diff --git a/CHANGES b/CHANGES index c7f7232..f3891ad 100644 --- a/CHANGES +++ b/CHANGES @@ -1,9 +1,13 @@ Since 3.2 * 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. diff --git a/scv.tex b/scv.tex index 49d438e..94234c4 100644 --- a/scv.tex +++ b/scv.tex @@ -361,7 +361,7 @@ Ji{\v r}\'i Lebl\\[3ex]} \today \\ -(version 3.2) +(version 3.3) \end{minipage}} %\addtolength{\textwidth}{\centeroffset} @@ -379,7 +379,7 @@ \bigskip \noindent -Copyright \copyright 2014--2019 Ji{\v r}\'i Lebl +Copyright \copyright 2014--2020 Ji{\v r}\'i Lebl %PRINT % not for the coil version @@ -2675,8 +2675,9 @@ \section{Derivatives} \end{remark} \begin{defn} -Let $U \subset \C^n$ be an open set. A mapping $f \colon U \to \C^m$ -is said to be holomorphic if each component is holomorphic. That +Let $U \subset \C^n$ be open. A mapping $f \colon U \to \C^m$ +is said to be \emph{holomorphic}\index{holomorphic mapping} +if each component is holomorphic. That is, if $f = (f_1,\ldots,f_m)$, then each $f_j$ is a holomorphic function. \end{defn} @@ -2713,9 +2714,9 @@ \section{Derivatives} For holomorphic functions the chain rule simplifies, and it formally looks like the familiar vector calculus rule. Suppose again -$U \subset \C^n$ and $V \subset \C^m$ are open sets and -$f \colon U \to V$, and $g \colon V \to \C$ are holomorphic. -Again name the variables +$U \subset \C^n$ and $V \subset \C^m$ are open, and +$f \colon U \to V$ and $g \colon V \to \C$ are holomorphic. +Name the variables $z = (z_1,\ldots,z_n) \in U \subset \C^n$ and $w = (w_1,\ldots,w_m) \in V \subset \C^m$. In formula \eqref{eq:chainrule} for the $z_j$ derivative, the $\bar{w}_j$ derivative of $g$ is zero and the $z_j$ derivative of @@ -3357,10 +3358,10 @@ \section{Cartan's uniqueness theorem} \begin{equation*} f(z) = z + f_k(z) + \sum_{j=k+1}^\infty f_j(z) , \end{equation*} -where $k \geq 2$ is an integer such that $f_j(z)$ is zero for all -$2 \leq j < k$. The degree-one homogeneous part is simply the vector $z$, +where $k \geq 2$ is an integer such that $f_2(z),f_3(z),\ldots,f_{k-1}(z)$ is zero. +The degree-one homogeneous part is simply the vector $z$, because -the derivative of the mapping at the origin is the identity. +the derivative of $f$ at the origin is the identity. Compose $f$ with itself $\ell$ times: \begin{equation*} f^\ell(z) = \underbrace{f \circ f \circ \cdots \circ f}_{\ell\text{ times}} @@ -4304,9 +4305,9 @@ \section{Tangent vectors, the Hessian, and convexity} \end{exbox} In particular, the exercise says that -given any domain $U \subset \C$ and any domain $V \subset \C$, the domain +for any domains $U \subset \C$ and $V \subset \C$, the set $U \times V$ is a domain of holomorphy in $\C^2$. The domains -$U$ and $V$, and therefore $U \times V$ can be spectacularly nonconvex. +$U$ and $V$, and hence $U \times V$, can be spectacularly nonconvex. But we should not discard convexity completely, there is a notion of \emph{pseudoconvexity}, which vaguely means ``convexity in the complex directions'' and is the correct notion to distinguish @@ -4314,23 +4315,23 @@ \section{Tangent vectors, the Hessian, and convexity} Let us figure out what classical convexity means locally for a smooth boundary. \begin{defn} \label{def:hypersurface} -A set $M \subset \R^n$ is a real +A set $M \subset \R^n$ is a \glsadd{not:Ck}% $C^k$-smooth \emph{\myindex{hypersurface}}% \index{Ck-smooth hypersurface@$C^k$-smooth hypersurface} if at each point $p \in M$, there exists a $k$-times continuously -differentiable