of rank r over X with total Chern class

c(E) = 1 + c1(E) + c2(E) + ... + cr(E)

i1 ir We consider the vectors I = (i1, ..., ir) of nonnegative integers and set cI = c1 ...cr and |I| = i1 + 2i2 + ... + rir

2.3.1. Definition: A polynomial P (c1, ..., cr) ∈ Q[c1, ..., cr] is called a weighted ho- mogeneous polynomial of degree d if

1/2 1/r d P (tc1, t c2, ..., t cr) = t P (c1, ..., cr) 13 for any positive t ∈ Q. In other words if

X i1 ir P = aI c1 ...cr |I|=d

The set of all weighted homogeneous polynomials of degree d will be denoted by

Pd.

The set Pd has a distinguished subset of elements, called Schur polynomials, which are defined as follows.

Denote by Λ(d, r) the set of all partitions of n by nonnegative integers ≤ r. Thus an element λ ∈ Λ(d, r) is of the form λ = (λ1, ..., λd) where

X r ≥ λ1 ≥ λ1 ≥ ... ≥ λd ≥ 0, λj = d

Each λ ∈ Λ(d, r) gives rise to a Schur polynomial Pλ of degree d defined as the d × d determinant

c c . . . c λ1 λ1+1 λ1+d−1

c c . . . c λ2−1 λ2 λ2+d−2 Pλ = ......

c c . . . c λd−d+1 λd−d+2 λd where by convention c0 = 1 and ci = 0 if i does not belong to [0, r]. In particular for λ = (d, 0, ..., 0) we get cd(E) and for λ = (1, 1, ..., 1) we get the d-th signed Segre

d class (−1) sd(E).

2.3.1. Proposition: The Schur polynomials form a basis for Pd.

14 Thus any degree d weighted homogeneous symmetric polynomial P can be written uniquely as X P = aλPλ λ∈Λ(d,r)

We will define two types of positive polynomials in Pd and discuss their properties.

First we need the definition of ample vector bundles.

2.3.2. Let L be a holomorphic line bundle over X. Then L is said to be ample if

n ⊗k ∗ n there exists a holomorphic embedding f : X → P such that L = f OP (1) for

n some positive integer k. Here OP (1) is the hyperplane line bundle of the projective space. If E is a vector bundle of higher rank, then E is said to be ample if the

associated hyperplane line bundle OP(E)(1) over P(E) is ample.

2.3.3. Definition: Let X be a projective manifold and E a holomorphic vector bundle over X. A polynomial P ∈ Pd is called numerically positive if the integral Z P (c1(E), ..., cr(E)) X is strictly positive for any ample vector bundle of rank r ≥ d over any projective manifold X and any d dimensional irreducible subvariety Y in X.

Examples of numerically positive polynomials are the Schur polynomials defined above, and in fact according to the main result in [FL], the cone of numerically positive polynomials coincides with the cone generated by the Schur polynomials.

Let us now define the second notion of positivity for polynomials in Pd, which we call Griffiths-positive. We need some preparations. 15 Let us denote the Chern roots of E by γ1, ..., γr, i.e we have formally

r 1 + c1(E)t + ... + cr(E)t = (1 + γ1t)...(1 + γrt)

Write also P = ⊕dPd. Then one can see that P is isomorphic to the ring of polyno- mials in γ1, ..., γr which are invariant under the action of the permutation group Sr.

∗ ∗ ∗ Let us denote that ring by P . Denote also the degree d polynomials in P by Pd so

∗ ∗ that we have the decomposition P = ⊕Pd

∗ Note that for I = (i1, ..., id), γI = γi1 ...γid has weight d, hence Pd consists of all invariant polynomials X P (γ) = pI γI

I=(i1,...,id)

Let now B = (Bρσ) be a variable r × r matrix, 1 ≤ ρ, σ ≤ r, and let γρ = Bρρ.

Then Griffiths proved that the ring of polynomials in γ1, ...γr invariant under the permutation group Sr is isomorphic to the ring I of polynomials in Bρσ invariant under B 7→ MBM −1 for any invertible matrix M ∈ GL(r, C). This result together

∗ with the isomorphism between P and P implies that the graded ring I = ⊕Id is isomorphic to the ring P of weighted homogeneous polynomials of Chern classes.

2.3.2. Lemma: Any P (B) ∈ Id can be written as

X P (B) = pρ,π,τ Bρπ(1)ρτ(1) ...Bρπ(d)ρτ(d)

π,τ∈Sd,ρ=(ρ1,...,ρd)

Proof. Let f0(B) = 1 and for d ≥ 1

X fd(B) = sgn(τ)Bρ1ρτ(1) ...Bρdρτ(d) τ∈Sd,1≤ρ1<ρ2<...<ρd≤r

16 Then I is just the set of polynomials in f0(B), f1(B), ..., fr(B). This is because of the isomorphism between P∗ and I and the fact that

X fd(γ) = γρ1 ...γρd ρ1<...<ρd is the d-th elementary symmetric function of γ1, ..., γr. Note that fd(B) ∈ Id because

r X r−d det(B + tI) = fd(B)t d=0 Since 1 X f (B) = ( )2 sgn(τ)sgn(π)B ...B d d! ρπ(1)ρτ(1) ρπ(d)ρτ(d) π,τ∈Sd,ρ=(ρ1,...,ρd) we get the desired representation.

2.3.4. Definition: A polynomial P ∈ Id is said to be Griffiths positive if P is not identically zero and the coefficients pρ,π,τ in the above lemma have the form

X pρ,π,τ = λρ,jqρ,j,πq¯ρ,j,τ j into a finite sum, where λρ,j are nonnegative real numbers.

