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{\bf Mirror Symmetry and Algebraic Geometry}
\medskip
by David A. Cox and Sheldon Katz
\bigskip
\medskip
{\bf Errata for the {\it first\/} printing as of March 28, 2006:}
\medskip
Page xviii: Add the following at the bottom of the page: ``PS:\ A list
of typographical errors for the book can be found at the web site
\medskip
\hskip20pt {\tt http://www.cs.amherst.edu/\~{}dac/ms.html}''
\medskip
Page 16, first display: In the summation, replace ``$d=1$'' with
``$d=0$''
\medskip
Page 26, bottom display: In the first line of the display, replace
``$770te^{t_1}$'' with ``$770t_1e^{t_1}$''
\medskip
Page 32, third bullet on lines $-19$ and $-18$: One comment to add
here is that for a toric variety $X$, the implication ``$X$ is
simplicial $\Rightarrow$ $X$ is an orbifold'' is elementary, while the
converse ``$X$ is an orbifold $\Rightarrow$ $X$ is simplicial'' is
deeper. A proof can be found in \emph{Rational smoothness and fixed
points of torus actions} by M.\ Brion, Transformation Groups {\bf 4}
(1999), 127--156.
\medskip
Page 34, lines 9 and 10: Replace $k\Delta$ with $k\Delta\cap M$
(twice).
\medskip
Page 34, line 11: Replace $l\Delta$ with $l\Delta\cap M$ and
$(k+l)\Delta$ with $(k+l)\Delta\cap M$.
\medskip
Page 40, line 13: Replace ``$\lambda_{n+1}/(D_{v_{n+1}}\cdot
C_\sigma)$'' with ``$(D_{v_{n+1}}\cdot
C_\sigma)/\lambda_{n+1}$''
\medskip
Page 59, immediately after (4.9), add the sentence:
"In (4.9) $H^{n-1}(Z_f,\C)$ has a canonical mixed Hodge structure and
$Gr^W_{n-1}H^{n-1}(Z_f,\C)$ as usual denotes the associated graded piece
of the weight filtration with a pure Hodge structure of weight $n-1$."
\medskip
Page 61, line 5 of Section 4.2: Replace ``$\mathbb{R}^5$'' with
``$\mathbb{R}^4$''
\medskip
Page 79, line 9: Replace ``boundary point of $S$'' with ``boundary
point of $S$ at which the Gauss-Manin connection has a regular
singularity (this is automatic if the family of threefolds extends to
a flat family over the boundary point)''
\medskip
Page 79, line 10: Replace ``$\mathcal{F}^0$'' with ``$\mathcal{F}^3$''
\medskip
Page 93, line $-2$: Replace ``$(\Delta\cap M)\times\{1\})$'' with
``$(\Delta\cap M)\times\{1\}$''
\medskip
Pages 98--99: Proposition 5.5.4 is incorrect as stated. Below you
will find numerous changes which fix the statement of the
proposition on page 98. Also, parts of the proof on page 99 are
seriously flawed. Fixing these would require considerable additions.
For this reason, the fixes given below prove only a special case of
the result. A complete proof can be found at the web site mentioned
in the errata to page xviii.
\medskip
Page 98, line 8: In (5.43), replace
``$\lambda_{r+i}^{c_1(\mathcal{L}_i)\cdot \beta_j}$'' with
``$(-\lambda_{r+i})^{c_1(\mathcal{L}_i)\cdot \beta_j}$''
\medskip
Page 98, lines $-2$ and $-1$: Replace Proposition 5.5.4 with the
following:
\bigskip
\hskip20pt {\scshape Proposition 5.5.4.}~\emph{ The formal function
$\big(\prod_{i=1}^k \lambda_{r+i}^{-1}\big)\, \tilde I$ satisfies the
$\mathcal{A}$-system associated to~$($5.42\/$)$ with
$\hat\b=(0,\dots,0,-1,\dots,-1) \in N\times\Z^k$.}
\bigskip
Page 99, line 2: Replace ``As we discussed in Section 5.5.2, the''
with ``When $\hat{\b} = \vec{0}$, by Section 5.5.2, the''
\medskip
Page 99, line 5: Add the following new sentence: ``Then, if we switch
to the $Z_i$ for $\hat{\b} = (0,\dots,0,-1,\dots,-1)$, it follows that
the $Z_i\big((\prod_{i=1}^k \lambda_{r+i}^{-1})\tilde{I}\big) = 0$.
