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Tasty Bits of Several Complex Variables: Changes

May 20th 2025 edition (version 4.2):

This is a minor version to fix errata and a couple of minor changes (one new exercise).

1. Add footnote to Exercise 2.2.1 that it works for smooth functions too by using Taylor's theorem so that the reader doesn't think it is only for polynomials.
2. Add Exercise 2.5.16.
3. In 4.4, when defining \((p,q)\)-forms, note that the coefficients are smooth functions on some open set \(U .\)
4. In 4.5, in the first sentence defining complex affine subspace, avoid trying to be cute and short and just give the standard, though slightly longer definition. The original had the issue that we really should have said that \(c\) is in the image of \(M.\)
5. Exercise 4.5.3 changed due to an erratum, some extra hypothesis on the functions is necessary, such as their first derivatives vanishing at the origin.
6. In 4.6, when describing the relationship of Cousin II to Weierstrass, add a footnote mentioning that the reason we assume nonvanishing of the derivative is only for simplicity (we don't know anything about varieties in chapter 4 yet).
7. Before Exercise 5.2.1, cite Wiegerinck for the results stated without proof.
8. In Example 6.8.3, mention at the beginning that we made \(M\) to contain the hypersurface \(w=0 .\)
9. Add "principal part" definition and the Mittag-Leffler theorem to appendix B, since we do refer to it in the Cousin problems section.
10. Mention that the cohomology groups from differential forms are the de Rham cohomology and add that to the index. Also put the initial trivial map \(d\) into the complex for completeness.
11. Add footnote to the sentence introducing Stokes (Theorem C.6) about the naming.
12. A few other minor clarifications and minor improvements.
13. A few style and language fixes.
14. Fix errata.

June 3rd 2024 edition (version 4.1):

This is a minor new version incorporating some errata fixes, and several minor changes due to some feedback from Richard Lärkäng who taught with the book in the past semester, and a couple of other minor changes (some that I was planning to make for version 4.0 but somehow they slipped through the cracks).

1. Instead of the functional notation for plugging vectors into differential forms, use the angle bracket pairing notation that is more standard and since we don't really use this much at all, there is no reason to have a less standard notation. This affects one place in chapter 1, exercise 4.4.3, and one place in appendix C.
2. The notation \({\mathbb{C}} \otimes T_p M\) is changed to the more common \({\mathbb{C}} T_p M\) which also avoids a slight abuse of notation. Thanks to Richard Lärkäng for the suggestion.
3. Lemma 2.3.9, the statement about the Levi form doesn't really make sense if \(M\) is not a boundary, so change the statement to explicitly say that the inside is "above" the \(M\) in the new coordinates. Also explicitly mention this at the end of the proof. Thanks to Richard Lärkäng for the suggestion.
4. At the beginning of section 2.4, give a reference to the one-variable book as not every student may have had a treatment of harmonic and subharmonic functions in a one-variable course.
5. Add hints to Exercise 2.4.8.
6. Change Proposition 2.4.5, to be about the sub-mean-value property for all small enough discs (which is really the best way to prove the original statement in the first place), leaving the original statement as an "In particular". That also changes Exercise 2.4.11 by really making it a bit easier as it gives a way to do it (and change the hint to that exercise too). Thanks to Richard Lärkäng for the suggestion.
7. In Example 2.5.4, put a square on the norm of \(z\) as that makes the computation easier. Thanks to Richard Lärkäng for the suggestion.
8. In Theorem 2.5.6, add part (iv) which is just the conclusion of the second version of the Kontinuitatssatz as that just follows from the proof directly with no extra work. Thanks to Richard Lärkäng for the suggestion.
9. In Theorem 2.5.6, make part (ii) less wordy: "U is Hartogs pseudoconvex". I mean the definition is just above the theorem!
10. Mark Exercise 2.6.1 as Oka's lemma to make the connection to the "proof" above clear.
11. Change Exercise 4.4.1 to ask also to prove the Leibniz rule (which easily follows from the first part and the real Leibniz rule), but this is good to understand for the computations later in the section. Thanks to Richard Lärkäng for the suggestion.
12. Add Leibniz rule to Appendix C.
13. Simplify the wording of some of the theorems in Appendix E.
14. Fix errata.