The set of Griffiths positive polynomials is denoted by Π and we have the grading by degree:Π = ⊕Πd. Note that Πd is a convex cone and from the definition it follows

t ¯ that Πd.Πe ⊂ Πd+e. Roughly speaking P ∈ Id is positive if upon writing B = A A, then

X 2 P (B) = |Qλ(A)| λ where Qλ(A) is a polynomial of degree d in A.

∼ We will also denote by Πd the subset in Pd under the isomorphism P = I

Now let us show some examples of Griffiths positive polynomials. 17 d 1. If L is a line bundle over X, then Πd consists of λc1(L) for λ > 0

2 2. The positive polynomials of degree 2 are the polynomials αc1 + βc2 with α ≥ 0,

α + β ≥ 0 with at least one of the inequalities strict.

i1 ir P 3. If cI = c1 ...cr is a mixed Chern class, then any nontrivial polynomial αI cI

with nonnegative coefficients is Griffiths positive.

Let us prove the third statement above. Note that it suffices only to show that the d-th Chern class is positive for each d. Indeed the Chern class

X cd = γρ1 ...γρd ρ1<...<ρd corresponds to the polynomial

1 X f (B) = ( )2 sgn(τ)sgn(π)B ...B d d! ρπ(1)ρτ(1) ρπ(d)ρτ(d) π,τ∈Sd,ρ=(ρ1,...,ρd)

1 2 P So we can take λρ,j = ( d! ) and qρ,j,π = sgn(π). Thus cq > 0 and hence αI cI > 0 for any nontrivial, nonnegative combination. This concludes the proof.

Let P ∈ Q[c1, ..., cr] be a weighted homogeneous polynomial of degree d, where

2d ci has degree i. Thus P defines a cohomology class P (E) in H (X). If Y is any d-dimensional irreducible, subvariety of X, one can represent the cohomology class

R of P (E) by a differential form and take the integral Y P (E) over Y . In 1969, Grif- R fiths asked which polynomials P have the property that Y P (E) is always positive whenever the bundle E is ample, in other words what are the numerically positive polynomials for ample bundles. He speculated that the Griffiths positive polynomi- als constructed above are all numerically positive. Of course the question has two 18 aspects. The first one is on the class level and we just described it. The second one is in the form level which we undertake in this thesis. We will describe the necessary definitions and the exact statement later.

The first attempt to solve the Griffiths question was made by Gieseker and Grif-

fiths himself who proved that all the Segre classes are positive. Later Bloch and

Gieseker proved that cn(E) is positive, although no proof is known even today that does not use the Hard Lefschetz Theorem.

In class level the Griffiths question was answered completely by Fulton and Lazars- feld who described the cone of positive polynomials completely. Their main theorem in [FL] asserts that P is a positive polynomial of Chern classes exactly when the coefficients aλ, in expressing P as linear combination of Schur polynomials, are non- negative and at least one of them is strictly positive. They also proved that that this cone in fact coincides with the cone of Griffiths positive polynomials. Here we will only sketch the proof of this last statement.

2.3.3. Theorem: Let P be a nonzero weighted homogeneous polynomial of degree n in Q[c1, ..., cr]. Assume that P has the following representation in terms of the basis of Schur polynomials Pλ X P = aλ(P )Pλ λ∈Λ(n,r) with aλ(P ) ∈ Q. Then P is Griffiths positive if and only if each of the Schur coeffi- cients is nonnegative.

Proof. (Sketch:) From [UT] Lemma 2.1 we know that aλ(P ) ≥ 0 if and only if for

19 every λ ∈ Λ(n, r) and every k-dimensional subvariety Y of the Grassmannian GqE of q-quotients of E with rank q universal quotient bundle Q, one has

Z P (c(Q)) ≥ 0 Y

But since P (c(Q)) can be represented by a nonnegative (k, k)-form it follows that the

Schur coefficients aλ(P ) are nonnegative for a Griffiths positive polynomial P .

Conversely, for λ ∈ Λ(n, r), let Fλ(B) ∈ In denote the invariant polynomial corresponding to the Schur polynomial Pλ under the isomorphism I ≡ Q[c1, ..., cr].

We will show that Fλ is Griffiths positive. To this end, we use the fact that the partitions of n are in natural bijection with the irreducible representations of the symmetric group Sn (cf.[CR]). Given λ ∈ Λ(n, r), let φλ denote the corresponding representation and χλ its character. Then one has the formula

1 X F (B) = χ (τ)B ...B λ n! λ ρ1ρτ(1) ρnρτ(n) τ∈Sn,ρ which implies that Fλ is Griffiths positive.

20 CHAPTER 3

POSITIVE DIFFERENTIAL FORMS, PUSH FORWARD

AND POSITIVE VECTOR BUNDLES

3.1 Positive Differential Forms

Let (Xn, g) be an n-dimensional Hermitian manifold and let ω be the metric (Kahler) form of g.

3.1.1. Definition: A smooth (p, p) form α is said to be simple if it has a represen- tation

p α = i α1 ∧ α¯1 ∧ α2 ∧ α¯2 ∧ ... ∧ αp ∧ α¯p

where each αj is a smooth (1, 0) form and αj’s are linearly independent. A (p, p) form

β is said to be positive, if β ∧ α = fωn with f > 0 for any simple form α of bidegree

(n − p, n − p). It is said to be semipositive if f ≥ 0. This definition is independent of the choice of the metric.

Note that any simple form is semi-positive. Also notice that when we restrict

21 ourselves to Cn, the space of (p, p)-forms with constant coefficients is spanned by simple forms. This follows from the relation

4 1 X dz ∧ dz¯ = is(dz + isdz ) ∧ (dz + isdz ) j k 4 j k j k s=1

3.1.1. Lemma: Let X and Y be complex manifolds and let f : X → Y be a holo- morphic submersion. Then for any simple form ω on Y , f ∗ω is a simple form on

X.