\medskip
Page 99, line 7: Immediately before the sentence ``It follows that
...'', insert the following new sentence: ``As noted above, we have
$c_1(\mathcal{L}_i)\cdot \beta \ge 0$ for all $i$, and for simplicity,
we will also assume that $D_\rho\cdot \b \ge 0$ for all $\rho$.''
\medskip
Page 99, line 9: Replace (5.47) with the following new display:
\bigskip
(5.48) \hskip90pt ${\displaystyle \Box_{\b}=\prod_{\rho}
\partial_{\rho}^{D_\rho\cdot\b}
-\prod_{i=1}^k\partial_{r+i}^{c_1(\cL_i)\cdot\b},}$
\bigskip
Page 99, line $-10$: Replace the formula for $\Box'_\b$ with the
following
\[
\Box'_\b =\prod_{\rho} \lambda_\rho^{D_\rho\cdot\b}\ \
\prod_{i=1}^k \lambda_{r+i}\ \ \Box_{\b}\ \ \prod_{i=1}^k
\lambda_{r+i}^{-1}
\]
\smallskip
Page 99, lines $-8$ to $-5$: Replace display (5.51) with the
following:
\[
\begin{array}{rcl}
\Box'_\b & = & \prod_{\rho} \delta_{\rho}(\delta_{\rho}-1)\cdots
(\delta_\rho-{D_\rho\cdot\b}+1)\\[4pt]
&& - q^\b \prod_{i=1}^k (-1)^{c_1(\cL_i)\cdot\b} \prod_{i=1}^k
(\delta_{r+i}-1)\cdots(\delta_{r+i} - c_1(\cL_i)\cdot\b)
\end{array}
\]
\smallskip
Page 100, lines $-3$ to $-1$. Delete these lines.
\medskip
Page 101, line 4: Delete ``It follows that \dots (5.37)'' and replace
with the following
\bigskip
This is a function of $q_1$, which by (5.43) is $-z$, where $z$
is defined in (5.36). It follows that $y_0,\dots,y_4$ are solutions
of the fifth order GKZ equation (5.37), provided we replace $z$ with
$-q_1$. The sign difference between $q_1$ and $z$ will be discussed
in Section~6.3.3.
\bigskip
Page 106, line 1: Replace ``$\nabla(e_{3-p})$'' with
``$\nabla_\delta(e_{3-p})$''
\medskip
Page 106, immediately after the display on line 21: Add the new
sentence ``Due to a constant of integration, the above expression for
$q$ is only defined up to a multiplicative constant. We fix this
choice by requiring that $q$ is of the form $q = z\exp(f(z))$, where
$f(z)$ is holomorphic with $f(0) = 0$.''
\medskip
Page 137, line $-6$: Replace ``induces isomorphism'' with ``induces an
isomorphism''
\medskip
Page 138, line 11: Insert the new sentence ``However, [{\bf Szendroi2}]
shows that Conjecture~6.2.8 is false in general.''
