Lecture notes on several complex variables - math.tamu.edu

1 Math 650, fall 2007 Texas A&M UniversityLecture notes on several complexvariablesHarold P. BoasDraft of December 7, 2007 Contents1 Power series .. Integral representations .. Partial differential equations .. Geometry .. 32 Power Domain of convergence .. Characterization of domains of convergence .. Natural boundaries .. Summary .. 93 Real convexity .. Convexity with respect to a class of functions .. Polynomial convexity .. Linear and rational convexity .. Holomorphic convexity .. Pseudoconvexity .. The Levi problem .. The Levi form .. Applications of the problem.

2 Solution of the -equation on smooth pseudoconvex domains .. 42ii1 IntroductionAlthough Karl Weierstrass studied holomorphic functions of two variables already in thenineteenth century, the modern theory of several complex variables may be dated to theresearches of Friedrich (Fritz) Hartogs (1874 1943) in thefirst decade of the parts of the theory of holomorphic functions the maximum principle, forexample are essentially the same in all dimensions. The most interesting parts of thetheory of several complex variables are the features that differ from the one-dimensional theory is illuminated by several complementary points of view:power series, integral representations, partial differential equations, and geometry.

3 Themulti-dimensional theory reveals striking new phenomena from each of these points Power seriesA one- variable power series converges inside a certain discand diverges outside theclosure of the disc. The convergence region for a two-dimensional power series, however,can have infinitely many different shapes. For instance, the largest open set in which theseries n=0 m=0znwmconverges is the unit bidisc{(z, w) :|z|<1 and|w|<1}, whilethe series n=0znwnconverges in the unbounded hyperbolic region where|zw|< theory of one-dimensional power series bifurcates intothe theory of entire func-tions and the theory of functions on the unit disc. In higher dimensions, studying powerseries already leads to function theory on infinitely many different types of natural question, to be answered presently, is to characterize the domains that areconvergence domains for multi- variable power a two- variable power series whose convergence domain is the unitball{(z, w) :|z|2+|w|2<1}.

4 Hartogs discovered that a function holomorphic in a neighborhood of the boundaryof the unit bidisc automatically extends to be holomorphic on the interior of the bidisc;one can prove this property by considering one- variable Laurent series on slices. Thus,1A student of Pringsheim, Hartogs belonged to the Munich school of mathematicians. Because of theirJewish heritage, both Pringsheim and Hartogs suffered greatly under the Nazi regime in the , a wealthy man, managed to buy his way out of Germany into Switzerland, where hedied at an advanced age in 1941. The situation for Hartogs, however, grew ever more desperate,and in 1943 he chose to end his own life by an overdose of barbiturates rather than to be sent to adeath Introductionin dramatic contrast to the situation in one variable , thereare domains inC2on whichall holomorphic functions extend to a larger domain.

5 A natural question, to be answeredpresently, is to characterize thedomains of holomorphy, that is, the natural domains ofexistence of holomorphic discovery of Hartogs also shows that holomorphic functions of several variablesnever have isolated singularities and never have isolated zeroes, in contrast to the one- variable (z, w) be a polynomial in two variables . Show that if the zero set ofpis compact, thenpis Integral representationsThe one- variable Cauchy integral formula for a holomorphicfunctionfinside a simpleclosed curveCsays thatf(z) =12 i Cf(w)w remarkable feature of this formula is that the kernel (w z) 1is both universal(independent of the domain) and holomorphic in the free variable .

6 There is no suchformula in higher dimensions! There are integral representations with a holomorphickernel, but they depend on the domain, and there is a universal integral representation,but its kernel is not holomorphic. There is a huge literatureabout constructing andanalyzing integral representations for various special types of the special case of a polydisc, one can simply iterate theCauchy integral. Areasonable working definition of holomorphic function isa function on a domain inCnthat is holomorphic in each variable separately and continuous in all variables holomorphic in this sense on the closed unit polydisc, then iterating the Cauchyintegral shows thatf(z) =(12 i)n |w1|=1.

7 |wn|=1f(w1, .. , wn)(w1 z1) (wn zn)dw1.. dwnwhen the pointzwith coordinates (z1, .. , zn) is in the interior of the polydisc. (Theassumed continuity offguarantees that this integral makes sense and can be evaluatedin any order by Fubini s theorem.) By the same arguments as inthe single- variable case,this iterated Cauchy formula suffices to establish standard local properties of holomor-phic functions. For example, holomorphic functions are infinitely differentiable, satisfythe Cauchy-Riemann equations in each variable , obey a localmaximum principle, andadmit local power series expansions. Moreover, a normal limit of holomorphic functionsis a multi-dimensional version of Hurwitz s theorem: the normal limitof nowhere-zero holomorphic functions is either nowhere zero or identically Partial differential equationsThe one-dimensional Cauchy-Riemann equations are a pair ofreal partial differentialequations for a pair of functions (the real and imaginary parts of a holomorphic function).

8 InCn, there are still two functions, but there are 2nequations. Thus whenn >1,the inhomogeneous Cauchy-Riemann equations form an overdetermined system; hencethere is a necessary compatibility condition for solvability. This feature is a significantdifference from the one- variable the inhomogeneous Cauchy-Riemann equations are solvable inC2(or in higherdimension), there is a solution with compact support in caseof compactly supporteddata. Whenn= 1, however, it is not always possible to solve the inhomogeneous Cauchy-Riemann equations while maintaining compact support. The Hartogs phenomenon canbe interpreted as a manifestation of this GeometryAccording to the one- variable Riemann mapping theorem, every bounded simply con-nected planar domain is biholomorphically equivalent to the unit disc.

9 In higher dimen-sion, there is no such simple topological classification of biholomorphically equivalentdomains. Indeed, the unit ball inC2and the unit bidisc inC2are holomorphicallyinequivalent way to understand intuitively why the situation changesin dimension 2 is torealize that there is extra room in the tangent space. InC2, there is room for one-dimensional complex analysis to happen in the tangent spaceto the boundary of adomain. Indeed, the boundary of the bidisc contains pieces of one-dimensional complexaffine subspaces, while the boundary of the two-dimensional ball does not contain anysuchanalytic , the zero set of a (not identically zero) holomorphic function inC2is a one-dimensional complex variety, while the zero set of a holomorphic function inC1is azero-dimensional variety (that is, a discrete set of points).

10 There is a mismatch between the dimension of the domain and the dimension of therange of a multi- variable holomorphic function. One might,however, expect an equidi-mensional holomorphicmappingto be analogous to a one- variable holomorphic too there are surprises. For instance, there exists a biholomorphic mapping from allofC2onto a proper subset ofC2whose complement has interior points. Such a mappingis called aFatou-Bieberbach image of a Fatou-Bieberbach map cannot have a bounded Power seriesExamples in the introduction showed that the domain of convergence of a multi-variablepower series can have various shapes; in particular, the domain need not be a convex , there is a special kind of convexity property that characterizes the theory requires some notation.