December 9th 2023 edition (version 4.0):

This is a major new edition with some new material and many smaller changes throughout. Important: Exercise numberes may have changed!.

Larger changes in general:

• New section 4.4 on the solvability of the \(\bar{\partial}\)-problem in the polydisc.
• New section 4.5 on the extension from an affine subspace and its relation to being a domain of holomorphy as an application of the \(\bar{\partial}\)-problem.
• New section 4.6 on Cousin problems.
• New short appendix on some analysis bits including some useful measure theory results.
• Many minor changes (see below), mostly in chapters 1, 2, 4, and 6, includes new exercises (66 new exercises), new figures.
• A couple of new results in existing sections, another version of the Kontinuitätssatz in section 2.1, prove the existence of regular points (and hence density) of a subvariety in 6.5, and prove that the singular set of a hypervariety is a subvariety in section 6.6.
• New font, more readable (shorter lines).
• Fix errata.

Detailed list of the smaller changes.

• Move the definition of \((p,q)\)-form and its derivative to section 4.4 as that's where it's needed first.
• Add a note about \(\log \lvert f \rvert\) being harmonic after Theorem 0.1.3
• Add equivalent definition of "locally bounded" to the footnote.
• In proof of Proposition 1.1.3, emphasize that (1.1) means \(f\) is continuous on \(U,\) not just \(\Delta,\) to avoid worrying about measurability. Improve the proof to only use the standard Leibniz rule for Riemann integrals.
• Add Exercise 1.1.8 on boundedness and extension through a point, and add relevant note to the corollary of Hartogs figure.
• Add a note after Theorem 1.2.1 to note that continuity up to the boundary is not really necessary.
• Improve the wording of the definition of Reinhardt domain.
• Add Exercises 1.2.3 and 1.2.4. This renumbers the exercises in 1.2.
• Make Exercise 1.2.9 (domain of convergence is a complete Reinhardt domain) into a proposition (keeping the exercise, but just stating it as a proposition), that flows better. It renumbers the following Theorems/Propositions/Lemmas in 1.2.
• After the identity theorem, note the difference with the one variable identity theorem, and give a quick example of why it is not so simple.
• Exercise 1.2.12 changed to use the unit ball instead of the polydisc.
• When introducing the Jacobian put "holomorphic" in parentheses and mention that we will not say so explicitly if the mapping is holomorphic.
• Add Exercise 1.3.7 (vector-valued maximum principle)
• In the proof of Lemma 1.4.7, emphasize where bounded is being used.
• In the proof of Theorem 1.5.1, make the plugging into \(f_k\) a bit more specific with displayed equations. Also introduce "higher order terms" earlier, and use it in the end instead of making up new names for them.
• Note that all Reinhardt domains are circular after Corollary 1.5.2.
• New Exercise 1.5.10.
• New Exercises 1.6.1, 1.6.2, 1.6.3, and 1.6.6. Renumbers rest of the exercises in 1.6.
• At the end of Example 1.6.3, make the zero set be the same as that for the first two functions in the example.
• Rename 2.1 to "Domains of holomorphic & holomorphic extension"
• New Exercises 2.1.4 and 2.1.5, renumbers rest of the exercises in 2.1
• New Exercise 2.1.15
• New Theorem 2.1.7, the extension version of the continuity principle, the Kontinuitatssatz. Name the two version "first version" and "second version."
• Adds Exercises 2.1.16, 2.1.17, and 2.1.18
• Remove Exercise 2.2.1 as it's now covered by 2.1.4, and add a different Exercise 2.2.1 after definition 2.2.2.
• After Exercise 2.2.2, after the computation showing that \(X_p f\) depends only on values on \(M,\) explicitly note what a smooth function on a nonopen set is.
• Add Proposition 2.2.6 about where tangent spaces are taken. Renumbers following propositions, definitions, and examples in 2.2.
• Change Lemma 2.2.9 to Proposition to be consistent with 2.3 where the analogous result is a Proposition.
• Add the explicit form of \(T_0 M\) into the statement of Proposition 2.2.9.
• Add blurb about translations and matrices preserving convexity and add Exercise 2.2.8 to prove it, renumbers the rest of the exercises in 2.2.
• Add Exercise 2.2.11
• Pull out the definition of the complexified real derivative to the before the statement of Proposition 2.3.1.
• Rename 2.3 to "Holomorphic vectors, Levi form, pseudoconvexity"
• Add Exercises 2.3.2 and 2.3.3, renumbers the rest in 2.3.
• Make Proposition 2.3.2 be more general, not just for biholomorphisms.
• Add a quick remark after Proposition 2.3.3 to notice how the z and w variables are handled in \(\varphi .\)