\medskip
Page 138, line below (6.24): Replace ``Strictly speaking, we should
take the quotient of this by the automorphisms of $V$. However,
$\mathrm{Aut}(V)$'' with ``We should take the quotient of this by
$\mathrm{Aut}_{\mathrm{toric}}(V)$, the subgroup of automorphisms of
$V$ which preserve $H^2_{\mathrm{toric}}(V)$. Then
$\mathrm{Aut}_{\mathrm{toric}}(V)$''
\medskip
Page 138, 3 lines below (6.24): Replace ``$\mathrm{Aut}(V)$'' with
``$\mathrm{Aut}_{\mathrm{toric}}(V)$''
\medskip
Page 146, line $-26$: Replace ``$D_6 \sim D_1 - 2 D_2$'' with ``$D_6
\sim D_3 - 2 D_1$''
\medskip
Page 147, line 20: Replace ``has rank 2'' with ``has rank 4''
\medskip
Page 170, line 1: Replace ``stacks are'' with ``the
stacks of interest to us are''
\medskip
Page 170, line 2: Replace ``to the category of sets'' with ``to the
category of groupoids''
\medskip
Page 170, line 3: After ``as we go along.'', insert the new sentence
``For simplicity of exposition, we will abuse terminology by refering
to stacks as if they were functors from schemes to sets.''
\medskip
Page 172, three lines above (7.11): Replace ``$\mathbb{P}_1$'' with
``$p_1$''
\medskip
Page 183, display in part (ii) of Definition 7.1.9: Replace
``$\cup\ldots\cup$'' with ``$\cup\cdots\cup$''
\medskip
Page 184, line 3 of Section 7.2: Replace ``$J$-holomorphic maps in
symplectic geometry'' with ``$J$-holomorphic maps in symplectic
geometry, due to [{\bf Gromov}],''
\medskip
Page 193, line 2: Replace ``$[X] \in H^0(X,\mathbb{Q})$ is the
fundamental class of $X$'' with ``$1_X = [X] \in H^0(X,\mathbb{Q})$
is the Poincar\'e dual of the fundamental class of $X$''
\medskip
Page 193, line 15: Replace ``$\pi_{n*}$'' with ``$\pi_{n!}$''
\medskip
Page 194, line 4: Replace ``$q_{n_2+1}$'' with ``$q_{1}$''
\medskip
Page 194, line 5: Replace
``$(C,p_1,\ldots,p_{n_1},q_1,\ldots,q_{n_2})$'' with
``$(C,p_1,\ldots,p_{n_1},q_2,\ldots,q_{n_2+1})$''
\medskip
Page 194, line 15: Replace ``$(C_1 \cup
C_2,p_1,\dots,p_{n_1},p,p_{n_1+1},\dots,p_n,q)$'' with \hfill\break
``$(C_1 \cup C_2,p_1,\dots,p_{n_1},p,q,p_{n_1+1},\dots,p_n)$''
\medskip
Page 194, line 2 of the Reduction Axiom: Replace ``$\Delta_a$'' with
``$T_a$''
\medskip
Page 194, line 5 of the Reduction Axiom: Replace ``$\phi$'' with
``$\psi$''
\medskip
Page 197, line 11: Replace ``$\overline{M}_{0,n}(X,\beta)$'' with
``$\overline{M}_{0,3}(X,\beta)$''
\medskip
Page 197, line 12: Replace ``$2 \dim X + 2$'' with ``$4 \dim X + 2$''
\medskip
Page 201, line $-7$: Replace ``Fundamental Class Axiom'' with ``Point
Mapping Axiom''
\medskip
Page 202, line 5 after Conjecture 7.4.3: Replace ``are all are'' with
``are all''
\medskip
Page 203, equation (7.48): In the summation, replace ``$d=1$'' with
``$d=0$''
\medskip
Page 205, line $-11$: replace ``$d=2$'' with ``$k=2$''
\medskip
Page 206, line 18: A fully rigorous proof of the enumerative
significance of $17,601,000$ can be found in \emph{Node polynomials
for families: methods and applications} by S.\ Kleiman and R.\ Piene,
Math.\ Nachr.\ {\bf 271} (2004), 69--90.
