# Tasty Bits of Several Complex Variables: Changes

December 23rd 2020 edition (version 3.4):

• Definition of piecewise-$C^1$ boundary is now precise.
• In Cauchy-Pompeiu use the derivative in $\bar{\zeta}$ rather than $\bar{z}$. It means the same thing, and it matches the proof.
• Better definition of the set of poles.
• Minor other wording improvements and clarifications throughout.
• Fix errata.

May 28th 2020 edition (version 3.3):

• An extra example of the density of regular points after 1.6.2
• In 2.2, add sentence about possibly leaving out "real" before hypersurface, but try to use "real" everywhere if needed.
• State Exercise 2.3.13 in a simpler way asking for the function to just be holomorphic in $W$. It is asking for something slightly stronger, but that is the way to really solve it anyway, and it should be easier this way
• In Exercise 2.3.17, there is no need to assume f is not identically zero and $U$ is a domain.
• Add hint to Exercise 2.4.7, it was probably a tiny bit too hard if we do not have all the harmonic function machinery.
• Improve wording of Exercise 6.8.3 to be more precise.
• Rewrite bits of the basic notations appendix.
• In the definition of $C^1$ path assume that derivative is nonzero.
• More reasonable wording of the argument principle.
• Several minor clarifications.
• A bunch of other minor improvements in language and grammar.
• Fix errata.

October 1st 2019 edition (version 3.2):

• Move Exercise 2.2.5 just past Example 2.2.7 since that makes a bit more sense (no numbers were changed, but it appears one page later).
• In Exercise 5.2.6, let $U$ be either the unit ball or the unit polydisc. The original was way too hard and didn't immediately apply to help with 5.2.8
• Add the statement of completeness of the Bergman space into Lemma 5.2.1 as that makes more sense, and essentially it just moves where the QED symbol appears.
• Use "zero" consistently instead of mixing "zero" and "root" in chapter 6.
• Minor improvements in wording, style, and grammar.
• Fix errata.

May 21st 2019 edition (version 3.1):

• Minor clarifications, style and grammar fixes throughout.
• In analytic continuation definition explicitly state that the path begins at p, although it seems implied.
• Fix errata.

May 6th 2019 edition (version 3.0):

The main goals of this revision are:

• Fill in some more of the details to make the course more accessible.
• Add more figures where appropriate.
• Reorganize some of the more confusing sections, to make the flow more logical.
• Reword some of the results.
• Make some results that were just mentioned in text or exercises into proper propositions/theorems.
• Fix any issues found along the way, and simplify a few proofs.
• Add one or two concepts (Levi-flat and solution of the Levi problem for Reinhardt domains), but really, there's not much new material in this edition, mostly just more motivation for what we are doing, and hopefully clearer (and cleaner) exposition.
• Add appendices on basic theorems in one complex variable, on Stokes theorem and differential forms (without proofs), and algebra.
The primary changes are listed below. These are only the somewhat larger changes. Lots of small minor improvements have been made throughout.
• New slightly different font
• Theorems, exercises, and even some sections have been renumbered due to some of the reordering, the new material, and the new exercises. This was unavoidable.
• A bunch of places used "domain" where "open set" is just as good. So try to use domain only when connectedness is needed.
• Slightly more general Cauchy-Pompeiu
• Add Re/Im notation in 0.1
• Make difference quotient in 0.1 look like the one in 1.1 for consistency.
• Add definition $C^k$ and $C^\infty$ and smooth in 0.1 since its needed all over.
• Include "minimum principle" in the max principle for harmonic functions in 0.1
• Add a bit more algebra detail and make sure to point out the rings are commutative.
• Move all the $O(\ell)$ explanation into one place, and explain the difference between the more standard big-oh notation vs our shorthand
• Add picture of unit ball in two dimensions
• Fill in some more details in 1.1.
• Add more on the multiindex notation as it is used to hopefully make the computations less mysterious.
• Use the "reduce to one variable proof" as the main one for converse of the power series theorem, and add one more alternative proof with an exercise
• Make the Cauchy estimates and computation of coefficients into a proposition in 1.2
• Say "local maximum" for the maximum principle ... why not.
• Makes more sense to define complete Reinhardt domains with closed polydiscs, so do that
• In 1.3, move the implicit function theorem to the end of the section (past chain rule for mappings and the proposition on the determinant of the Jacobian), it is much more natural to prove it then.
• Add figure to proof of Rothstein's theorem in 1.4
• Add a little bit of explanation of the computation for $f^\ell$ in 1.5
• Add an exercise to 1.5
• Add examples for analytic sets in 1.6.
• Avoid unnecessary use of Sard's theorem in the theorem "one-to-one implies biholomorphic" (was 1.6.3)
• Add a short blurb about convexity right at the beginning of 2.1
• Improve figure for Hartogs figure.
• Add part b) to exercise 2.2.2 for any function vanishing on $M$
• Add a blurb about tangent space being derivatives along $M$ and give a short example
• Split up the numbered definitions in 2.2 a bit more.
• Add proposition for the "inertia of Levi form" does not depend on the defining function and relevant exercise now just says to prove this proposition.
• Similarly, add the "inertia is preserved" to the change of variables theorem (was 2.3.7) instead of just making it a remark after the theorem
• Add discussion in 2.3 of the Levi form in terms of graph coordinates before the Lewy hypersurface example
• Add figure of the Lewy hypersurface
• Simplify the lemma normalizing the Levi form. There is really no reason for the derivatives.
• Add definition of Levi-flat to end of 2.3 and add related exercises
• Computation of the complex Hessian under change of coordinates is done with different variable names on source and target for more clarity
• Add graph of Poisson kernel for a few $r$
• Add a figure for $n=1$ for definition of subharmonic function where it is just the convex function
• Make the exercise about the (weakly/strongly) pseudoconvex points more logical, and include 3 different domains
• Move the solution of Dirichlet problem up in 2.4 so that it can be used.
• Add a remark about how pluriharmonic and plurisubharmonic functions are in some sense the correct several complex variable generalizations of affine linear and convex functions.
• Rework and add exercises in 2.4
• Add a note about convexity and example of the hull in section 2.5.
• Rework and add exercises in 2.6
• State the solution of the Levi problem as a theorem in 2.6 even though we do not prove it.
• Add definition of logarithmically convex complete Reinhardt domain and a series of exercises that prove the Levi problem for complete Reinhardt domains after Cartan-Thullen
• Rename chapter 3 to "CR functions" from "CR geometry" since we really did quite a bit of "CR geometry" in chapter 2 already, in chapter 3 it is CR functions specifically that we are interested in.
• Add a bit more explanation of the complexification and the abuse of notation for treating $\bar{z}$ as an independent variable in the context of real-analytic functions.
• In 3.2, explain taking derivatives of functions only defined on a hypersurface, and add a relevant exercise.
• In the proposition that a restriction of a holomorphic function is a smooth CR function, use the directly definition, it is easier.
• In 3.2 add the "reality condition" the complexified graph proposition, and with it fix the gap showing that the vector fields given give the CR vector fields
• In the complexified graph proposition, define complexification of $M$ and also state that functions vanishing on $M$ vanish on the complexification, which is left as an exercise.
• Add a short example of the complexified graph result.
• Add a short one dimensional example of extending a function from a circle.
• Make Severi's theorem a "Theorem" rather than just "Proposition"
• In 3.2, prove the proposition "zero on the boundary means zero inside" including in one dimension (using Radó), and add a figure.
• Add a couple of exercises in 3.2, renumbering others
• Split off the $d(f \wedge dz) = 0$ claim as a lemma before Baouendi-Treves
• Add two figures in the proof of Baouendi-Trèves
• Simplify proof of Lewy a tiny bit, and improve notation a bit.
• Add remark with a reference to Lewy's example to end of 3.4.
• Explicitly say that the exterior derivative in $z$ and $\bar{z}$ is the same as in $x$ and $y$ before the proof of Pompeiu.
• Add figure to proof of Cauchy-Pompeiu
• Remove an exercise in 4.3 that was just wrong.
• Add new exercises to 4.3
• Reorganize 6.1, including renaming it to "The ring of germs", adding exercises, and reordering some of the other exercises to be more logical.
• Add figure for Weierstrass preparation theorem, and add some detail to the proof.
• Explicitly state that roots of P are inside D for the Weierstrass preparation
• Add more complete explanation of what symmetric functions are
• Add explanation of why simply connectedness is important for global definition of roots after Proposition 6.2.6
• Add some exercises in 6.2
• Split 6.2 into two sections
• Add some examples in 6.2
• Add a note on the division theorem from algebra
• Add exercises in what is now 6.3
• Describe better what geometrically distinct is and define geometrically unique explicitly
• Make the D in the dependence on roots propositions and the discriminant theorem be a domain (bounded in the case of the discriminant theorem
• Improve (strenghten) the statement of the theorem on the existence of discriminant, add a figure to the proof and give a more detailed proof of the slightly stronger theorem.
• Rename what was 6.3 (now 6.4) to the "Properties of the ring of germs"
• Move the ideal definitions and exercises to what is now 6.4.
• Add a couple of simple examples in what was 6.4 (now 6.5).
• Add note (and exercise) on well-definedness of dimension, and some rewording in what was 6.4 (now 6.5).
• Add exercises in what was 6.4 (now 6.5)
• What was 6.4.1 is now a separate section (that was really an oversight, that should have been a separate section to begin with)
• Add examples in what is now 6.5, and move the cusp figure up to where the example first appears now
• Simplify definition of regular point to jive with what was used previously in 1.6 (just relabeling coordinates)
• Add exercises in what is now 6.5
• Split out the hypervariety results to a separate section in chapter 6
• Use $\zeta$ instead of $w$ in the Segre variety section to avoid confusion
• Add example and exercise in Segre variety section
• Add a very short Appendix A on basic notation, which may not be totally standard
• Add Appendix B on one complex variable stuff
• Add Appendix C on differential forms and Stokes' theorem
• Add Appendix D on basic algebra notation and results
• And many other minor (and some not so minor) clarifications and improvements that were either too small to note down, or that I forgot to note.