function $r \colon V \to -\R$, defined in a neighborhood $V$ of $p$ with nonvanishing derivative +differentiable function $r \colon V \to \R$ +with nonvanishing derivative, defined in a neighborhood $V$ of $p$ such that $M \cap V = \bigl\{ x \in V : r(x) = 0 \bigr\}$. The function $r$ is -called the \emph{\myindex{defining function}} (at $p$). +called the \emph{\myindex{defining function}} of $M$ (at $p$). -An open set (or domain) $U$ with +An open set (or domain) $U \subset \R^n$ with \emph{$C^k$-smooth boundary}% \index{Ck-smooth boundary@$C^k$-smooth boundary} is a set where $\partial U$ is a $C^k$-smooth hypersurface, -and for every $p \in \partial U$ there is a defining +and at every $p \in \partial U$ there is a defining function $r$ such that $r < 0$ for points in $U$ and $r > 0$ for points not in $U$. @@ -4350,9 +4351,7 @@ \section{Tangent vectors, the Hessian, and convexity} What we really defined is an \emph{\myindex{embedded hypersurface}}. In particular, in this book the topology on the set $M$ will be the subset -topology. - -For simplicity, in this book we generally deal with smooth +topology. Furthermore, in this book we generally deal with smooth (that is, $C^\infty$) functions and hypersurfaces. Dealing with $C^k$-smooth functions for finite $k$ introduces technicalities that make certain theorems and arguments unnecessarily difficult. @@ -4380,7 +4379,8 @@ \section{Tangent vectors, the Hessian, and convexity} In $\C^n$ a hypersurface defined as above is a \emph{\myindex{real hypersurface}}, to distinguish it from a complex hypersurface that would be the zero set of -a holomorphic function. +a holomorphic function, although we may at times leave out the word ``real'' +if it is clear from context. \begin{defn} For a point $p \in \R^n$, the set of \emph{tangent vectors}\index{vector} $T_p \R^n$ is given by @@ -4403,10 +4403,12 @@ \section{Tangent vectors, the Hessian, and convexity} $T_p \R^n$ and $T_q \R^n$ are distinct spaces. \glsadd{not:evalpartial}% -An object $\frac{\partial}{\partial x_j}\Big|_p$ is a linear functional +An object +$\frac{\partial}{\partial x_j}\big|_p$ +is a linear functional on the space of smooth functions: When applied to a smooth function $g$, it gives -$\frac{\partial g}{\partial x_j} \Big|_p$. Therefore, $X_p$ is also such a +$\frac{\partial g}{\partial x_j} \big|_p$. Therefore, $X_p$ is also such a functional. It is the directional derivative from calculus; it is computed as $X_p f = \nabla f|_p \cdot (a_1,\ldots,a_n)$. @@ -4467,9 +4469,9 @@ \section{Tangent vectors, the Hessian, and convexity} Show that $T_pM$ is independent of which defining function we take. That is, prove that if $r$ and $\tilde{r}$ are defining functions for $M$ at $p$, then -$\sum_j a_j \frac{\partial r}{\partial x_j} \Big|_p = 0$ +$\sum_j a_j \frac{\partial r}{\partial x_j} \big|_p = 0$ if and only if -$\sum_j a_j \frac{\partial \tilde{r}}{\partial x_j} \Big|_p = 0$. +$\sum_j a_j \frac{\partial \tilde{r}}{\partial x_j} \big|_p = 0$. \end{exercise} \end{exbox} @@ -4590,9 +4592,8 @@ \section{Tangent vectors, the Hessian, and convexity} (3a+4b)\frac{\partial}{\partial y_2}\big|_0$, where we let $(y_1,y_2)$ be the coordinates on the target. You should check on some test function, such as -$\varphi(y_1,y_2) = \alpha y_1 + \beta y_2$, that this satisfies the -definition. - +$\varphi(y_1,y_2) = \alpha y_1 + \beta y_2$, that the definition above is +satisfied. \medskip @@ -5034,7 +5035,7 @@ \section{Holomorphic