 # On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety. (AM-157)

Mark Green
Phillip Griffiths
Pages: 224
https://www.jstor.org/stable/j.ctt24hrts

1. Front Matter
(pp. i-iv)
(pp. v-2)
3. Chapter One Introduction
(pp. 3-21)

In this work we shall define the tangent spaces$T{{Z}^{n}}(X)$and$T{{Z}^{1}}(X)$to the spaces of$0$-cycles and of divisors on a smooth,$n$-dimensional complex algebraic variety$X$. We think it may be possible to use similar methods to define$T{{Z}^{p}}(X)$for all codimensions, but we have not been able to do this because of one significant technical point. Although the final definitions, as given in sections 7 and 8 below, are algebraic and formal, the motivation behind them is quite geometric. This is explained in the earlier sections; we have chosen to present the exposition in...

4. Chapter Two The Classical Case When $n = 1$
(pp. 22-30)

We begin with the case$n = 1$, which is both suggestive and in some ways misleading. Most of the material in this chapter is standard but will help to motivate what comes later. We want to define the tangent to an arc$z(t)=\sum\limits_{i}{{{n}_{i}}{{x}_{i}}(t)}$in the space${{Z}^{1}}(X)$of$0$-cycles on a smooth algebraic curve$X$. Later on we will more precisely define what we mean by such an arc—for the moment one may think of the${{x}_{i}}(t)$as being given in local coordinates by a Puiseaux series in$t$.

Thedegree$\deg z(t)=:\sum\limits_{i}{{{n}_{i}}}$is constant in$t$...

5. Chapter Three Differential Geometry of Symmetric Products
(pp. 31-41)

Let$X$be a smooth variety of dimension$n$. We are interested in the geometry of configurations of$m$points on$X$, which we represent as effective$0$-cycles$z={{x}_{1}}+\cdots +{{x}_{m}}$of degree$m$. Set theoretically such configurations are given by the$m$-fold symmetric product${{X}^{(m)}}=\underbrace{X\times \cdots \times X}_{m}/{{\Sigma }_{m}}$where${{\Sigma }_{m}}$is the group of permutations.

We will be especially concerned with arcs of$0$-cycles, given by a regular mapping$B\xrightarrow{z}{{X}^{(m)}}$from a smooth (not necessarily complete) curve into${{X}^{(m)}}$. If$t$is a local uniformizing parameter on$B$, such an arc may be thought of as...

6. Chapter Four Absolute Differentials (I)
(pp. 42-53)

Given a commutative ring$R$and subring$S$, one defines theKähler differentialsof$R$over$S$, denoted$\Omega _{R/S,}^{1}$to be the$R$-module generated by all symbols of the form$adb,\quad \quad a,b\in R$subject to the relations$\caption{(4.1)}\ \left\{ \begin{array}{l} d(a+b)=da+db \\ d(ab)=adb+bda \\ ds=0\text{ if }s \in S. \\ \end{array} \right.$

In this chapter$R$will be a field$k$of characteristic zero, a polynomial ring over$k$, or a local ring${\mathcal{O}}$with residue field$k$of characteristic zero. From$d(a+a)=2da$it follows that$d2=0$, and in fact$dp=0 \quad \quad p\in \mathbb{Z},$and then from$d({{q}^{-1}})=-{{q}^{-2}}dq=0$we have$d(p/q)=0 \quad \quad p,q\in \mathbb{Z}.$

The Kähler differentials$\Omega _{R/\mathbb{Q}}^{1}$are calledabsolute differentials. We shall be primarily concerned with...

7. Chapter Five Geometric Description of $\underline{\underline{T}}{{Z}^{n}}(X)$
(pp. 54-60)

Following a standard method in differential geometry, we will describe the tangent space$TZ_{\left\{ x \right\}}^{n}(X)=:{{T}_{\left\{ x \right\}}}{{Z}^{n}}(X)=Z_{\left\{ x \right\}}^{n}(X)/{{\equiv }_{{{1}^{\text{st}}}}}$where${{\equiv }_{{{1}^{\text{st}}}}}$is an equivalence relation given by a subgroup of$Z_{\{x\}}^{n}(X)$that we will callfirst order equivalence.

The coordinate description of${{\equiv }_{{{1}^{\text{st}}}}}$is as follows: Denoting by$P$the space of coefficients of Puiseaux series of arcs in${{X}^{(m)}}$reducing to$mx$at$t = 0$, we have a mapping$\Omega _{X/\mathbb{Q},x}^{q}\to \Omega _{P/\mathbb{Q},z}^{q-1}$given by${\caption{(5.1)}}\ \varphi \to \tilde{I}(z,\varphi )$where$\tilde{I}(z,\varphi )$is the provisional universal abelian invariant associated to an arc$z(t)={{x}_{1}}(t)+\cdots +{{x}_{m}}(t)$in${{X}^{(m)}}$with all${{x}_{i}}(0)=x$. This map depends on the choice of parameter$t$and choice of...

