Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Other similar general results can be derived that

link the behavior of the structure functions with the

properties of the stable distributions corresponding

to the type of flow singularities expected in the limit

of infinite Reynolds number.

The common feature of the two cases just

described is the presence of strong structures that

live for long tim es because viscosity stabilizes

them. They are therefore more common than

what could be expected on purely statistical

grounds. They are responsible for the tails of the

probability distributions of the v elocity derivatives,

but they are not the only intermittent features of

turbulent flows. The increase of the flatness in

Figure 3 below r 50 is clearly connected with

the presence of the coherent vortices, but even for

larger separations there is a smooth evolution of

(4) that suggests that the formation of intense

structures is a gradual process that takes place

across the inertial range. Much less is known

about those hypothetical inertial structures than

about the viscous ones.

We can now recast the problem of intermittency

in Navier–Stokes turbulence into geometric terms.

The defining empirical observation for that system is

that the energy dissipation given by eqn [4] does not

vanish even in the infinite Reynolds number limit in

which  !0. This means that the flow has to

become singular as

jrujL



 Re

1=2

. The strict

similarity approximation assumes that those singu-

larities are uniformly distributed across the flow, but

the ex perimental evidence just discussed shows that

this is not true. The singularities are distributed

inhomogeneously, and the inhomogeneity develops

across the inertial cascade. The problem of inter-

mittency is to characterize the geometry of the

support of the flow singularities in the limit of

infinite Reynolds number.

In the absence of detailed physical mechanisms

for the dynamics of the inertial range, most

intermittency models are based on plausible pro-

cesses compatible with the invariances of the

inviscid Euler equations. The precise power law

given in eqn [9] for the structure functions depends

on the strict similarity hypothesis [3], but the f act

that it is a power law only depends on the scaling

invariances of the equations of motion. The

energies and sizes of the eddies in the inertial

range are too small for the integral scales of the

flow to be relevant, and too large for the viscosity

to be important. They therefore have no intrinsic

velocity or length scales. Under those conditions,

any f unction of the velocity which depends on

a length has to be a power. Consider a quantity

with dimensions of velocity, such as u( r) = S(n)

1=n

which is a function of a distance such as r.

On dimensional grounds we should be able to

write it as

uðrÞ¼UFð Þ½14

where  = r=L, and L and U(L) are arbitrary length

and velocity scales. The value of u(r) should not

depend on the choice of units, and we can

differentiate eqn [14] with respect to L to give

u ¼ðdU=dLÞF ðÞUL

1

ðdF= dÞ¼0 ½15

which can only be satisfied if

d

¼ F )F  



½16

and  = L(dU=dL)=U is con stant. This suggests

generalizing eqn [9] to

SðnÞr

ðnÞ

½17

where the exponents are empirically adjusted. Only

(3) = 1 can be derived directly from the Navier–

Stokes equations. Equation [17] implies that (n)

satisfies a power law with exponent (n) n(2)=2.

In Figure 3, for example, the flatness follows a

reasonably good power law outside the viscous

range, consistent with (4)  2(2) 0.12. The

anomalous behavior near the viscous limit, and

similar limitations at the largest scales, mean that

only very high Reynolds number flows can be used

to measure the scaling exponents, and that the range

over which they a re measured is never very large.

Moreover, the integrand of the higher-order struc-

ture functions peaks at the extre me tails of the

probability distributions of the velocity differences,

which implies that very long experimental samples

have to be used to accumulate enough statistics to

measure the high-order exponents. For these and for

other reasons, the scaling exponents above n & 810

are poorly known. This is unfortunate because we

will see later that some of the most interesting

intermittency properties of the velocity field, such as

the nature of the flow singularities in the infinite

Reynolds number limit, depend on the behavior of

the (n) for large n.

Experimental values for the scaling exponents are

given in Table 1. They are generally smaller than the

ones predicted by the strict similarity approxima-

tion, implying that the moments of the velocity

differences decrease with the separation more slowly

than they would if they were self-similar, and

suggesting that new stronger structures become

important as the scale decreases.

Note that we have included in the table values for

odd-order powers. Up to now we have not specified

Intermittency in Turbulence 147

which velocity component is being analyz ed, but

most experiments refer to the one in the direction

of the separation. That is the easiest case to

measure, specially if time is used as a surrogate

for distance, and those PDFs are not symmetric

even in isotropic turbulence. Negative increments

are more common than positive ones because of the

extra energy required to stretch a vortex, and the

effect is clearly visible in the distributions in

Figure 2. Those longitudinal odd-order structure

functions do not vanish, and their scaling expo-

nents are the ones given in the table. The transverse

structure functions are those in which the velocity

component is normal to the separation, and their

odd-order moments vanish by symmetry in iso-

tropic turbulence. There has been a lot of discus-

sion about whether the longitudinal scaling

exponents of even orders differ from the transverse

ones. Early results suggested that the latter are

lower than the former, undermining the case for

intermittency theories based on si milarity argu-

ments, and suggesting that a more mechanistic

approach was needed. The present consensus

seems to be that both sets of exponents are

equal, but that there are residual effects of low

Reynolds numbers and of flow anisotropy that are

difficult t o avoid experimentally. The question is

still open.