October 11th 2018 edition (version 2.4):

1. Change definition of holomorphic to be the more standard one ($f$ locally bounded and complex differentiable in each variable, giving the Cauchy-Riemann definition only for continuously differentiable functions as an alternative) this doesn't require a deeper one-variable result (that would not appear in a basic one-variable book). Exercise 1.1.4 now makes more sense with this definition.
2. Be a bit more formal with the definition of one-variable holomorphic function, and define it with the complex derivative to make the one variable section aligned with the new definition in the first chapter.
3. Be more explicit and careful with "uniformly absolute" convergence of series and when meaning "uniformly absolute convergence" always state it that way (even though we say that the only type of convergence in several variables is absolute, so it can't mean anything else).
4. Add definition of $O(\ell)$ when first used.
5. All links are https now.
6. Minor clarifications and style fixes throughout.
7. Fix errata.

June 27th 2018 edition (version 2.3):

1. Fix Exercise 1.2.11, and add part c).
2. Fix Exercise 1.4.1, and add part c).
3. As people have different opinions about what "strongly (pseudo)convex" means for unbounded domains, only define the term for bounded.
4. Improve exposition very slightly throughout.
5. Fix errata.

November 29th 2017 edition (version 2.2):

1. Reword the hypotheses of proposition 6.2.5, 6.2.6 and theorem 6.2.7
2. Move exercise 4.3.4 to section 4.2 so renumbered to 4.2.4. It makes a lot more sense right after 4.2.3. (Thanks to John Treuer for the suggestion)
3. Improvements and fixes in language and style in a number of places.
4. Fix errata.

March 21st 2017 edition (version 2.1):

2. Fix wording of exercise 6.4.16, $\ell_j$ are not needed
3. Fix definition of meromorphic functions to be the standard one, noting the deep result of Oka that shows what localy a quotient means globally a quotient on domains in ${\mathbb C}^n$.
4. Very minor improvements in style and exposition in a few places
5. License changed to dual CC-BY-SA and CC-BY-NC-SA
6. Fix errata.
7. Fix few minor typos

May 5th 2016 edition (version 2.0):

• Add chapters 5 (integral kernels) and 6 (varieties).
• Split the harmonic (plurisubharmonic) section (2.4) from Hartogs pseudoconvexity (new 2.5). That of course moves 2.5 to 2.6.
• Add Rado's theorem to the section on harmonic functions section 2.5.
• Add a few corollaries to Hartogs phenomenon.
• Make implicit function theorem a theorem rather than just an exercise.
• Add exercises throughout, and reorder some exercises to fix dependencies between them.
• Do some minor reordering in the early chapters.
• Add more detail about orientation in ${\mathbb C}^n$.
• Add diagram for proof of the Riemann extension theorem.
• Add diagram for proof of the tomato can principle.
• Add diagram for proof of the local nature of Hartogs pseudoconvexity (Lemma 2.5.7)
• Add diagram for proof of Lewy extension theorem.
• Add more historical context in several places.
• Lots of small minor improvements in style and exposition, some minor reordering and reworking of proofs and a few extra clarification in many places.
• Add more terms to the index.
• Fix errata.

November 24th 2015 edition:

• Fix errata and several typos.

August 21st 2015 edition:

• Improvements in exposition, a few better pagebreaks.
• Replace "vanishes to order k" with "is O(k)" as the former seems to be not uniformly agreed upon if it is O(k) or O(k+1), wheras the latter is unambiguous.
• Add definition of "automorphism" and "automorphism group"
• Fix errata and several typos.

November 19th 2014 edition:

Improvements in exposition in a bunch of places and also fix errata.

September 2nd 2014 edition:

• Minor improvements in exposition.
• Add exercises 1.2.9 and 2.4.25 (which moves 2.4.25 to number 2.4.26).

May 1st 2014 edition:

First version