Proof. By our assumption, the holomorphic map f is surjective, and has maximal rank everywhere on X. So for any given set of linearly independent local (1, 0)-forms

∗ ∗ ∗ α1, α2, ..., αk on Y , the pull-backs f α1, f α2, ..., f αk are still linearly independent

(1, 0)-forms, so the pull back by f of a simple form on Y

∗ ∗ ∗ ∗ ∗ f (α1 ∧ α¯1 ∧ ... ∧ αk ∧ α¯k) = f α1 ∧ f α1 ∧ ... ∧ f αk ∧ f αk is again a simple form on X.

In order to understand positive forms better we will prove equivalent conditions for positivity given by the next proposition.

3.1.2. Proposition Let X be a complex manifold and let η be a (p, p)-form on X.

Then the following are equivalent.

1. η is positive

2. If W is any complex submanifold of dimension p then the restriction η|W is

equal to a volume form of W multiplied by a positive function 22 3. η is real and for any x ∈ X and any linearly independent (1, 0) type tangent

vectors V1, ..., Vp at x, it holds

p2 (−i) η(V1, ..., Vp, V 1, ..., V p) > 0

In particular for any (p, 0)-form ϕ, ip2 ϕ ∧ ϕ¯ ≥ 0. Locally if we take a frame

{ϕ1, ..., ϕn} of (1, 0)-forms, then any (p, p)-form ψ can be expressed as

p2 X ψ = i fIJ¯ϕI ∧ ϕ¯J¯

where ϕI = ϕi1 ∧ ... ∧ ϕip if I = (i1, ..., ip), and the sum runs through all multi indices

I ⊂ {1, 2, ..., n} of length p. Then ψ is real if and only if the matrix (fIJ¯) is Hermitian and ψ is positive if and only if (fIJ¯) is positive definite everywhere. From this one can see that the wedge product of positive (p, p) form and positive (q, q) form is a positive (p + q, p + q) form as long as p + q ≤ n.

A cohomology class γ ∈ H2k(X, C) is said to be positive if

Z γ > 0 Z for any k dimensional analytic subvariety Z ⊂ X. Of course if a closed (p, p) form is positive, then its cohomology class is also positive however the converse need not hold in general.

23 3.2 Push Forward of Differential Forms

Let M and N be oriented differentiable manifolds of respective dimensions m and n and let f : M → N be a proper submersion. That is f is surjective and has surjective differential everywhere, and the fibers are compact. Write r = m − n.

For any smooth (p + r)-form u on M, we claim that there exists a unique smooth p-form v on N such that the equality Z Z u ∧ f ∗ϕ = v ∧ ϕ M N holds for any smooth (n − p)-form ϕ on N with compact support.

The uniqueness part of the claim is clear: if v and v0 both satisfy the above

R 0 integral equation, then N (v − v ) ∧ ϕ = 0 for any (n − p)-form ϕ on N with compact support. By taking ϕ = ρ[∗(v − v0 )], where ρ is a smooth nonnegative function with compact support and and ∗ is the Hodge ∗- operator, we see that v − v0 must be zero everywhere, since Z Z Z 0 = (v − v0 ) ∧ ϕ = (v − v0 ) ∧ ρ[∗(v − v0 )] = ρ|v − v0 |2dV N N N where dV is a volume form.

To see the existence of such a form v, by the uniqueness we may restrict ourselves to a small neighborhood of a point in N. In other words we may assume that N admits a global coordinate system {x1, ..., xn} and f is just the projection map from

M = N × F onto its first factor, with F a compact orientable differentiable manifold of dimension r = m − n. Of course the orientation of F has to be chosen in such a way that the product of the orientations of F and N gives the orientation of M. 24 R Fix a top degree nowhere vanishing form ξ on F such that F ξ = 1. For any multi-index I = (i1, i2, ..., ip), where 1 ≤ i1 < i2 < ... < ip ≤ n, let us denote as usual

ˆ dxI = dxi1 ∧ dxi2 ∧ ... ∧ dxip and |I| = p. Also denote by I the complement index and

σI = ±1 such that dxI ∧ dxIˆ = σI dx1 ∧ dx2 ∧ ... ∧ dxn.

When we fix a coordinate system (x1, ..., xn) on N, clearly there exist (p+r −|I|)- forms ψI on M such that

X ∗ u = ψI ∧ f dxI |I|≥p P Given any smooth (n−p)-form ϕ on N with compact support, write ϕ = |I|=p hIˆdxIˆ.

We have

∗ X ∗ X ∗ ∗ u ∧ f ϕ = ( ψJ f dxJ ) ∧ ( f hIˆf dxIˆ) |J|≥p |I|=p

X ∗ ∗ ∗ = ( ψI f hIˆσI ) ∧ f (dx1 ∧ ... ∧ dxn) = η ∧ f (dx1 ∧ ... ∧ dxn) |I|=p Therefore, we get

Z Z Z Z ∗ ∗ u ∧ f ϕ = η ∧ f (dx1 ∧ ... ∧ dxn) = ( η)dx1 ∧ ... ∧ dxn M N×F N {x}×F Z X Z = ( ( ψI )hIˆσI )dx1 ∧ ... ∧ dxn N |I|=p {x}×F R P If we let vI (x) = {x}×F ψI for each |I| = p and v = vI dxI then

X X X v ∧ ϕ = vI dxI ∧ hJˆdxJˆ = vI hIˆσI dx1 ∧ ... ∧ dxn |I|=p |J|=p |I|=p so Z Z Z X ∗ v ∧ ϕ = hIˆσI dx1 ∧ ... ∧ dxn = u ∧ f ϕ N N |I|=p M Clearly v is a smooth form on N. This establishes the existence of v and the claim is proved. 25 We will call this form v the push forward of u via f and will denote it by f∗u.