8. Chapter Six Absolute Differentials (II)
(pp. 61-83)

In differential geometry the tangent space to a manifold maybe defined axiomatically in terms of an equivalence relation on arcs. It would of course be desirable to do the same for the tangent space to the space of cycles. Denoting by ≡ the equivalence we would like to define on an arc$z(t)$in$Z_{\left\{ x \right\}}^{n}(X)$, it should have the following properties:

(i) If${{z}_{1}}(t)\equiv {{\tilde{z}}_{1}}(t)$and${{z}_{2}}(t)\equiv {{\tilde{z}}_{2}}(t)$, then${{z}_{1}}(t)+{{z}_{2}}(t)\equiv {{\tilde{z}}_{1}}(t)+{{\tilde{z}}_{2}}(t);$

(ii)$z(\alpha t)\equiv \alpha z(t)$for$\alpha \in \mathbb{Z}$.

(iii) If$z(t)$and$\tilde{z}(t)$are two arcs in$\text{Hilb}_{0}(X)$with the same tangent in$T\ \text{Hilb}_{0}(X)$, then$z(t)\equiv \tilde{z}(t)$.

(iv) If$z(t)=\tilde{z}(t)$as arcs...

9. Chapter Seven The ${{\mathcal{E}}xt}$-definition of $T{{Z}^{2}}(X)$ for $X$ an Algebraic Surface
(pp. 84-99)

In Chapter 5 we have given the geometric description${\caption {(7.1)}}\ \underline{\underline{T}}{{Z}^{2}}(X)=\underset{x\in X}{\mathop{\bigoplus }}\,\underline{\underline{\text{Hom}}}{}^o (\Omega _{X/\mathbb{Q},x}^{2}/\Omega _{\mathbb{C}/\mathbb{Q}}^{2},\Omega _{\mathbb{C}/\mathbb{Q}}^{1})$for the tangent space to the space of$0$-cycles on a smooth algebraic surface.¹ This geometric description was in turn based on representing arcs in${{Z}^{2}}(X)$in local coordinates by Puiseaux series, taking the absolute differentials of these expressions, and then extracting what is invariant when one changes local uniformizing parameters. In invariant terms, one represents an arc in${{Z}^{2}}(X)$by a diagram$Z\subset X\times B,\quad \dim\ B=1,$and uses a pull-back, push-down construction to define the tangent map as a pairing${{T}_{{{s}_{0}}}}B\otimes (\Omega _{X/\mathbb{Q},x}^{2}/\Omega _{\mathbb{C}/\mathbb{Q}}^{2})\to \Omega _{\mathbb{C}/\mathbb{Q}}^{1}.$

In either case we are giving arcs in the space...

10. Chapter Eight Tangents to Related Spaces
(pp. 100-149)

In the preceding chapter we have given the formal definition$\caption{(8.1)}\ \underline{\underline{T}}Z^{2}(X)=\underset{\left\{ \begin{matrix} Z\text{ codim }2 \\ \text{subscheme} \\ \end{matrix} \right.}{\mathop{\lim }}\,{\mathcal {E}}xt_{{{\mathcal {O}}_{X}}}^{2}({{\mathcal {O}}_{Z}},\Omega _{X/Q}^{1})$of the tangent sheaf to the space${{Z}^{2}}(X)$of$0$-cycles on a smooth algebraic surface$X$. For the study of the infinitesimal geometry of${{Z}^{2}}(X)$—especially the subspace$Z_{\text{rat}}^{2}(X)$of$0$-cycles that are rationally equivalent to zero—it is important to define and study the tangent sheaf to several related spaces. This is the objective of the first two sections of this chapter, and in the last section we will relate these to$\underline{\underline{T}}{{Z}^{2}}(X)$and arrive at the main result of this work.

In Appendix A...

11. Chapter Nine Applications and Examples
(pp. 150-185)

Let$X$be a smooth algebraic curve and$\tau =\sum\limits_{i=1}^{d}{({{x}_{i}},{{\tau }_{i}})}$a configuration of points${{x}_{i}}\in X$(assumed for simplicity of exposition to be distinct) and tangent vectors${{\tau }_{i}}\in {{T}_{{{x}_{i}}}}X$. Classically Abel’s differential equations (cf. ) dealt with the question

(9.1.)When is$\tau$tangent to a rational equivalence?

The answer, which emerged from the work of Abel, Jacobi, Riemann and others in the nineteenth century is

The necessary and sufficient condition that$\tau$be tangent to a rational equivalence is that for every regular differential$\omega \in {{H}^{0}}(\Omega _{X/\mathbb{C}}^{1})$${\caption{(9.2)}}\ \left\langle \omega,\tau \right\rangle =:\sum\limits_{i=1}^{d}{\left\langle \omega ({{x}_{i}}),{{\tau }_{i}} \right\rangle }=0.$

It is the higher dimensional analogue of this statement that we shall be discussing, focusing on the...

12. Chapter Ten Speculations and Questions
(pp. 186-194)

There are three main questions:

(10.1)Can one define$T{{Z}^{p}}(X)$in general?

The technical issue that arises in trying straightforwardly to extend the definitions given in the text for$p = n, 1$concerns cycles that are linear combinations of irreducible subvarieties$Z=\sum\limits_{i}{{{n}_{i}}{{Z}_{i}},}$where some${{Z}_{i}}$may not be the support of a locally Cohen-Maculay scheme. Similar issues are not unfamiliar—for example, in duality theory—and we can see no geometric reason why the question (10.1) should not have an affirmative answer.

The second main question is:

(10.2)For$p =n, 1$can one define$T{{Z}^{p}}(X)$axiomatically?

This issue has been raised several times in the text....

13. Bibliography
(pp. 195-198)
14. Index
(pp. 199-200)