Multiplicative Models

The most successful phenom enological models for

the geometry of intermittency are based on the

concept o f a multiplicative cascade. Consider some

flow property v, such as the locally averaged

energy transfer rate by eddies of size r

,which

cascades into smaller eddies of size r

kþ1

which is

some fraction of r

. Denote by p

)the

probability distribution of the value of v at the

step k of the cascade.

Assume that the cascade is Markovian in the sense

that the probability distribution of v

depends only

on its value in the previous step,

kþ1

ðv

kþ1

Þ¼

ðv

kþ1

; k Þp

ðv

Þdv

½18

This is in contrast to some more complicated

functional dependence, such as on the values of v

in some extended spatial neighborhood, or on

several previous cascade stages. This assumption

intuitively implies that v

kþ1

evolves faster, or on a

smaller scale, than v

, and that it is in some kind of

equilibrium with its precursor. If the cascade is

deterministic in that sense, v

can be represented as

a product

¼ q

k1

...q

½19

in which the factors q

= v

k1

are statistically

independent of each other.

If the underlying process is invariant to scaling

transformations, the transition probability density

function has to have the form

ðv

kþ1

Þ¼v

1

wðq

kþ1

; k Þ½20

The multiplicative model works most naturally

for positive variables, and we will assume that

to be the case in the following, but most results

can be generalized to arbitrary distributions. We

will also assume for simplicity that all the

cascade steps are equivalent, so that the distribu-

tion w(q) of the multiplicative facto rs is indepen-

dent o f k, and depends only on our choice for

kþ1

Local deterministic self-similar cascades lead

naturally to intermittent distributions, in the sense

that the high-order flatness factors for v

become

arbitrarily large as k increases. It follows from eqns

[18]–[20] that the nth order moment for p

can be

written as

ðnÞ¼



ðÞd ¼S

ðnÞS

ðnÞ

½21

where S

(n) is the nth order moment of the

multiplicative factor q,andn is any real number

for which the integral exists. If we define flatness

factors as in eqn [7], we can rewrite eqn [21] as



ðnÞ¼

ðnÞ

ðnÞ

½22

It follows from Chebichev’s inequality that

SðnÞSðn  2ÞSð 2ÞSðn  4ÞSð2Þ

... ½23

Table 1 Longitudinal scaling exponents

Order Experimental Strict similarity

20:.70  .:01 0.667

3 1.00 1

41:.30  .:03 1.333

51:.56  .:04 1.667

61:.79  .:03 2.000

71:.99  .:10 2.333

82:.22  .:05 2.667

The values on the second column are averages from different

experiments, and the standard deviations reflect scatter among

experiments. The third column is the value from the strict

similarity equation [9].

148 Intermittency in Turbulence

from where

1  ð4Þ ð6Þ... ½24

which is true for any distribution of positive

numbers. Equality only holds for trivial distributions

concentrated on a single value. The product in eqn

[22] therefore increases without bound with the

number of cascade steps, and the flatness factors

diverge.

It is tempting to substitute k in [21] by a

continuous variable, in which c ase the PDFs form

a continuous semigroup generated b y infinitesimal

scaling steps. This leads to beautiful t heoretical

developments, but it is not necessarily a good idea

from the physical point of view. For example, while

it might be reasonable to a ssume that the properties

of an eddy of size r depend only on those of the

eddy of size 2r from which it derives, the same

argument is weaker when applied to eddies of

almost equal sizes. We will r estrict ourselves here

to the discrete case.

Limiting Distributions

The multiplicative process just described can be

summarized as a family of distributions p

) such

that the probability density for the product of two

variables is

pðv

Þ¼p

þk

ðv

þk

Þ½25

and it is natural to ask whether there is a limiting

distribution for large k. We know that, in the case of

sums, rather than products, such distributions tend

to be Gaussian under fairly general conditions, and

the first attempt to analyze [25] was to reduce it to a

sum by defining

z ¼ k

1

logðv

Þ½26

The argument was that z would tend to a Gaussian

distribution, and that the limiting distribution for v

would be lognormal. This was soon shown to be

incorrect. The central part of the distribution

approaches lognormality, but the tails do not,

because the central limit theorem says nothing

about their behavior. The family of lognormal

distributions is a fixed point of eqn [25], but it is

unstable, and it is only attained if the individual

generating distributions are themselves lognormal.

The lognormal distribution has moments

ðnÞ¼expðan þ bn

Þ½27

which are conserved under [21], so that the product

of lognormally distributed variables stays lognormal.