The push forward satisfies the following properties

1. f∗(du) = d(f∗u)

∗ 2. (f∗u) ∧ τ = f∗(u ∧ f τ)

Proof. (of 2.) Let ϕ be any smooth form with compact support. Then

Z Z Z Z ∗ ∗ ∗ ∗ f∗(u ∧ f τ) ∧ ϕ = u ∧ f τ ∧ f ϕ = u ∧ f (τ ∧ ϕ) = f∗u ∧ τ ∧ ϕ N M M N

and hence the result follows.

3.3 Positive Vector Bundles

We say that a line bundle L over X is positive if it admits a Hermitian metric whose curvature is a positive (1, 1) form. Recall also that L is said to be ample if there exists

n ⊗k ∗ n a holomorphic embedding f : X → P such that L = f OP (1) for some positive

n integer k. Here OP (1) is the hyperplane line bundle of the projective space. Since

n the standard Fubini-Study metric for OP (1) has positive curvature and the metric naturally induce a metric on L through pull-back (restriction) and the k’th root, we know that L is ample implies L is positive. The converse is also true: any positive line bundle is ample. This is exactly the statement of Kodaira embedding theorem.

Now we define the positivity concepts for vector bundles.

3.3.1. Definition: Let E be a vector bundle over X. Then E is said to be ample if

the associated line bundle OP(E)(1) over P(E) is ample. 26 For sections u, v in E we define the (1, 1) form Θuv¯ by

r X k Θuv¯ = Θi hkjuiv¯j i,j,k=1 P P where u = uiei and v = viei under a frame {e1, ..., er} of E. Then Θuv¯ is a global (1, 1) form on X independent of the choise of e.

3.3.2. Definition: A Hermitian bundle is said to be positive in the sense of Griffiths, or Griffiths positive if iΘvv¯ is a positive (1, 1) form for any x ∈ X and any 0 6= v ∈ Ex.

Note that if we choose a local frame {e1, e2, ..., er} for E at x then we have

j Θ(E)ej e¯j = Θj. So the diagonal entries of the curvature matrix Θ of a positive vector bundle E are all positive differential forms. In particular the first Chern form

C1(E) which is the trace of E is a positive (1, 1)-form.

Before we illustrate the Griffiths positivity in examples let us examine the be- haviour of vector bundles under quotients and subbundles. Let E be a Hermitian vector bundle of rank r and F a subbundle of E equipped with the restriction metric.

Consider the exact sequence

0 → F → E → Q → 0

Choose a unitary frame {e1, ..., er} for E whose first s elements form a unitary frame for F . By abuse of notation we will also denote by ej the image of ej in Q for s + 1 ≤ j ≤ r. Then of course {es+1, ..., er} is a unitary frame for Q and under these frames the matrices of connection and curvature are given by     θ β Θ(F ) + β ∧ β∗ ∗  F    θE =   , Θ(E) =    ∗   ∗  −β θQ ∗ Θ(Q) + β ∧ β 27 where β = (βij) is a matrix of (1, 0) forms called the second fundamental form of F . P For any section u of F , if u = ujej we have

s X ¯ iΘ(F )uu¯ = iΘ(E)uu¯ − i ξj ∧ ξj j=1

Ps Ps ¯ where ξj = k=1 ukβki. Since i j=1 ξj ∧ ξj is a sum of simple positive forms it is nonnegative. Thus we get the following proposition.

3.3.1. Proposition: The curvature of a subbundle is less than or equal to the cur- vature of the ambient bundle and the curvature of the quotient bundle is great than or equal to the curvature of the ambient bundle.

As an application we get that any globally generated bundle is nonnegative. To be precise recall that a bundle is said to be globally generated (or generated by its

0 global sections) if for any x ∈ X and any v ∈ Ex there is an element σ of H (X,E) with σ(x) = v. In other words if there is a surjection

⊕m OX → E → 0

0 where m = dimC H (X,E). Since the curvature of the flat metric on the trivial bundle is zero the claim follows.

3.3.3. Curvature of The Tautological Line bundle Now we examine the rela-

tionship between the curvatures of E and the tautological line bundle OP(E)(−1) The

Hermitian metric h on E naturally induces a Hermitian metric on OP(E)(−1). We

28 ∗ claim that for any p ∈ X and any v 6= 0, v ∈ Ep the curvature of L = OP(E)(−1) at the point (p, [v]) ∈ P(E) is given by

1 Θ(L)| = Θ(E∗) − ω (p,[v]) |v|2 vv¯ FS

r−1 where ωFS is the Fubini-Study metric on the fiber, P(Ep) ≡ P .

Let z = (z1, ..., zn) be local holomorphic coordinates in a neighborhood U ⊂ X

∗ of p so that p = 0 and e = {e1, ..., er} be a holomorphic frame of E in U. we can

∗ P moreover assume that v = er(p). If a point s ∈ E |U is expressed as s = siei, we

∗ take (z1, ..., zn, s1, ..., sr) as local coordinate system in E . As a point set OP(E)(−1)

∗ coincides with E outside the zero section, so (z1, ..., zn, s1, ..., sr) can be used as local coordinates for OP(E)(−1) outside its zero section. From the choice of e, at the point v we have s = ... = s = 0, s = 1. Setting t = si we take (z, t) = 1 r−1 r i sr

(z1, ..., zn, t1, ..., tr−1) as local coordinates of P(E) near (p, [v]). Note that in this case

σ = t1e1 + ... + tr−1er−1 + er is a nowhere zero holomorphic section of OP(E)(−1) near the point (p, [v]). We have

r 2 X |σ(z, t)| = tit¯jhi¯j(z) i,j=1 where tr = 1 and hi¯j = h(ei, ej) is the matrix of metric under the frame e. Chang- ing the local coordinates (z1, ..., zn) and the local holomorphic frame {e1, ..., er} if necessary, we may assume that

3 hi¯j(z) = δij − Ri¯jαβ¯zαz¯β + O(|z| )

29 2 Therefore, (∂|σ| )|0 = 0 and

r−1 n ¯ 2 X ¯ X (∂∂|σ| )|0 = dti ∧ dti − Rrrα¯ β¯dzαdz¯β i=1 α,β=1

Thus r−1 ¯ 2 X X Θ(L)|0 = −(∂∂ log |σ| )|0 = Rγγαβ¯ dzα ∧ dz¯β − dtj ∧ dt¯j α,β j=1 1 Note that the first sumation is exactly |v|2 Θ(E)vv¯ and the second one the Fubini-Study metric on P(Ep).

