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LaTeX gen-p-l
style
\documentclass{book}
\usepackage{amssymb,amsfonts,gen-p-l}
We shall generally use the notation
of [Gu]; a brief review
follows. Let $M$ be a simply
connected (and compact, connected)
K\"{a}hler manifold, of complex
dimension $n$, whose nonzero
integral cohomology groups are of
even degree and torsion-free. We
choose a basis $b_0,b_1,\dots,b_s$
of $H^\ast(M;{\mathbf Z})$,
such that $b_1,\dots,b_r$ form a
basis of $H^2(M;{\mathbf Z})$.
The Poincar\'{e} dual basis of $H_\ast(M;{\mathbf
Z})$ will be
denoted by $B_0,B_1,\dots,B_s$. The
dual basis of
$H^\ast(M;{\mathbf Z})$ with respect
to the intersection form $(\
,\ )$ will be denoted by $a_0,a_1,\dots,a_s$.
Thus, we have
$(a_i,b_j) = \langle a_i,B_j\rangle
= \langle b_i, A_j \rangle =
\delta_{ij}$. We shall choose
$b_0=1$, the identity element of
$H^\ast(M;{\mathbf Z})$, so that
$B_0$ is the fundamental homology
class of $M$; sometimes we write
$B_0=M$.
For homology classes (or
representative cycles of such
classes ---
we blur the distinction) $X_1,\dots,X_i$,
when $i\ge 3$, the
notation $\langle X_1 \vert \dots \vert
X_i \rangle_D$ will denote
the {\lq\lq}usual{\rq\rq} genus 0
Gromov-Witten invariant obtained
using moduli spaces of {\lq\lq}stable
rational curves with $i$
marked points{\rq\rq}. For the
definition and properties of
$\langle X_1 \vert \dots \vert X_i\rangle_D$
we refer to
[Fu-Pa] and chapter 7 of [Co-Ka]
(where the standard
notation $\langle
I_{0,i,D}\rangle(x_1,\dots,x_i)_{0,D}$
is used).
Here, $D$ is an element of
$\pi_2(M)$, so we may write
$D=\sum_{i=1}^r s_i A_i$, and we
shall assume as in [Gu] that
the homotopy class $D$ contains
holomorphic maps ${\mathbf C}
P^1\to M$ only when $s_i\ge 0$ for
all $i$.
It is necessary to issue a warning
at this point. In \S 7 of
[Gu], the notation $\langle X_1 \vert
\dots \vert
X_i\rangle_D$ had a different
meaning, namely the intersection
number $\vert{\rm Hol}_D^{X_1,p_1}
\cap \dots \cap {\rm
Hol}_D^{X_i,p_i} \vert$, where the
points $p_1,\dots,p_i$ are
fixed. To avoid confusion the latter
will be denoted by $\langle
X_1 \vert \dots \vert X_i \rangle^{fix}_D$
in the present article.
For $i=3$, the two definitions
agree. For $i\ge 4$, they are (in
the words of [Fu-Pa]) solutions to
two different enumerative
problems, and they have somewhat
different properties.
A general element of $H^\ast(M;{\mathbf
C})$ will be denoted by
$\hat{t} = \sum_{i=0}^s t_i b_i$.
Since elements of
$H^2(M;{\mathbf C})$ play a special
role, we reserve the symbol
$t$ for a general element of
$H^2(M;{\mathbf C})$, i.e.
$t=\sum_{i=1}^r t_i b_i$.
The {\lq\lq}large{\rq\rq} quantum
product on the vector space
$H^\ast(M;{\mathbf C})$ is defined
by
\[
\langle a\circ_{\hat{t}} b, C\rangle
= \sum\limits_{D,k\ge 0}
\frac{1}{k!} \langle A\vert B \vert
C \vert \hat{T}\leftarrow k
\rightarrow \hat{T} \rangle_D
\]
where (as the notation indicates)
the Poincar\'{e} dual $\hat T$
of $\hat t$ appears $k$ times in the
general term of the series.
The $D=0$ term is special, because
$\langle A\vert B \vert C \vert
\hat T \leftarrow k \rightarrow \hat
T \rangle_0$ is zero unless
$k=0$. Using this fact, and the {\lq\lq}divisor
rule{\rq\rq}
$\langle A \vert B \vert C \vert T \leftarrow
k \rightarrow T
\rangle_D= \langle A \vert B \vert C
\rangle_D \langle t,
D\rangle^k$, we see that the {\lq\lq}small{\rq\rq}
quantum product
$a{\circ_{t}} b$ is equal to the
quantum product which was used in
[Gu]. We shall not be concerned with
the question of
convergence of infinite series like
this; we shall assume that the
series converges in a suitable
region or simply treat it as a
formal series. Each of $\circ_{\hat{t}}$
and ${\circ_{t}}$ endows
$H^\ast(M;{\mathbf C})$ with the
structure of a commutative
algebra (over ${\mathbf C}$) with
identity element $b_0=1$.
The (large) quantum product on the
vector space $H^\ast(M;{\mathbf
C})$ is determined by giving all
quantum products of the basis
elements $b_i$; these in turn are
determined by the following
function, which is called the
Gromov-Witten potential:
\[
\Phi(\hat{t})= \sum\limits_{D,k\ge
3} \frac{1}{k!} \langle \hat{T}
\leftarrow k \rightarrow \hat{T} \rangle_D.
\]
This may be regarded as a generating
function for the
Gromov-Witten invariants; it is
rather unwieldy, of course, and
one of the main themes of the
subject is the fact that there are
alternative expressions for it.
Because of the linearity of
Gromov-Witten invariants, we have
\[
{\frac{\partial}{\partial t_i}} \langle
\hat{T} \vert A \vert B
\vert \dots \rangle = \langle B_i \vert
A \vert B \vert \dots
\rangle
\]
and hence
\[
\langle b_i \circ_{\hat{t}} b_j, B_k
\rangle = \frac
{{\partial}^3} {{\partial} t_i
{\partial} t_j {\partial} t_k }
\Phi = b_i b_j b_k \Phi
\]
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