• In Exercise 2.3.4 ask to show that the new full Hessian matrix is also Hermitian.
• New Exercises 2.3.5-2.3.7
• In 2.3 add a bit more on the relationship of the positivity/eigenvalues of of the real and complex Hessians. (mainly the exercises)
• The real Hessian in terms of the \(z\)s and \(\bar{z}\)s has now the block rows flipped, that is really the correct way to look at it so that it is hermitian.
• When saying how does a complex linear change of variables acts on the Hessian, mention that we mean by the derivative \(D_{\mathbb{C}} A .\)
• In the discussion before Theorem 2.3.8, use L instead of H for the complex Hessian for consistency.
• In Exercise 2.3.13, require that the dimension is at least two, otherwise it is too trivial.
• Improve the discussion after Narasimhan's lemma (2.3.10)
• Change Exercise 2.3.14 to address an erratum: First show that the \(a\) and \(b\) matrices can be made symmetric.
• In the proof of Theorem 2.3.11, when writing out the analytic discs, mention that \(\xi\) is in the disc.
• After the proof also mention that the proof could be done with those discs using the first version of the continuity principle (new in this edition).
• Rename 2.4 to "From harmonic to plurisubharmonic functions"
• Make Exercise 2.4.11 into a proposition as we use it without mentioning it much often. Leave the proof still as an exercise. Renumbers rest of the propositions in 2.4.
• Move Exercise 2.4.27 up and make it Exercise 2.4.22 as it makes a lot more sense at that point. Renumbers the following exercises.
• Make the composition of holomorphic and psh function being psh an explicit corollary as it is important in the exposition. Still leave the proof as an exercise.
• Clarify the proof of Radó's theorem.
• Start 2.5 with a better introduction to why we consider hulls.
• In 2.5, add a footnote about why we won't define convex functions on arbitrary domains.
• In Exercise 2.5.15, it is much easier to use exercise 2.5.9 than the "hint" in text, so replace that hint.
• In Exercise 2.5.13, require that the disc only goes out of \(U\) at the origin, which is really the only reasonable way to do this.
• In the proof of Theorem 2.5.8, label the estimates and reword the proof little bit to make it clearer.
• Move Exercise 2.6.13 (Behnke-Stein) past Cartan-Thullen. It is much easier then.
• In the proof of Cartan-Thullen, don't use strict inequalities which aren't needed
• In 3.1, when looking at convergence of real power series, use open product of intervals to start with for consistency.
• Example 3.1.8, remove the square, it's not necessary
• Be more precise in exercise 4.1.3 to mention that the wanted function is continuous up to the boundary (so that it has boundary values).
• Rename 4.2 to be a bit more descriptive.
• Ask a little bit more in 4.2.2, that only the second form allows the differentiation under the integral.
• New part b) to Exercise 5.2.3.
• After definition 6.1.3, use A and B for the sets not X and Y, and also mention set complement for germs. So ask about complement in Exercise 6.1.7.
• Define "unit" in 6.2
• Add Figure 6.1 to proof of the Weierstrass preparation theorem.
• Reword the proof of Weierstrass preparation theorem somewhat to be clearer, and add power sum to the index.
• Add Figure 6.3 to Exercise 6.2.4 (the square root example).
• Mention that the formulas relating the symmetric functions and power sums are called the Newton identities or Girard-Newton formulas.
• After the Weierstrass preparation theorem when we remark it is a generalization of the implicit function theorem, say it is a k-valued solution rather than k solutions, as that could be misinterpreted.
• Before the Weierstrass division theorem, give the one-variable version of it.
• Add a remark about how the preparation and division theorems are really theorems about formal power series.
• At the beginning of section 6.3, when talking about geometrically distinct zeros, use \(\zeta\) for the (single) variable instead of \(z_n .\)
• In the proof of Proposition 6.3.1, mention that WLOG we are assuming that \(\alpha(0)=0 .\)
• Clarify the proof of Theorem 6.3.3, explicitly write out the argument principle, use \(\gamma_j\) instead of \(\gamma'\) as that could be misinterpreted, and make the figure in the proof more correct as the curves are picked at \(p'\) not any \(z'.\)
• Make the definition of ideal in 6.4 clearer.
• Add a quick definition of irreducible just above Theorem 6.4.2 for completeness. Also state Theorem 6.4.2 for nonunits only as to avoid having to think of what does it mean for a unit.
