# Tasty Bits of Several Complex Variables: Errata

This page lists major problems in the various editions. Simple typos, misspellings, and such are not listed here. Also not listed are things that were correct but simply deserved improvement. Of course, older editions may suffer from recently discovered errata as well.

June 27th 2018 edition (version 2.3):

• The definition of holomorphic functions (Definition 1.1.2) is correct (so it's not really an erratum), but it is not quite compatible with most introductory one variable books. This was not on purpose. I will change the definition to be the more standard one using the complex derivative in each variable, and when using Cauchy-Riemann we will assume $C^1$. Very little is lost by the reader just assuming that $f$ is continuously differentiable in the definition.
• In the footnote on page 82, the polarization identity is only for the real one (only the real part). It should say $4 \langle z,w \rangle = {\|z+w\|}^2-{\|z-w\|}^2 +i \bigl( {\|z+iw\|}^2 - {\|z-iw\|}^2 \bigr)$. What's there is true for real $z$ and $w$.

November 29th 2017 edition (version 2.2):

• Exercise 1.2.11 part b) is only true if $n=2$. In the next update there will be a new part c) which asks to find a counterexample if $n=3$. (Thanks to Liz Vivas for pointing this out)
• Exercise 1.4.1 part b) is wrong: There can be a one-real-dimensional curve of such points, but there can be no neighborhood of $p$. A better way to state the problem is to prove that if $\Delta$ is an anlytic disc, prove that $\partial \mathbb{B}_n \cap \Delta$ is nowhere dense in $\Delta$. (Thanks to Alekzander Malcom for pointing this out a couple of years ago, and I forgot to mark it down, and now thanks to Trevor Fancher for finding this issue again.)
• Exercise 6.4.21: an analytic disc in ${\mathbb C}^2$ is only a local variety (the word "local" was missing), since it is not necessarly closed in ${\mathbb C}^2$.
• Exercise 6.5.1: replace "hypersurface" with "submanifold".
• In the discussion of Segre variety on page 133, where it says "(and nowhere else)", that is meant only if $\widetilde{r}$ is a defining function in the same sense as $r$.
• In Example, 6.5.3, when setting $\bar{z}=\bar{w}=0$ we forgot to actually set $\bar{z}=0$.

March 21st 2017 edition (version 2.1):

Thanks to John Treuer for pointing out the following errata:

• On page 45, in the $2 \times 2$ Hessian at the top, the lower right derivative should be $\frac{\partial^2 r}{\partial y^2}$.
• On page 77, about midway down, the series is written $\sum_{\alpha} \frac{1}{\alpha} \cdots$ when it should be $\sum_{\alpha} \frac{1}{\alpha !} \cdots$. This is just the Taylor series for $f$.
• On page 80, $f_d$ is the degree $d$ homogeneous part (or $f_j$ is the degree $j$ homogeneous part).
• On page 85, in the proof of proposition 3.2.6, "If both $M$ and $f$ are real-analytic", the $f$ should be a $\phi$, since we're trying to prove that $f$ is real analytic.
• On page 87, last line in proof of 3.2.10, "If $X_{q_k} f = 0$ for" should be "Then $X_{q_k} f = 0$ for"
• On page 100, exercise 4.1.3 is wrong (a bonus exercise is to find a counterexample to it as is currently stated using the unit disc as a domain), the correct hypothesis is not that $\int_{U} \frac{\frac{\partial f}{\partial \bar{z}}(\zeta)}{\zeta-z} \, dA(\zeta) = 0$ for every $z \in \partial U$. Also take $U$ to be bounded.
• On page 100, in exercise 4.1.4, the $\varphi$ in the integrals should be $f$.
• On page 116, in exercise 6.1.4 a), $I$ should be $(f)$.
• On page 118, right after the statement of the Weierstrass Preparation Theorem, "degree $m$ monic polynomial" should be "degree $k$ monic polynomial".
• On page 120 and 121, Propositions 6.2.5, 6.2.6, and Theorem 6.2.7. $\mathbb D$ and $D$ are both used. The theorems work for any disc $D$, not just the unit disc.
• On page 121, in the first line of the proof of Theorem 6.2.7, "Let $U_m$ be the subset of $U$" should be "Let $U_m$ be the subset of $U'$".
Other errata:
• On page 34, in exercise 1.6.2, the holomorphic function had better be defined in a domain (a connected set).
• On page 115, formula (5.2), on the right hand side it should be $\sum_j \varphi_j(z)\overline{\varphi_j(\zeta)}$.
• On page 117, in the definition of Weierstrass polynomial, the sum (the $\Sigma$) goes to $k-1$ (it was missing the upper limit).
• On page 130, an overzealous spellchecker corrected hypersurfaces to hypocrites.
• On page 135, in the Diederich-Fornaess theorem, there is no need to talk about the dimension of $X$ in the statement of the theorem.