The moments in eqn [27] are generated by the

recursive relation

ðnÞ¼

ðn þ 3ÞS

ðn þ 1Þ

ðnÞS

ðn þ 2Þ

¼ 1 ½28

with suitable conditions for n < 2. Under [21],

(n) = Q

(n), and it is clear that only when all

the Q

(n) are exactly equal to 1 do they continue to

be so under multiplication. Otherwise, any Q

initially larger than 1 tends to infinity after enough

cascade steps, while any one initially smaller than 1

tends to 0. Only an exactly lognormal distribution

of the generating factors results in a lognormal

limiting distribution, and even small errors lead to

very differen t patterns of moments. This contrasts

with the situation for sums of random variables, in

which the Gaussian distribution is not only a fixed

point, but also has a very large basin of attraction.

Multifractals

The problem with using the transformation [26] to

find the limiting distribution of a multiplicative

process is not so much the technique of analyzing

the statistics of products in terms of those of sums,

but the inappropriate use of the central limit

theorem. It can be bypassed by using instead the

theory of large deviations of sums of random

variables. The key result is obtained by expanding

the characteristic function of p

when k  1, and

states that

ðv

Þ



2k



1=2

k½ðzÞz

½29

where z is defined as in [26] and , which plays the

role of an entropy, is a smooth function of z. Primes

stand for derivatives with respect to z. Let us define

as the point where



 

ðz

Þ¼n ½30

which corresponds to the location of the maximum

of  þ nz. The entropy  can be computed from the

moments of the transition probability density. Using

Laplace’s method to expand the nth moment of p

we obtain

ðnÞ¼

1

kðnþ1Þz

ðv

Þdz







1=2

kð

þnz

½31

from where, using [21],



 log S

ðnÞ¼ðz

Þþnz

½32

The essence of Laplace’s approximation is that, for

k  1, most of the contribution to the integral in

eqn [31] comes from the neighborhood of z

, so that

Intermittency in Turbulence 149

it makes sense to consider each such neighborhood

as a separate ‘‘component’’ of the cascade.

The geometric interpretation of this classification

into components as a multifractal was developed in

the context of three-dimensional homogeneous

turbulence. We have up to now assumed very little

about the nature of each cascade step, but it is

natural in turbulence to interpret it as the process in

which eddies decay to a smaller geometric scale. The

argument works for any variable for which scale

similarity can be invoked, but we have seen that

most experiments are done for the magnitude of the

velocity increments across a distance r.Ifwe

assume for simplicity that r

kþ1

= e, so that

= exp(k), eqns [26] and [29] can be written as

¼ðr

z

; p

ðz

Þðr



½33

The multifractal interpretation is that the ‘‘compo-

nent’’ indexed by n, whose velocity increments are

‘‘singular’’ in terms of r with exponent z

, lies on a

fractal whose volume is proportional to its prob-

ability, and which therefore has a dimension

D(z

) = 3 þ 

Note that eqn [32] implies that the scaling

exponents in eqn [17] can now be expres sed as

ðnÞ¼log S

ðnÞ¼

½34

There was an enumeration there of several things

which are equivalent: the expon ents, the spectra, the

distribution, and the limiting distribution p

(v)–

univocally determine each other. Note however that

different quantiti es have different scaling exponents.

For example, it follows from eqn [6] that, if the

scaling exponents for the local dissipation are



(n), the exponents for u would be



u

(n) = n=3 þ 

(n=3).

Some properties can be easi ly derived from the

previous discussion. If we assume, for example, that

the multiplicative factor q is bounded above by q

which is reasonable for many physical systems, eqn

[26] implies that z

 log q

. In fact, if the transition

probability behaves near q

as w(q)(q

 q)



, the

scaling exponents tend to



¼ n log q

ð þ 1Þlog n þ Oð1Þ½35

for n  1. In the case in which w(q) has a

concentrated component at q = q

, the log n is

missing in eqn [35]. In all cases, the singularity

exponent of the set associated with n !1 is

= log q

, because the very high moments are

dominated by the largest possible multiplier. In the

case of a concentrated distribution the dimension of

this set approaches a finite limit, but otherwise

DðnÞð þ 1Þlog n ½36

which becomes infini tely negative. This should not

be considered a flaw. The set of events which only

happen at isolated points and at isolated instants has

dimension D = 1 in three-dimensional space, and

those which only happen at isolated instants, and

only under certain circumstances, have still lower

negative dimensions. Sets with very negative dimen-

sions are however extremely sparse, and are difficult

to characterize experimentally.

The multifractal spectrum of the velocity differ-

ences in three-dimensional Navier–Stokes turbulence

has been measured for several flows in terms of the

scaling exponents, and appears to be universal. The

probability distribution w(q) of the multipliers has

also been measured directly, and agrees well with

the values implied by the exponents. It is also

approximately independent of r, although not

completely, perhaps due to the same experimental

problems of anisotropy and limited Reynolds

number which plague the measurement of the

scaling exponents. There has been extensive theore-

tical work on the consequences of imposing various

physical constraints on the multipliers, specially the

conservation requirement that the average value of

the diss ipation has to be conserved across each

cascade step. Several simple models have been

proposed for the transition distribution which

approximate the experimental exponents well, but

the relation lacks specificity. Models that are very

different give very similar results, and it is impos-

sible to choose among them using the available data.