3.3.2. Proposition A Griffiths positive vector bundle is ample.

Proof. From the above curvature computation we see that if E is Griffiths positive

then OP(E)(1) has positive curvature, hence by the Kodaira’s theorem it is ample.

But this by definition means that E is ample.

When Griffiths defined positive vector bundles he also conjectured that these bundles have positive Chern forms. He was able to show this for rank two vector bundles on surfaces. In fact the following weak version of the Griffiths conjecture was prove after succsessful works of Gieseker, Bloch-Gieseker and Fulton-Lazarsfeld.

3.3.3. Theorem: If X is a projective manifold and E an ample vector bundle over

X then the Schur polynomials are all positive. In particular all Chern classes are positive.

As mentioned above, if the Chern forms are positive, then the Chern classes are positive but the converse need not be true.

Now let us recall the special case proved by Griffiths himself. 30 3.3.4. Theorem (Griffiths): If X is a projective surface and E a Griffiths positive vector bundle of rank two over X, then the Chern form C2(E) is positive.

Proof. Let   Θ1 Θ2  1 1  Θ(E) =    1 2  Θ2 Θ2

2 We shall show that the second Chern form C2(E) = (1/2πi) det Θ is positive. Taking

¯ t a unitary frame {e1, e2} at a point we may assume that Θ+Θ = 0. Then Θ(E)e1e¯1 =

1 2 1 2 Θ1 > 0, Θ(E)e2e¯2 = Θ2 > 0. Since Θ1 > 0 and Θ2 > 0 we may choose a coframe

1 2 ω1, ω2 of X such that Θ1 = ω1 ∧ ω¯1 + ω2 ∧ ω¯2 and Θ2 = αω1 ∧ ω¯1 + βω2 ∧ ω¯2 where

α, β > 0. Thus   ω ∧ ω¯ + ω ∧ ω¯ η  1 1 2 2  Θ =     −η¯ αω1 ∧ ω¯1 + βω2 ∧ ω¯2 P where η = ij hijωi ∧ ω¯j. Letting dV = ω1 ∧ ω¯1 ∧ ω2 ∧ ω¯2

1 C (E) = ( )2 det Θ = (α + β − h h¯ − h h¯ + h h¯ + h h¯ )dV 2 2πi 11 22 22 11 12 12 21 21

¯ ¯ 2 2 By the Schwarz inequality h11h22 + h22h11 ≤ |h11| + |h22| , so

1 C (E) = ( )2 det Θ ≥ (α − |h |2 + β − |h |2 + |h |2 + |h |2)dV 2 2πi 11 22 12 21

2 2 So it will suffice to prove that α > |h11| , β > |h22| .

Let t ∈ C and consider the vector v = te1 + e2.

2 1 2 ¯ 1 2 Θvv¯ = |t| Θ1 + tΘ1 + tΘ2 + Θ2 =

2 2 (|t| + 2<(th11) + α)ω1 ∧ ω¯1 + (|t| + 2<(th22) + β)ω2 ∧ ω¯2 31 +(th12 + th21)ω1 ∧ ω¯2 + (th21 + th12)ω2 ∧ ω¯1

Since E is Griffiths positive, Θvv¯ is a positive (1, 1)-form for any t ∈ C. In particular

2 |t| + 2<(th11) + α > 0

¯ 2 2 Taking t = −h11, we get α > |h11| . Similarly, β > |h22| . This completes the proof of Griffiths’ theorem on the positivity of C2.

If one tries to employ the same method even to prove the positivity of the form

2 C1(E) − C2(E) one ends up with polynomials of degree four which makes it very complicated. This is the exact same reason why it is difficult to apply the method of

Griffiths to higher dimensional manifolds.

Finally we conclude this section by defining the strongest notion of positivity for vector bundles, the so called Nakano positivity. First recall that E is Griffiths positive if for any nonzero section v of E the (1, 1) form iΘvv¯ is positive, i.e. if τ is a nonzero vector field of type (1, 0) then Q(v ⊗ τ, v ⊗ τ) = iΘvv¯(τ, τ¯) > 0. The Nakano positivity is defined by declaring that the quadratic form Q is positive definite on

E ⊗ TX . That is a vector bundle E over X is said to be Nakano positive if it admits a Hermitian metric such that Q(χ, χ) > 0 for any nonvanishing section χ of E ⊗ TX .

Then any Nakano positive vector bundle is Griffiths positive since iΘvv¯(τ, τ¯) is just the restriction of Q to the set of sections of E ⊗ TX of the form v ⊗ τ.

32 CHAPTER 4

POSITIVITY OF THE SEGRE FORMS

4.1 Proof of the Main Theorem

In this section we will prove our main theorem. The idea is to show that the signed

k Segre forms (−1) Sk can be obtained via a push-forward of positive differential forms.

For this reason we take Φ to be the first Chern form of the associated line bundle

OP(E)(1) over P(E). If E is equipped with a metric h which makes it Griffiths positive, then as we have seen before the metric h induces a metric with positive curvature on

s OP(E)(1). Therefore if E is Griffiths positive, the forms Φ will be positive for any positive integer s. The proof of the main theorem consists of two steps. First we show

r+k−1 k that the push forward of the form Φ lies in the cohomology class of (−1) Sk.