May 5th 2016 edition (version 2.0):

• On page 22, the definition of meromorphic functions is nonstandard. The standard definition is that $F$ is the quotient of holomorphic functions locally. That the definitions are equivalent for domains in ${\mathbb{C}}^n$ follows via a deep result of Oka, and our definition would be misleading when generalizing to complex manifolds. So the next version will use the standard defition. Thanks to Debraj Chakrabarti for pointing this out.
• In chapter 5, page 108 at the end of proof of the Bochner-Martinelli formula we wrongly assume that $f$ is holomorphic, and while that's an important case the theorem is stated for smooth functions. In the last estimate therefore we also get terms of the form $\frac{\partial f}{\partial \zeta_j}(\zeta)(\bar{\zeta}_j-\bar{z}_j)$ which can also be bounded by $Mr$, so the estimate still follows. Thanks to Anirban Dawn for pointing it out.
• Exercise 6.4.16, the numbers $\ell_j$ were not needed, assume they are all 1.

November 24th 2015 edition:

• On page 8, when it says why we can swap the order of summation, it says it is because of uniform convergence, when it should say uniformly absolutely convergent. Similar error is due to cut and paste on page 16.
• In exercise 1.2.4, the $f$ should be holomorphic in a neighborhood
• In exercise 1.4.2, X and Y should just be assumed locally compact Hausdorff, and for the extension $f(\infty) = \infty$.
• In proof of theorem 1.6.3, 1) the $\varphi$ need only be defined for $\xi = 0$. 2) The last equality shouldn't include $\det Dh$ since that does not make sense, it is $\det D\varphi$. (also $\varphi$ and $\phi$ were both used by mistake here)
• In Definition 2.1.1, the set $V$ should be assumed nonempty.
• In Definition 2.2.3, replace $T_pM$ with $T_p\partial U$.
• On top of page 44, there is a $\frac{1}{2}$ missing in front of the Hessian in the equation that should be $y = \frac{1}{2} x^t H x + E(x)$. Similarly a few lines down.
• Page 49, the the computation about 2/3 of the way down, when we plugged in for $w$ in $-\operatorname{Im} w$, the minus sign disappeared.
• Page 54, in the proof of Theorem 2.3.10 (tomato can principle). In the first displayed equation in the proof, the second $z_1$ is missing a conjugate sign. In the second displayed equation, the $\lambda$ mysteriously disappeared from the right hand side.
• Page 70, Exercise 2.5.7, clearly the points $p$ should be in $\partial U$.
• In Exercise 4.3.2, $f$ should be defined on $\partial U$ not $U$.

August 21st 2015 edition:

• On page 13, the boundary of the bidisk is $\partial {\mathbb{D}}^2 = (\partial {\mathbb{D}} \times \overline{{\mathbb{D}}}) \cup (\overline{{\mathbb{D}}} \times \partial {\mathbb{D}})$. The closures were missing.
• Also on page 13, Exercise 1.1.3. Part b) is not correct as stated. I will just remove part b) from next version.
• On page 24, in Rothstein's theorem and also in Exercise 1.4.4, the hypothesis of "holomorphic" is missing.
• On page 35, Exercise 2.1.2, it should say every connected component is a domain of holomorphy as the intersection could be disconnected.
Thanks to Jianou Zhang for pointing out most of these.

November 19th 2014 edition:

• Exercise 1.2.9: The sequence of functions must be nowhere zero as in the standard corollary to Hurwitz from one variable, otherwise there are easy counterexamples.
• Exercise 1.5.1: There is an extra not, that is, "if U is not unbounded" should be "if U is unbounded". The hint hopefully makes it clear what was meant.
• Page 45, definition of $T_p^{(1,0)}M$, the right hand side should have intersection with $T_p^{(1,0)} {\mathbb C}^n$ of course, not M. Same for (0,1) vectors.
• Page 31, at the end of proof of Theorem 1.6.1, Exercise 1.6.3 is used and was forgotten to be mentioned, that is the complement of $g^{-1}(0)$ is connected.

September 2nd 2014 edition:

These are mostly minor typos. There were also a bunch of english typos or hard to understand sentences that are now fixed.

• In Example 2.3.6, for $z_j$ (little $z$) the $j$ runs from $1$ to $n-1$, not $n$.
• Theorem 2.4.14 as stated only makes sense for proper subsets of ${\mathbb C}^n$, that is the boundary of $U$ better be nonempty.
• In proof of Theorem 2.4.16 The $j$ should run from 2 to $n-1$.
• In proof of Proposition 3.1.5, clearly $f$ cannot be defined where $z\zeta = -1$, not 1.