Multiplicative cascades and the resulti ng inter-

mittency are not limited to Navier–Stokes turbu-

lence. The equations of motion have only entered

the discussion in this section through the assumption

of scaling invariance. Multifractal models have in

fact been proposed for many chaotic systems, from

social sciences to economics, although the geometric

interpretation is hard to justify in most of them. It is

also important to realize that the fact that a given

process can in principle be described as a cascade

does not necessari ly mean that such a description is

a good one. Neither does a cascade imply a

multiplicative process. For each particular case, we

need to provide a dynamical mechanism that

implements bot h the cascade and the trans ition

multipliers. In three-dimensional Navier–Stokes

turbulence, the basic transport of energy to smaller

scales and to higher gradients is vortex stretching.

The differential strengthening and weakening of the

vorticity under axial stretching and compression

also provide a natural way of introducing the self-

similar transition probabilities of the local dissipation.

Examples of nonintermittent cascades abound.

We have already mentioned that the vorticity in

150 Intermittency in Turbulence

decaying two-dimensional turbulence gets concen-

trated into stable vortex cores which eventually

block the decay. The resulting enstrophy distribu-

tion is highly intermittent, but it is not well

described by a multifractal. Conversely, forced

two-dimensional turbulence is dominated by an

inverse energy cascade to larger scales, which is not

intermittent.

In addition, the intermittency of some systems is

not a small-scale effect. Turbulent mixing of a

passive scalar, which is the key process in

turbulent heat transfer a nd in the atmospheric

dispersion of pollutants, is an extremely intermit-

tent phenomenon. The gradients of the scalar tend

to be very localized, but they concentrate in sheets,

narrow in thickness but otherwise extended. Some

progress has recently been made on a simplified

model due to Kraichnan for this problem, which is

the linear stirring of a passive scalar by a random

noise with delta correlation. I ts statistics have been

computed analytically, but the constraints of

linearity and of uncorrelated forcing are strong,

and the same methods do not appear to be

extensible to mixing by real turbulence (see

Lagrangian Dispersion (Passive Scalar)). Another

problem in w hich intermittency is confined to

large-scale surfaces is the motion of a three-

dimensional pressureless gas, which has been used

as a model for hypersonic turbulence and for the

large-scale evolution of dark matter in the early

universe.

In summary, intermittency is a fascinating property

of many random systems, including three-dimensional

Navier–Stokes turbulence, which interferes, sometimes

strongly, with their description by simple cascade

models. Significant advances have been made in its

quantitative kinematic analysis. In some cases we also

have a qualitative understanding of its roots. But in very

few cases do we understand it well enough to make

quantitative predictions.

See also: Ergodic Theory; Incompressible Euler

Equations: Mathematical Theory; Lagrangian Dispersion

(Passive Scalar); Turbulence Theories; Vortex Dynamics;

Wavelets: Applications.

Further Reading

Feller W (1971) An Introduction to Probability Theory and Its

Applications, 2nd edn., vol. 2, pp. 169–172 and 574–581.

New York: Wiley.

Frisch U (1995) Turbulence. The Legacy of A.N. Kolmogorov,

pp. 159–192. Cambridge: Cambridge University Press.

Jime´nez J (1998) Small scale intermittency in turbulence.

European Journal of Mechanics B: Fluids 17: 405–419.

Jime´nez J (2001) Intermittency and cascades. Journal of Fluid

Mechanics 409: 99–120.

Lanford OE (1973) Entropy and equilibrium states in classical

mechanics. In: Lenard A (ed.) Statistical Mechanics and

Mathematical Problems, Lecture Notes in Physics, vol. 20,

pp. 1–113. Berlin: Springer.

Nelkin M (1994) Universality and scaling in fully developed

turbulence. Advances in Physics 43: 143–181.

Paladin G and Vulpiani A (1987) Anomalous scaling laws in

multifractal objects. Physics Reports 156: 147–225.

Pope SB (2000) Turbulent Flows, pp. 167–173 and 254–263.

Cambridge: Cambridge University Press.

Schroeder M (1991) Fractals, Chaos, Power Laws, Sect. 9.

New York: W.H. Freeman.

Sreenivasan KR and Stolovitzky G (1995) Turbulent cascades.

Journal of Statistical Physics 78: 311–333.

Vassilicos JC (ed.) (2001) Intermittency in Turbulent Flows.

Cambridge: Cambridge University Press.

Intersection Theory

A Kresch, University of Warwick, Coventry, UK

Introduction

Intersection theory is the theory that governs the

rigorous definition of intersections of cycles. This

can take place in a variety of mathematical context s,

for instance, the intersections of two cycles on an

oriented manifold in algebraic topology, of two

currents on a differentiable manifold in differential

geometry, or of two subvarieties on a nonsingular

algebraic variety in algebraic geometry.