vectors, the Levi form, and pseudoconvexity} One can also define vector fields in these bundles. Let us describe $\C \otimes T_pM$ -for a real smooth hypersurface $M \subset \C^n$. +for a smooth real hypersurface $M \subset \C^n$. Let $r$ be a real-valued defining function of $M$ at $p$. A vector $X_p \in \C \otimes T_p\C^n$ is in @@ -5672,12 +5673,12 @@ \section{Holomorphic vectors, the Levi form, and pseudoconvexity} eigenvalues of $D^*HD$ is the same as that for $H$. The eigenvalues might have changed, but their sign did not. We are only considering $H$ and $D^*HD$ on a subspace. In linear algebra -language, suppose $D$ is invertible and consider a subspace $T$ and its -image $DT$. Then the inertia of $H$ restricted to $DT$ is the same +language, consider an invertible $D$, a subspace $T$, and its image $DT$. +Then the inertia of $H$ restricted to $DT$ is the same as the inertia of $D^*HD$ restricted to $T$. -Let $M$ be a smooth hypersurface given by $r=0$, then $f^{-1}(M)$ is -a smooth hypersurface given by $r \circ f = 0$. +Let $M$ be a smooth real hypersurface given by $r=0$, then $f^{-1}(M)$ is +a smooth real hypersurface given by $r \circ f = 0$. The holomorphic derivative $D = Df(p)$ takes $T_{p}^{(1,0)}f^{-1}(M)$ isomorphically to $T_{f(p)}^{(1,0)}M$. @@ -6137,7 +6138,7 @@ \section{Holomorphic vectors, the Levi form, and pseudoconvexity} \end{exbox} A hyperplane is the ``degenerate'' case of normal convexity. -There is also a flat case of pseudoconvexity. A smooth hypersurface +There is also a flat case of pseudoconvexity. A smooth real hypersurface $M \subset \C^n$ is \emph{\myindex{Levi-flat}} if the Levi form vanishes at every point of $M$. The zero matrix is positive semidefinite and negative semidefinite, so both sides of $M$ are pseudoconvex. @@ -6155,7 +6156,7 @@ \section{Holomorphic vectors, the Levi form, and pseudoconvexity} \end{exercise} \begin{exercise} -Let $f \in \sO(U)$ for some domain $U \subset \C^n$, $f \not\equiv 0$. +Consider $f \in \sO(U)$ for an open $U \subset \C^n$. Let $M = \bigl\{ z \in U : \Im f(z) = 0 \bigr\}$. Show that if $df(p) \not=0$ for some $p \in M$, then near $p$, $M$ is a Levi-flat hypersurface. @@ -8760,7 +8761,7 @@ \section{CR functions} \begin{exbox} \begin{exercise} -Suppose $M \subset \C^n$ is a smooth hypersurface +Suppose $M \subset \C^n$ is a smooth real hypersurface and $f \colon M \to \C$ is a CR function that is a restriction of a holomorphic function $F \colon U \to \C$ defined in some neighborhood $U \subset \C^n$ of $M$. Show that $F$ is unique, @@ -8770,7 +8771,7 @@ \section{CR functions} \begin{exercise} Show that there is no maximum principle of CR functions. In fact, find a -smooth hypersurface $M \subset \C^n$, $n \geq 2$, and a smooth CR function +smooth real hypersurface $M \subset \C^n$, $n \geq 2$, and a smooth CR function $f$ on $M$ such that $\sabs{f}$ attains a strict maximum at a point. \end{exercise} @@ -9514,7 +9515,7 @@ \section{Approximation of CR functions} \subimport*{figures/}{cylinder-bt.pdf_t} \end{myfig} -We orient $D$ in +Orient $D$ in the standard way as if it sat in the $(x,t)$ variables in $\R^n \times \R$. Stokes' theorem says \begin{equation*} @@ -10106,7 +10107,7 @@ \section{Extension of CR functions} \end{exercise} \begin{exercise} -A smooth hypersurface $M \subset \C^3$ is defined by $\Im w = \sabs{z_1}^2-\sabs{z_2}^2 + O(3)$ +A smooth real hypersurface $M \subset \C^3$ is defined by $\Im w = \sabs{z_1}^2-\sabs{z_2}^2 + O(3)$ and $f$ is a real-valued