Second we show that the push forward is invariant under the action AΘA−1 which assures that it must be a combination of the Chern forms. These two results together

k+r−1 k imply that the push forward of Φ is in fact (−1) Sk(E).

We first need a lemma which asserts that push forward of positive forms is still positive.

33 4.1.1. Lemma: Let X and Y be compact complex manifolds of respective dimensions m and n and let f : X → Y be a holomorphic fibration without singular fibers. If u is a positive (p + r, p + r) form on X, then f∗u is a positive (p, p) form on Y where as before r = m − n.

Proof. First let u be a top degree positive form. Denote the volume form of Y by dVY , and let ω be the Kahler form of a Hermitian metric on X. Since f is of maximum

r ∗ rank everywhere on X, it is not hard to see that ω ∧ f dVY is a positive (m, m)-form

r ∗ on X. So we can write u = g(ω ∧ f dVY ) for some positive function g on X. We have Z r f∗u = ( gω )dVY F

R r is a positive form since ( F gω ) is a positive function on Y .

Now let u be any positive form on X and let τ be any simple positive form on

∗ Y such that f∗u ∧ τ is a top degree form on Y . Then u ∧ f τ will be a positive top

∗ degree form on X and hence so is f∗(u ∧ f τ) = (f∗u) ∧ τ. But this by definition implies that f∗u is positive.

Now we prove the following proposition which will imply that the push forward

k+r−1 k of Φ and (−1) Sk(E) are cohomologous.

4.1.2. Theorem: Let E be a rank r Hermitian vector bundle over a projective man-

ifold X of dimension n and L = OP(E)(1) be the associated line bundle on P(E) an assume morover that r ≥ n. If Φ = C1(L), then Z Z r+k−1 ∗ k Φ ∧ π (η) = (−1) Sk(E) ∧ η P(E) X 34 for any 0 ≤ k ≤ n and any closed (n − k, n − k) form η on X, where π denotes the projection π : P(E) → X.

Proof. We prove this by induction on k. If k = 0 then using fiber integration the above equation reduces to

Z Z r−1 π∗(Φ ) ∧ η = η X X for any closed (n, n) form η on X. But this is true since

Z r−1 r−1 π∗(Φ ) = Φ = 1 π−1(x) and hence the result holds for k = 0. Now suppose that the result has been proved for all k0 < k. We have the Grothendieck relation in the form level

r ˆ r−1 ˆ r−2 ˆ Φ + C1Φ + C2Φ + ··· + Cr = dξ

ˆ ∗ ∗ where Cj = π (Cj(E )) and ξ is a globally defined (2r − 1) form. Multiplying with

Φk−1 ∧ π∗(η) for some closed (n − k, n − k)-form η on X and taking the integral over

P(E) we get

Z Z Z k+r−1 ∗ ˆ k+r−2 ∗ k−1 ˆ ∗ Φ ∧ π (η) + C1 ∧ Φ ∧ π (η) + ... + Φ ∧ Cr ∧ π (η) P(E) P(E) P(E) Z = Φk−1 ∧ dξ ∧ π∗(η) P(E) By the induction hypothesis

Z Z ˆ k+r−j+1 ∗ ∗ Cj ∧ Φ ∧ π (η) = Sk−j(E ) ∧ Cj ∧ η P(E) P(E)

35 and hence

Z Z Z Z k+r−1 ∗ ∗ k−1 ∗ Φ ∧π (η)+ C1 ∧Sk−1(E )∧η +···+ Ck ∧η = Φ ∧dξ ∧π (η) P(E) X X P(E)

But

∗ ∗ ∗ ∗ Sk(E ) + C1(E )Sk−1(E ) + ... + Ck(E ) = 0 and hence we get

Z Z Z k+r−1 ∗ k−1 ∗ ∗ Φ ∧ π (η) = Φ ∧ dξ ∧ π (η) + Sk(E ) ∧ η P(E) P(E) X

Therefore to complete the proof we must show that

Z Φk−1 ∧ dξ ∧ π∗(η) = 0 P(E)

But this is true by Stokes’ theorem on P(E) since Φk−1 ∧ π∗(η) is closed, as both Φ and η are closed, thus Φk−1 ∧ dξ ∧ π∗η is exact.

Note that if one is able to prove the above proposition for any smooth form η then by the uniqueness of the push forward this would imply that the signed Segre form is indeed the push forward. The above proposition implies only that they lie in

r+k−1 the same cohomology class. Indeed the push forward π∗(Φ ) satisfies

Z Z r+k−1 ∗ r+k−1 Φ ∧ π η = π∗(Φ ) ∧ η P(E) X for any smooth form η and from the above proposition we have

Z Z r+k−1 ∗ k Φ ∧ π η = (−1) Sk(E) ∧ η P(E) X 36 for any closed form η. Therefore for any closed form η we have Z r+k−1 k [π∗(Φ ) − (−1) Sk(E)] ∧ η = 0 X

r+k−1 k Let us prove that this implies that π∗(Φ ) − (−1) Sk(E) is an exact form. To simplify the notation consider the following more general situation. Let u be a closed

R form such that X u ∧ η = 0 for any exact form η. We will prove that u is exact.

Using the Hodge decomposition write u = α + dϕ + d∗ψ with α harmonic. Then since ∗α and α are both closed

0 = (u, α) = ||α||2 + (dϕ, α) + (d∗ψ, α) = ||α||2 hence α = 0.