In algebraic geometry the theory is especially well

developed (Fulton 1998). A cycle on an algebraic

variety (or scheme) is a formal linear combination of

irreducible closed subvarieties. These are subject to

an equivalence relation called rational equivalence.

For every rational function on every subvariety, its

zero set is deemed rationally equivalent to its poles

(with appropriate multiplicities).

As an example, in the complex projective plane

, any two lines are rationally equivalent since

the ratio of two linear forms will vanish on one line

and have a pole along the other. Similarly, a curve

of degree d is rationally equivalent to d lines. Any

two points in CP

can be joined by a line (a copy of

Intersection Theory 151

), and a rational function on CP

can be chosen

to vanish at one point and have a pole at the other.

The groups of cycles modulo rational equivalence,

known as Chow groups, are

ðCP

ÞﬃZ; generated by the fundamental

class ½CP



ðCP

ÞﬃZ; generated by the class of a line

ðCP

ÞﬃZ; generated by the class of a point

Two distinct lines ‘

and ‘

meeting at a point p have

this point as their intersection-theoretic product:

½‘

½‘

¼½p½1

Intersection theory must also provide a self-intersection

[‘

]  [‘

]. Because ‘

and ‘

are rationally equivalent,

this must also be the class of a point, but symmetry

precludes the choice of a distinguished point on ‘

Instead, [‘

]  [‘

] is declared to be the rational

equivalence class of a point on ‘

, an element of

(‘

) rather than a specific cycle. This example

illustrates that intersections cannot generally be defined

on the level of cycles.

Algebraic Intersection Products

Refined Intersections

For a general nonsingular variety X, say of dimen-

sion m,ifU and V are subvarieties of X of respective

dimensions c and d, then there is a refined

intersection product

½U½V2CH

cþdm

ðU \ VÞ½2

The traditional definition of the intersection

product is based on two ideas. First, given two

cycles that intersect properly, which by definition

means that no component of their intersection has

codimension less than the sum of the codimensions

of the given cycles, the intersection product should

be a formal sum of these compone nts, each with a

multiplicity that correctly reflects the geometry of

the intersection. Second, given two arbitrary cycles,

it should be possible to replace one of them by a

rationally equivalent cycle which intersects the other

properly.

While these ideas are simple, it took several

decades for them to be carried out successfully.

The case of curves on a surface meeting at a point

was understood in the nineteenth century. General-

izing the classically understood canonical divisor

class on a variety, work in the 1930s by Sever i,

Todd, and others showed that there are groups of

equivalence classes of cycles in which canonical

invariants of higher degrees can be defined (in

modern language, higher Chern classes of the

tangent bundle). Weil’s foundations for algebraic

geometry of the 1940s included a study of intersec-

tions of cycles. It was not until the 1950s that the

notion of Chow groups was formalized and inter-

section theory was properly developed in this

context. Chevalley, Chow, Samuel, Severi, and

others contributed essential components of the

theory. In an interesting parallel devel opment, an

intersection theory based on intersection multipli-

cities in algebraic topology was put forth by

Alexander and Lefschetz in the 1920s, a decade

before the introduction of the cup product in

cohomology.

Deformation to the Normal Cone

In the 1970s, Fulton and MacPherson established a

construction of the intersection product in algebraic

intersection theory that does not require moving

cycles into general position. To accomplish this, they

used an elegant geometric construction known as

deformation to the normal cone.

Let i : X ! Y be an embedding of codimension d

of nonsingular varieties. Let V be a subvariety of Y

of dimension k whose intersection with X is of

interest. We may view X as the zero set of a section s

of some algebraic vector bundle E on Y.By

ðy;Þ7!ð

1

sðyÞ;Þ

we have a map of the product of Y with the

punctured affine line, Y  (A

n {0}), into E  A

We denote the closure of the image by M



Y.An

alternative, more intrinsic description is in terms of

the blowup construction of algeb raic geometry:



Y ¼ Bl

Xf0g

ðY  A

Geometrically, M



Y has a copy of Y over each  6¼ 0

and a copy of the normal bundle N

Y over  = 0. This

is the key construction that Fulton and MacPherson

make use of. The same construction applied to V,that

is, the closure of V  (A

n {0}) in M



Y,hasover0a

sort of singular normal bundle known as the normal

cone

X\V

V  N

X\V

One of the properties of Chow groups is that they

are unchanged upon pullback to the total space of a

vector bundle (apart from the obvious dimension

shift). The refined intersection of V with X, denoted

[V], is defined to be the unique element of

kd

(X \ V) whose pullback to N

Y is equal to

X\V

V].