\medskip
Page 206, equation (7.54): The Clemens conjecture for $d = 10$ is
proved in \emph{Rational curves of degree 10 on a general quintic
threefold} by E.\ Cotterill, Comm.\ in Algebra, {\bf 33} (2005),
1833--1872. This, plus the paper mentioned in the previous erratum,
give a complete proof that $n_{10} = 6\times 17,601,000 +$ number of
rational curves of degree $10$ on the quintic threefold.
\medskip
Page 207, line 5, replace ``$17,601,600$'' with ``$17,601,000$''
\medskip
Page 218, line 16 (this is the line following the second display):
Replace ``$\int_V T^i \cup T_j = \delta_{ij}$'' with ``$\int_X T^i
\cup T_j = \delta_{ij}$''
\medskip
Page 233, line $-4$: replace ``commutative" with ``supercommutative"
\medskip
Page 235, line $-18$: Replace ``coefficient of $q^\beta$ in $T_i*T_j$
has degree $\mathrm{deg}(T_i) + \mathrm{deg}(T_j) + 2\int_\beta
\omega_X$, just as we noted for the small quantum product in the proof
of Proposition 8.1.5.'' with ``coefficient of $T^\ell q^\beta$ in
$T_i*T_j$ has degree $\mathrm{deg}(T_\ell) -\big(\mathrm{deg}(T_i) +
\mathrm{deg}(T_j) + 2\int_\beta \omega_X\big)$. The argument is
similar to the proof of Proposition 8.1.5.''
\medskip
Page 240, line $-3$ to $-5$: Replace this paragraph with the
following: ``The potential function $\Phi$ we studied earlier
satisfies the conditions of Proposition 8.4.1. This is why the big
quantum product makes $H^*(X,\mathbb{C})$ into a Frobenius algebra.''
\medskip
Page 244, line $-2$: Replace ``$q^\beta$ is defined as in (8.40) to
be'' with ``$q^\beta = e^{2\pi i \int_\beta \omega}$, where as in
(8.40),''
\medskip
Page 246, line 17: Replace ``exponents are positive'' with ``exponents
in the $q_j$ are positive''
\medskip
Page 275, line $-7$: Delete the word ``compact''
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Page 275, line $-6$: Delete ``with $EG$,"
\medskip
Page 275, line $-3$: Replace ``and $EG = \pi_1^* S \otimes \cdots
\otimes \pi_n^* S$,'' with ``and the vector bundle obtained from $EG$
by the natural action of $(\mathbb{C}^*)^n$ on $\mathbb{C}^n$ is
$\pi_1^* S \oplus \cdots \oplus \pi_n^* S$,''
\medskip
Page 276, line $-15$: Replace ``such that the action of $G$ lifts to
an action of $E$'' with ``together with a lifting of the action on $X$
to an action on $E$''
\medskip
Page 276, line $-10$: Replace ``$\mathrm{Euler}_T(E)$'' with
``$\mathrm{Euler}_G(E)$''
\medskip
Page 276, line $-5$: Replace
``$\mathcal{O}(\lambda_1)\oplus\cdots\oplus \mathcal{O}(\lambda_1)$''
with ``$\mathcal{O}(\lambda_1)\oplus\cdots\oplus
\mathcal{O}(\lambda_n)$''
\medskip
Page 277, line 3: Replace ``smooth manifold'' with ``compact complex
manifold''
\medskip
Page 277, line 14: Replace ``assertion" with ``assertions"
\medskip
Page 277, line $-3$: Replace ``$H_T(\p^r)$'' with ``$H_T^*(\p^r)$''
\medskip
Page 278, line 3: Replace ``$c_r(E_T^*)$'' with ``$(-1)^rc_r(E_T^*)$''
\medskip
Page 278, line after (9.5): Replace ``is has'' with ``has''
\medskip
Page 280, first display: replace ``$Z_i$" with ``$Z_j$", ``$a_i$" with
``$a_j$", and ``$N_i$" with ``$N_j$".
\medskip
Page 283, line $-19$: replace ``$(f_{C_v},C_v)$" with ``$C_v$".