• Put the definition of radical into the main text rather than as a footnote for more consistency (and add it to the index).
• Use \(V_p(I)\) instead of \(V(I)\) as we are doing the germ notion and that's more consistent.
• Add two basic exercises about varieties to 6.5, so exercises are renumbered.
• Simplify the statement in 6.5.4.
• Define the dimension of a subvariety and the concept of pure dimension using only regular points, which seems a little ontologically simpler. Also simplifies the last exercise in 6.5.
• Add the intersection of two complex manifolds to example 6.5.6, and add the relevant figure. Also rename X and Y to U and M so that the naming in the example makes more sense.
• Move the statement of the theorem about \(Z_f\) to section 6.5 from 6.6 (now Theorem 6.5.9), and add the generalization of this result (that regular points exist and are thus dense) to arbitrary varieties, Lemma 6.5.10.
• Add the statement about dimension of singularity being less than the dimension of the variety to Theorem 6.5.11 (which is left without proof).
• In 6.6, what is now Theorem 6.6.2 had a gap (must use that that regular points are dense, so fix this gap).
• Reorganize section 6.6, and add a theorem (Theorem 6.6.5) proving that the singular set of a hypervariety is a subvariety, makes things more complete at least for hypervarieties. Add a figure to go with Theorem 6.6.5 and a couple of exercises to finish the proof. Also a figure to the parabola example. Also one exercise was removed due to an erratum. That all renumbers the exercises in the section.
• Reword the Local parametrization theorem a little bit and explicitly say that E is the discriminant set.
• Exercise 6.7.9 was missing a conclusion to prove. Make it into proving an if and only if for irreducibility, which is really what was meant.
• Add a short note mentioning the resolution of singularities being a more general version of Puiseux.
• Add a figure to Example 6.8.2.
• In section 6.8, add a figure for the definition of the Segre variety.
• In Appendix B, introduce and use winding numbers and describe how things are connected to our statements. That makes things more compatible with most one-variable books.
• Improve Figure C.1 a little.
• In Appendix C add a short note on how to evaluate \(k\)-forms.
• In Appendix D define integral domain and properly assume it for the definitions of PID and UFD.
• Add reference to my one variable book in the reading.
• When computing the Hessian use the vertical bar to evaluate, it may be clearer
• Figures now have numbers, captions, float, and have explicit references in text.
• Replace many j's with k's or similar for aesthetic reasons. Some other letters are changed to be more consistent too.
• Use Hermitian form or Hermitian quadratic form where appropriate rather than sesquilinear which should only be called that if both arguments can vary independently.
• Some figures have been mildly cleaned up.
• Shorten a couple of section titles slightly.
• Tighten the language and improve clarity in many places.
• Fix errata.

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.
• Add more examples throughout.
• Reorganize some of the more confusing sections, to make the flow more logical.
• Reword some of the results.
• Add many exercises.
• 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 exercises 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
• Add exercises 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 exercises in 1.3
• Add figure to proof of Rothstein's theorem in 1.4
• Add exercises 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 exercises to 2.1
• 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 exercises to 2.3.
• 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
• Clarify/simplify the proof of Rado's theorem and add a figure.
• Add a note about convexity and example of the hull in section 2.5.
• Add exercises in 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 exercises in 3.1
• 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
• Add note at end of 3.2 about the "PDE problem" we are about to solve
• 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 exercises in 3.4
• 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
• Add list of notations
• Add Zariski-Samuel to the list of further reading
• 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):

1. Add exercise 6.4.21
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:

• Add exercises 1.6.5, 1.6.6, 1.6.7 about Jacobian conjecture.
• 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

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