Since du = 0 we get dd∗ψ = 0. Therefore ||d∗ψ||2 = (dd∗ψ, ψ) = 0 and thus d∗ψ = 0, which implies that u is exact.

k+r−1 4.1.3. Theorem: The push forward form π∗(Φ ) is identically equal to the signed

k Segre form (−1) Sk(E) on X for any 1 ≤ k ≤ n.

i Proof. Recall that Φ = C1(OP(E)(1)) = 2π Θ(OP(E)(1)) is a global (1, 1) form on P(E),

∗ and at p = (x, [v]) ∈ P(E), where v ∈ Ex, we have

i 1 Φ = (− Θ + ω ) 2π |v|2 vv¯ FS

∗ where Θ is the curvature of E , and ωFS is the Fubini-Study metric on the fiber

∗ ∼ r−1 ∗ P(Ex) = P , induced from the metric on Ex.

k Since ωFS = 0 for k ≥ r, we have by the binomial formula that, for any 1 ≤ k ≤ n,

k i X k + r − 1 1 Φk+r−1 = ( )k+r−1 (−1)j ( Θ )j ∧ ωk+r−1−j 2π j |v|2 vv¯ FS j=0 37 When we push forward this form, we are integrating over the fibers of π : P(E) → X, so only the last term in the right hand side survives:

  Z k+r−1 i k+r−1 k k + r − 1 1 k r−1 π∗Φ = ( ) (−1) ( 2 Θvv¯) ∧ ωFS 2π k ∗ |v| [v]∈P(Ex)

∗ Fix a point x ∈ X and let {e1, ..., er} be a local unitary frame of E near x. For v = Pr v e , write U = {[v] ∈ (E∗)|v 6= 0} and t = vi , 1 ≤ i ≤ r. Then i=1 i i P x r i vr

∼ r−1 ∗ U = C is an open dense set of the fiber P(Ex) and t = (t1, ..., tr−1) is its coordinate.

On this fiber, we have

r 1 X dt ∧ dt¯ ( Θ )k ∧ ωr−1 = ( Θ t t¯ )k |v|2 vv¯ FS ij i j (1 + |t|2)k+r i,j=1

2 2 2 where |t| = |t1| +...+|tr−1| , dt = dt1 ∧dt2 ∧· · ·∧dtr−1, and we wrote for convenience

k+r−1 tr = 1. Plug this into the expression for π∗Φ , we get that at x ∈ X,

k+r−1 π∗Φ =

  r r Z k + r − 1 i X X ti ··· ti t¯i ··· t¯j (−1)k ( )k Θ ··· Θ 1 k 1 k dt ∧ dt¯ i1j1 ikjk 2 k+r k 2π t∈ r−1 (1 + |t| ) i1,...,ik=1 j1,...,jk=1 C r r k + r − 1 X X = (−1)k B Θ ··· Θ k I,J i1j1 ikjk i1,...,ik=1 j1,...,jk=1 where i Z t ··· t t¯ ··· t¯ k i1 ik i1 jk ¯ BI,J = ( ) 2 k+r dt ∧ dt 2π r−1 (1 + |t| ) t∈C

k and I = (i1, ..., ik), J = (j1, ..., jk) both belong to the set {1, 2, ··· , r} .

For any permutation σ ∈ Sym(k), write σ(I) = (iσ(1), ..., iσ(k)). Then it is easy to see that BI,J = 0 unless J = σ(I) for some σ ∈ Sym(k), and any BI,σ(I) ∈ Q.

k+r−1 i Therefore π∗Φ = P ( 2π Θ) becomes a homogeneous polynomial of degree k in the 38 i ∗ entries of the curvature 2π Θ of E , with rational coefficients. On the other hand the

k+r−1 push forward π∗Φ is a global (k, k)-form on X, independent of the choice of local frames of E∗. That is, the polynomial P is invariant under the change A 7→ AΘA−1 for any A ∈ GL(r, C). Therefore, it must be a polynomial of the Chern forms,

k+r−1 π∗Φ = f1(C1,C2, ..., Cr), where f1 ∈ Pk is a weighted homogeneous polynomial

k of degree k with rational coefficients. Of course (−1) Sk = f2(C1,C2, ..., Cr) is also a weighted homogeneous polynomial of the Chern forms of E and we proved in the previous theorem that their difference, f = f1 − f2 is a closed global (k, k)-form on

X which represents the trivial cohomology class.

Note that [f] = 0 regardless of what bundle E we begin with. In particular if we

⊗x1 ⊗xr choose E = H ⊕ · · · ⊕ H , where H is an ample line bundle on X and x1, ..., xr

2k are positive integers, we get [f] = f(C1, ..., Cr) = 0 in H (X). But

k f(C1(E), ..., Cr(E)) = h(x1, ..., xr)c1(H)

where h ∈ Q[x1, ..., xr] is homogeneous of degree k, so we get h(x1, ..., xr) = 0 for any positive integers x1, ..., xr. By the homogenity of h we have

h(x1, ..., xr) = 0

+ for all x1, ..., xr ∈ Q . So the polynomial h must be trivial, proving that f1 = f2.

This establishes the fact that

k+r−1 k π∗Φ = (−1) Sk(E) for any 1 ≤ k ≤ n. 39 As a direct consequence of 4.1.2 and 4.1.3, we obtain the positivity of the signed

Segre forms for Griffiths positive vector bundles.

4.1.4. Theorem: Let X be a projective manifold and let E be a Griffiths positive holomorphic vector bundle on X. Then the Segre forms are positive.

Proof. Since E is Griffiths positive, the hyperplane line bundle OP(E)(1) is positive and hence Φk+r−1 is a positive (k + r − 1, k + r − 1) form. Since π : P(E) →

k+r−1 X is a holomorphic submersion with compact fibers, it follows that π∗(Φ ) =

k (−1) Sk(E) is a positive form.

4.2 Semipositivity

In their remarkable paper [DPS], Demailly, Peternell and Schneider extended the result of Fulton and Lazarsfeld to a semi-positive neffness case. To be precise they showed that under the neffness definition of semi-positivity, the positive polynomials for nef bundles will be always nonnegative. In this section we will remark that a similar conclusion follows for the Chern forms if we allow our bundle to be only semi-positive in the Griffiths sense.