\medskip
Page 285, line $-12$: Replace ``different from $e$" with ``different from
$v$"
\medskip
Page 291, second display: Replace ``$Z^0(U_0\cap
U_1,f^*_3\mathcal{O}_{\mathbb{P}^1}(-1))$'' with ``$Z^1(U_0\cap
U_1,f^*_3\mathcal{O}_{\mathbb{P}^1}(-1))$''
\medskip
Page 291, 2 lines above the third display on page: Replace ``have of''
with ``have all of''
\medskip
Page 293, line $-6$: The erratum for page 206 gives a reference that
rigorously establishes the enumerative significance of $17,601,000$.
\medskip
Page 304, line $-7$: The list of axioms for gravitational correlators
should also include a Linearity Axiom and an Effectivity Axiom, which
are similar to the versions for Gromov-Witten invariants stated on
page 192.
\medskip
Page 305, line $-9$: Delete ``$c_1(\cL_i)^{d_i}\cup$'' and replace
``$\pi_*(\xi)$'' with ``$\pi_*(\prod_{i=1}^nc_1(\cL_i)^{d_i}\cup\xi)$''.
\medskip
Page 305, lines $-4$ and $-1$: Delete ``$(X,\beta)$'' once on line $-4$ and
three times on line $-1$.
\medskip
Page 306, line 3: replace ``$\tau_1$'' with ``$\tau_{d_1}$''.
\medskip
Page 306, line 4: replace ``$\tau_n$'' with ``$\tau_{d_n}$''.
\medskip
Page 316, first line of display in Theorem 10.2.4: Replace
``$S(e^{-T_j}\cup T)$'' with ``$s(e^{-T_j}\cup T)$''
\medskip
Page 316, second line of display in Theorem 10.2.4: Replace
``$-S(T_j\cup T)$'' with ``$-s(T_j\cup T)$''
\medskip
Page 331, line 1: Replace ``the the'' with ``the''
\medskip
Page 336, line $-22$ (this is display (11.6)): Replace
``$\alpha_n(w_0,w_1)\mbox{\large )}$'' with
``$\alpha_n(w_0,w_1))\mbox{\large )}$''
\medskip
Page 336, lines $-20$ and $-19$: Replace ``We summarize this by
saying that'' with ``In particular,''
\medskip
Page 336, line $-18$: Replace ``without common factor.'' with
``without common factor, modulo scalars.''
\medskip
Page 353, lines $-11$ and $-9$: Replace ``$P$'' and ``$Q$'' with
``$\hat{P}$'' and ``$\hat{Q}$'' respectively
\medskip
Page 353, line $-3$: Replace ``$h^{-1}$'' with ``$\hbar^{-1}$''
\medskip
Page 381, second line of display (11.74): In the denominator of the
fraction, replace ``$\hbar - c_1(\mathcal{L}_1)$'' with ``$\hbar -
c$'', just as in display (11.52) on page 365.
\medskip
Page 398, line 23: Replace ``but these techniques do not apply'' with
``and subsequent work of A.~Elezi and B.~Kim has shown that these
techniques can be applied''
\medskip
Page 409, line $-3$: Replace ``smooth $(1,1)$-form'' with
``smooth, closed $(1,1)$ form''
\medskip
Page 437, index entry [AKl]: Replace ``A.\ Altmann'' with ``A.\ Altman''
\medskip
Page 444, line 11: Insert a new reference:
\medskip
[Gromov] M.\ Gromov, {\em Pseudoholomorphic curves in symplectic
manifolds\/}, Invent.\ Math.\ {\bf 82} (1985), 307--347.
\medskip
Page 449, line 1: Replace ``by irreducible'' with ``be irreducible''
\medskip
Page 449, line $-5$: Insert a new reference:
\medskip
[Szendroi2] B. Szendroi, {\it On a conjecture of Cox and Katz}, Math.\
Z., {\bf 240} (2002), 233--241, math.AG/0110166.
\medskip
\end{document}