smooth CR function on $M$. Show that $\sabs{f}$ does not attain a maximum at the origin. \end{exercise} @@ -10153,7 +10154,7 @@ \section{Extension of CR functions} \begin{exercise} Prove the third item in the Lewy extension theorem without the use of the tomato can principle. That is, prove in a more elementary -way that if $M \subset U \subset \C^n$ is a smooth hypersurface +way that if $M \subset U \subset \C^n$ is a smooth real hypersurface in an open set $U$ and $f \colon U \to \C$ is continuous and holomorphic in $U \setminus M$, then $f$ is holomorphic. \end{exercise} @@ -13133,7 +13134,7 @@ \section{Varieties} \label{sec:varieties} \bigr\} . \end{equation*} If $(X,p)$ is a germ and $X$ a representative, -we say the dimension of $(X,p)$ is the dimension of +the \emph{dimension} of $(X,p)$ is the dimension of $X$ at $p$. The dimension of the entire subvariety $X$ is defined to be @@ -13219,8 +13220,8 @@ \section{Hypervarieties} \label{section:hypervarieties} \begin{thm} \label{thm:codim1var} If $(X,p)$ is a germ of a pure codimension 1 subvariety, then -there is a germ holomorphic function $f$ at $p$ -such that $(Z_f,p) = (X,p)$. Furthermore, $I_p(X)$ is generated by $(f,p)$. +there is a germ of a holomorphic function $f$ at $p$ +such that $(Z_f,p) = (X,p)$. Further, $I_p(X)$ is generated by $(f,p)$. \end{thm} \begin{proof} @@ -13650,9 +13651,10 @@ \section{Irreducibility, local parametrization, and Puiseux theorem} \begin{exbox} \begin{exercise} -Suppose $(X,0) \subset (\C^2,0)$ is an irreducible germ defined +Consider an irreducible germ +$(X,0) \subset (\C^2,0)$ defined by an irreducible Weierstrass polynomial $f(z,w) = 0$ (polynomial in $w$) -of degree $k$. Prove that there exists a holomorphic $g$ such that +of degree $k$. Prove there exists a holomorphic $g$ such that $f\bigl(z^k,g(z)\bigr) = 0$ and $z \mapsto \bigl(z^k,g(z)\bigr)$ is one-to-one and onto a neighborhood of 0 in $X$. \end{exercise} @@ -13726,13 +13728,13 @@ \section{Segre varieties and CR geometry} \label{sec:crgeomcr} \begin{example} Let $M \subset \C^n$ be a smooth real hypersurface containing a complex hypersurface $X$ (zero set of a holomorphic function -with nonzero derivative), at some $p \in X \subset M$. +with nonzero derivative), at $p \in X \subset M$. Apply a local biholomorphic change of coordinates at $p$, so that in the new coordinates $(z,w) \in \C^{n-1} \times \C$, $X$ is given by $w=0$, and $p$ is the origin. The tangent hyperplane to $M$ at 0 contains $\{ w=0 \}$. -After rotating the $w$ coordinate (multiplying it by $e^{i\theta}$), +By rotating the $w$ coordinate (multiplying it by $e^{i\theta}$), we assume $M$ is tangent to the set $\bigl\{ (z,w) : \Im w = 0 \bigr\}$. In other words, @@ -14044,7 +14046,7 @@ \section{Segre varieties and CR geometry} \label{sec:crgeomcr} \begin{equation*} \Im w = g\bigl(\snorm{z}^2\bigr) . \end{equation*} -$M$ is a smooth hypersurface. +$M$ is a smooth real hypersurface. Consider $p = (1,0,\ldots,0) \in M$. For every $0 < s < 1$, let $q_s = (s,0,\ldots,0) \in M$ and $X_s = \bigl\{ (z,w) \in M : w = s \bigr\}$. Each $X_s$ is a local complex subvariety of dimension $n-1$ @@ -14056,7 +14058,7 @@ \section{Segre varieties and CR geometry} \label{sec:crgeomcr} \begin{exbox} \begin{exercise} -Find a smooth compact hypersurface $M \subset \C^n$ that contains a germ +Find a compact smooth real hypersurface $M \subset \C^n$ that contains a germ of a positive dimensional complex subvariety. \end{exercise} \end{exbox}