152 Intersection Theory

This single construction encompasses and inter-

polates between two extreme cases of intersections:

½V¼½X \ V when X and V

meet transversely

½3

½V¼c

ðN

YÞ\½V when V  X ½4

Equation [3] makes reference to trans verse i nte r-

section, a notion that is s tronger than proper

intersection. In situations when it applies, for

example, in eqn [1], it signifies t hat i nte rsect ion

operations behave as one might expect. Equation

[4] includes the self-intersecti on formula which s ays

that [X]  [X] is e qual to the top Chern class of

With this construction, which is well documented

in Fulton (1998), the general refined intersection in

eqn [2] is obtained by reduction to the diagonal. Let



denote the diagonal inclusion X ! X  X of the

nonsingular vari ety X. For subvarieties U and V of

X, we define

½U½V¼

½U  V½5

Equation [5] makes the Chow groups of X into a

ring, the Chow ring CH



(X), which is graded by

codimension by setting

ðXÞ¼CH

mk

ðXÞ

Links with Topology

Cycle Map to Homology

For algebraic varieties over the complex numbers,

there is a cycle map which links the Chow groups

with a topological homology group. If X is an

algebraic variety over C, then let H



(X) denote the

Borel–Moore homology of X, that is, the homology

of locally finite singular chains on X (viewed as a

topological space with the classical topology). If X is

embedded as a closed subset of an oriented

differentiable manifold M, then there are

identifications

ðXÞﬃH

ni

ðM; M n XÞ½6

where n is the dimension of M. There is a cycle class

map

ðXÞ!H

ðXÞ

which sends the class of each irreducible subvariety

Z of dimension k in X to its fundamental class

[Z] 2 H

(X).

Let M be an oriented differentiab le manifold of

dimension n and let X and Y be closed subsets of M.

Then the cup product H

(M, MnX)  H

(M, MnY)

! H

iþj

(M, Mn(X \ Y)) induces, via eqn [6],an

intersection product

ðXÞH

ðYÞ!H

iþjn

ðX \ YÞ

which is the topological analog of the refined

intersection product of eqn [5]. The products are

compatible via the cycle class map. The topology of

complex algebraic varieties and the compatibilities

between algebraic and topological intersections are

discussed in Fulton (1998). An interesting applica-

tion of this interplay of intersection theories is the

convolution product in Borel–Moore homology,

which is important in geometric representation

theory (see Chriss and Ginzburg (1997)).

Riemann–Roch Theorems

The classical Riemann–Roch theorem relates the

dimensions of linear systems on an algebraic curve

(algebraic quantities) with their degrees and

the cu rve’s gen us (topological quantities). The

Hirzebruch–Riemann–Roch theorem states that on

a nonsingular projective variety X,ifE is an

algebraic vector bundle on X and (E) denotes its

Euler characteristic (the alternating sum of the ranks

of the sheaf-theoretic cohomology groups), then

ðEÞ¼

chðEÞtdðT

Þ½7

where

denotes the degree of the zero-dimensional

component of the quantity that follows, and the Chern

character ch(E) and Todd class td(T

) are certain

standard universal polynomials of Chern classes.

Grothendieck had the inspired idea that eqn [7]

could be generalized to a covariance property for the

Chern character times the Todd class. If X and Y are

nonsingular varieties and f : X ! Y is a projective

morphism (or, more generally, a proper morphism),

then there is a well-defined push-forward f



Chow groups. There is also a kind of pus h-forward

for vector bundles. The Grothendieck group of

vector bundles on X, denoted K

(X), is the group

of formal linear combinations of vector bundles,

modulo the relations [E] = [E

] þ [E

] whenever E

a sub-bundle of E with quotient bundle E

. Every

coherent sheaf F has a well-defined class in K

(X),

namely, the alternating sum of [E

] where E



is any

finite resolution of F by vector bundles (locally free

sheaves). The push-forward f



[E] is defined as the

alternating sum of the classes in K

(Y) of the higher

direct images R



E. The Grothendieck–Riemann–

Roch theorem states that

chðf



½EÞ  tdðT

Þ¼f



ðchðEÞtdðT

ÞÞ ½8

Intersection Theory 153

in CH



(Y)  Q. Notice that eqn [7] represents the

case that Y is a point.

There is an even more general formulation valid

for singular varieties. It is necessary to work with a

homology version of the Grothendieck group,

namely, the Grothendieck group K

(X) of coherent

sheaves on X. The Baum–Fulton–MacPherson ver-

sion of the Grothendieck–Riemann–Roch theorem

prescribes transformations



: K

ðXÞ!CH



ðXÞQ ½9

which are covariant for proper morphisms. When

X is nonsingular, 

is given by the ‘‘Chern

character’’ times the ‘‘Todd class’’, and covariance

becomes eqn [8].

In the case of varieties over the complex numbers,

there is also a transformation from the algebraic

Grothendieck group K

(X) to a topological analog,

satisfying various compatibilities. The composition

with the homology Chern character gives Riemann–

Roch transformations K

(X) ! H



(X; Q) satisfying

properties akin to those of eqn [9].