A line bundles L over a projective manifold X is said to be nef if L.C ≥ 0 for any effective curve C in X. This definition is no longer valid for compact Kahler manifold

X because there may not be any curve in X. However we have the following lemma which characterizes neffness in terms of a curvature condition.

4.2.1. Lemma: Let X be a projective manifold and L a holomorphic line bundle 40 over X. Then L is nef if and only if for any ε > 0, there exists a hermitian metric h on L such that i Θ (L) ≥ −εω 2π h where ω is a fixed Kahler metric on X.

Proof. If the line bundle L satisfies the above curvature condition then L is nef since

Z Z i Z L · C = c1(L) = Θh(L) ≥ −ε ω ≥ 0 C C 2π C

Conversely assume that L is nef. Then for any ample line bundle A, the line bundle

A + kL will be ample from the Nakai-Moishezon criterion which we state below.

Choose a metric hA on A with positive curvature and set ω = iΘ(A). Since kL + A is ample it has a metric hkL+A of positive curvature. Then the metric hL =

1/k (hkL+A ⊗ hA−1 ) has curvature

1 −1 iΘ(L) = (iΘ(kL + A) − iΘ(A)) ≥ iΘ(A) k k and in this way the negative part can be made smaller than εω by taking k large enough.

4.2.2. Theorem: (Nakai-Moishezon Criterion) A line bundle on a projective manifold X is ample if and only if

Z k c1(L) > 0 Z for any k-dimensional subvariety Z in X.

41 Now we can define nef vector bundles E on a compact Kahler manifold X. It

will be defined in terms of the nefness of the line bundle OP(E)(1). Note that in general the manifold P(E) is not projective so we cannot take the original definition of nefness, but rather we should take the curvature condition as a definition when the base manifold is not projective.

4.2.1. Definition: A vector bundle E on X is said to be nef if the line bundle

OP(E)(1) over P(E) is nef.

Now it is easy to see that if E is a Griffiths semi-positive vector bundle then

E is nef. Indeed E is Griffiths semi-positive means that the (1, 1) form Θvv¯ =

Pr k i,j,k=1 Θi hkjviv¯j is semi-psitive, where v is a non-zero section of E. But we have

1 seen that Θ(OE(1))|(p,[v]) = |v|2 Θ(E)vv¯ + ωFS from which we see in fact that OP(E)(1) has a metric with non-negative curvature so OP(E)(1) is trivially nef. Moreover in this case the first Chern form of OP(E)(1) is a semi-positive (1, 1) form so to complete the picture we will show that the push forward of semi-positive forms are still semi- positive. Indeed let X be a compact complex manifold, α a semi-positive (p, p)-form and ϕ strictly positive (p, p)-form. Let also f : X → Y be a holomorphic fibration without singular fibers to some other compact complex manifold Y . The form α + εϕ will be strictly positive for ε > 0 and hence the push forward f∗(α+εϕ) will be strictly positive. Since the push forward is linear we can let ε go to zero hence deducing the semi-positivity of f∗(α).

42 4.3 Some Remarks For Further Research

To prove the positivity of the signed Segre forms it was sufficient to consider only the one dimensional quotiens of the vector bundle E. To obtain more general results and maybe to prove the Griffiths’ conjecture in full generality we believe that it is necessary to consider the Grassmannians. Recall that if V is a vector space, then the Grassmanian of rank q quotients of V is the set of all q codimensional subspaces of V . It is denoted by GqV . If E is a vector bundle over a manifold X, then the corresponding Grassmannian manifold is denoted by GqE. Over this manifold GqE we have the following universal exact sequence.

0 → S → π∗E → Q → 0 where π is the projection π : Gq(E) → X. If E has a metric which makes it Griffiths positive, then this metric induces a metric on Q which makes it only Griffiths semi- positive, but it can be shown that det Q is always positive, therefore Q ⊗ det Q will be positive. Thus for example considering the case q = 2 one can try to push forward

7 the positive forms C2(Q ⊗ det Q) and C1 (Q).

43 BIBLIOGRAPHY

[BG] Bloch, S., Gieseker, D.,The positivity of the Chern classes of an ample vector bundle, Invent. Math. 12 (1971), 112–117 [CR] Curtis, Charles W.; Reiner, Irving Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI Interscience Publish- ers, a division of John Wiley and Sons, New York-London 1962 xiv+685 pp [D] Demailly,J.P., Multiplier ideal sheaves and analytic methods in algebraic geome- try, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), 1–148, ICTP Lect. Notes, 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001 [DPS] Demailly,J.P., Peternell, T., Schneider, M., Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295–345. [FL] Fulton, W., Lazarsfeld, R., Positive polynomials for ample vector bundles, Ann. of Math. (2) 118 (1983), no. 1, 35–60 [G] Griffiths, P., Hermitian differential geometry, Chern classes, and positive vector bundles, 1969 Global Analysis (Papers in Honor of K. Kodaira) pp. 185–251 Univ. Tokyo Press, Tokyo [Gi] Gieseker, David p-ample bundles and their Chern classes, Nagoya Math. J. 43 (1971), 91–116 [U] Umemura, Hiroshi Some results in the theory of vector bundles, Nagoya Math. J. 52 (1973), 97–128 [UT] Usui, Sampei; Tango, Hiroshi On numerical positivity of ample vector bundles with additional condition, J. Math. Kyoto Univ. 17 (1977), no. 1, 151–164 [YZ] Yau, S.-T.; Zheng, F. On a borderline class of non-positively curved compact Kahler manifolds, Math. Z. 212 (1993), no. 4, 587–599. [Z] Zheng, F., Complex differential geometry, AMS/IP Studies in Advanced Mathe- matics, 18. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2000. xii+264 pp. ISBN: 0-8218-2163-6

44