The Analytic Setting

The Atiyah–Singer index theorem stands as

an important generalization of the Hirzebruch–

Riemann–Roch theorem. The index of an elliptic

differential operator on a differentiable manifold

plays the role of the Euler characteristic, and is

equated with a topological quantity. One of the

consequences of the index theorem is the validity of

eqn [7] for general compact compl ex manifolds.

More in the domain of pure analysis is the

question of intersecting two currents on a differenti-

able manifold. Currents arise naturally out of

Chern–Weil theory. To each current is associated a

wave front, a subset of the cotangent bundle that

reflects the geometry of the singular set of the

current. A current can be pulled back to an

embedded submanifold whenever the embedding is

transverse to the wave front. By reduction to the

diagonal, this gives an intersectio n of two currents

with transverse wave fronts which reduces to the

usual wedge product in the case of smooth differ-

ential forms (see Ho¨ rmander (1990)).

Applications of Intersection Theory

Enumerative Geometry

Intersection theory has proved to be a useful tool in

diverse areas such as enumerative geometry, singular-

ity theory, and moduli problems. Enumerative pro-

blems have intrigued generations of geometers.

Chasles, Maillard, Schubert, and Zeuthen are among

the geometers of the second half of the nineteenth

century who solved an impressive array of problems,

including, as a notable example, Steiner’s five conics

problem to determine the number of plane conics

tangent to five given conics in general position.

In modern terms, the successful solution to an

enumerative problem involves setting up a space which

parametrizes the geometric objects being counted,

suitably compactified, and carrying out an intersec-

tion-theoretic computation on this space. Steiner’s

problem illustrates how ‘‘excess intersection’’ can

occur and cause difficulty. Inside the CP

of plane

conics, including degenerate conics, those tangent to a

given conic constitute a sextic hypersurface. So

= 7776 would appear plausible; this was, in fact,

the originally proposed solution. However, the most

degenerate conics, the double lines, all appear as limits

of families of conics tangent to any given conic. The

refined intersection of five of these sextics has a cycle

class of degree 4512 supported on the Veronese

surface of double lines. This leaves 3264, the correct

answer given by Chasles in 1864. The issue of

providing rigorous foundations for these kinds of

calculations was recognized by Hilbert, who set it as

the 15th of his 23 major mathematical problems

outlined in 1900. A good survey of early and modern

efforts in enumerative geometry can be found in

Kleiman and Thorup (1987).

Singularity Theory and Degeneracy Loci

In any situation where a geometric object is

described by parameters, there will be values of the

parameter at which the geometry changes qualita-

tively. The significance of this is evident in the space

of conics above. Singularity theory is concerned with

the loci in parameter spaces on which these

transitions can occur. Let  : Y ! P be a map of

differential manifolds, or of nonsingular algebraic

varieties, which is generally (but not everywhere)

submersive, so that there are singular fibers. Let d

denote the dimension of P, which can be considered

as a parameter space, and let c be the dimension of

Y. Consider the loci



ðÞ¼fy 2 Y jrkðT

y;Y

! T

ðyÞ;P

Þd  kg

of singularity theory. Thom made an influential

study of these in the 1950s, and Porteous in 1971

gave the following formula, now called the Thom–

Porteous formula:



ðÞ ¼ s

ððkþcdÞ

ð



 T

Þ½10

The symbol on the right is shorthand for

(kþcd,..., kþcd)

,thecasea

=  = a

= k þ c  d of

the Schur determinant s

,..., a

)

= det (s

þji

)

1i, jk

and for vector bundles E and F the s

(F  E)are

154 Intersection Theory

defined by the formula s(F E) =

(1)

(E)=

(1)

(F). In algebraic intersection theory, eqn

[10] has the precise meaning that when



()hasthe

expected codimension k(k þ c  d)inY (or is emp ty),

its cycle class is equal to the given polynomial in

Chern classes. The Thom–Porteous formula applies

to the degeneracy loci of arbitrary maps of vector

bundles E ! F. Degeneracy loci c onstitute an active

area of research in intersection theory, and there are

generalizations, for example, to cases where there

are more bundles or bundle maps with symmetry

(see Fulton and Pragacz (1998)).

Moduli Spaces

The parameter spaces that have appeared often admit

interpretations as moduli spaces. Moduli problems

start with geometric objects to be classified, and ask for

families of these objects over an arbitrary base space to

be represented as faithfully as possible by maps from

the base space to some space called a moduli space. For

enumerative applications it is most useful for the

moduli space to be compact. One of the principal

examples is the moduli of algebraic curves of given

genus g:forg  2, the moduli space of smooth curves

has a compactification



by stable curves, as

defined and studied by Deligne and Mumford. While

the M

are singular, the singularities are mild enough

to permit the definition of an intersection theory for

and



, as was done by Mumford in the 1980s.

More generally, if X is a complex projective variety,

Kontsevich’s spaces of stable maps



g, n

(X, )com-

pactify the moduli of genus g curves with n marked

points together with algebraic maps to X having image

in homology class  2 H

(X). These spaces, and some

high-powered intersection theory that takes place on

them, are vitally important in Gromov–Witten theory.

K-theory also provides an alternative approach to

intersection products in algebraic geometry.

Extensions and Related Theories

Motives and Higher Chow Groups

Intersection theory has evolved into a mature theory

with numerous extensions and offshoots. Many of

these are a result of endeavors to forge links with

other branches of mat hematics. One of the exten-

sions, higher Chow groups, has its roots in a basic

property of intersection theory, the excision prop-

erty, which states that if X is a variety and U  X an

open subvariety, with Z = XnU, then the inclusion

and restriction maps fit into a right exact sequence



Z ! CH



X ! CH



U ! 0

This is reminiscent of the long exact homology

sequence of a pair in algebraic topology. Indeed,

there is a corresponding long exact sequence of

Borel–Moore homology groups, but the elementary

algebraic theory lacks such a long exact sequence.

Bloch introduced higher Chow groups in the 1980s

to fill this gap. The theory, which is quite

complicated, provides groups CH



(X, j), with



(X,0)= CH



X, such that there is a long exact

sequence

!CH



ðU; j þ 1Þ!CH



ðZ; jÞ!CH



ðX; jÞ

! CH



ðU; jÞ!

These groups are closely connected to algebraic K-

theory and also to a related theory called motivic

cohomology.

Motives, a sort of universal cohomology theory

envisaged by Grothendieck, conjecturally form a

category which can be extended to a bigger category

of mixed motives that reflects mixed structures in

cohomology, such as mixed Hodge structures.

Recently, Voevodsky et al. (2000) have introduced

motivic cohomology groups which form an integral

part of a homotopy theory for algebraic varieties.

Voevodsky’s work, including a proof of the Milnor

conjecture of K-theory, earned him a Fields Medal

in 2002.

Arithmetic Intersection Theory

There is an arithmetic version of intersection theory

which applies to an arithmetic scheme X, which is,

informally, a scheme defined over every prime field

(all finite fields F

and also Q) in a consistent way.

This means that X can be base-extended to any

field. In situations where the complex variety X(C)

is nonsingular, there is an arithmetic Chow ring



(X), intr oduced by Gillet and Soule´ in 1990.

Elements of



(X) are equivalence classes of pairs

(Z, g) where Z is an algebraic cycle on X and g is

known as a Green current for Z, a current on X(C)

satisfying the relation

2



@g þ 

ZðCÞ

¼ ! ½11

for some smooth differential form ! satisfying some

conditions. Here, 

integration along Z(C). The point to notice is that

eqn [11] relates analysis (the Green current) and

algebra (the cycle) on X on one side with topology

on the other, as ! will be a closed form whose class

in de Rham cohomology is Poincare´ dual to [Z(C)].

Arithmetic intersection theory is used to define

arithmetic height functions. Height functions have

important applications to Diophantine problems, and

were an essential component of the proof by Faltings of

the Mordell conjecture, which earned him a Fields

Medal in 1986. Arithmetic intersection theory grew

Intersection Theory 155

out of an earlier theory of Arakelov, in which X(C)is

endowed with a Ka¨hler metric, and the form ! in eqn

[11] is required to be harmonic. The Arakelov Chow

group is only a ring when harmonic forms are closed

under wedge product, which is not the case generally

but which is true in some interesting cases, for example,

for Grassmannian varieties. Arakelov treated the case

of arithmetic surfaces, that is, the case when X(C)isan

algebraic curve (‘‘surface’’ refers to a second dimension

in the arithmetic direction), and introduced a pairing of

arithmetic divisors, in analogy with the usual pairing of

divisors on an algebraic surface. Arakelov’s work, its

subsequent generalizations, and more recent develop-

ments are covered in Faltings (1992).

Equivariant Theories and Stacks

Moduli problems such as those mentioned previously

are often best represented not by traditional varieties,

but by a more sophisticated sort of object called a

stack. Taking inspiration from Mumford’s intersec-

tion theory on M

, intersection theory on algebraic

stacks has grown into a mature theory in its own

right. Examples of stacks include orbifolds, for which

there is the Chen–Ruan (orbifold) cohomology theory

as well as an algebraic analog due to Abramovich,

Graber, and Vistoli (see Abramovich, et al. (2002)).

Another class of examples are quotient stacks of a

variety by the action of an algebraic group. In these

cases the Chow groups of the stack are equivariant

Chow groups, part of a rich theory modeled on

equivariant cohomology in algebraic topology.

Behrend (2002) provides a nice survey of stacks,

equivariant intersection theory, and their uses in

Gromov–Witten theory. The Bott residue formula is

an important tool in equivariant intersection theory

which is particularly well suited to making concrete

calculations, for example, in enumerative geometry.

A description with nice examples can be found in

Ellingsrud and Strømme (1996).

See also: Cohomology Theories; Hamiltonian Group

Actions; Index Theorems; K-Theory; Moduli Spaces:

An Introduction.