M2R Université Grenoble Alpes 2019-2020 Introduction to analytic geometry (course by Jean-Pierre Demailly)
Sheet number 1, 03/10/2019
1. Standard area and volume calculations in Rn and Cn. Here, Rn and Cn are equipped with their standard Euclidean/Hermitian structures, the Hermitian norm being denoted z7→ |z|.
(a) The Lebesgue measure inCn'R2n can be written as
dλ(z) = i
2dz1∧dz1∧. . .∧ i
2dzn∧dzn= in2
2n dz1∧. . .∧dzn∧dz1∧. . .∧dzn,
wherezj =xj+iyn andCn 'R2n is oriented (conventionally) by coordinates(x1, y1, . . . , xn, yn).
(b) Surface element of Euclidean spheres
Show that the degreen−1 differential form onRn defined by
dσ(x) = 1 R
n
X
j=1
(−1)j−1xjdx1∧. . .∧ddxj∧. . .∧dxn (where b∗ means a term∗ omitted),
induces on the sphereS(0, R)⊂Rn the Euclidean area measure.
In the case of Cn, show that a similar role can be played by
dσ(z) = in2 2n
2 R
n
X
j=1
(−1)j−1zjdz1∧. . .∧dzcj ∧. . .∧dzn∧dz1∧. . .∧dzn.
Hint. Computedr(x)∧dσ(x) withr(x) = (P
|xj|2)1/2 (resp. r(z) = (P
zjzj)1/2); compare with dλ.
(c) Ifx=ruis the usual polar decomposition inRnr{0} (withr =|x| ∈R+ and u= |x|x ∈Sn−1, show that dλ(x) =rn−1dr∧dσ(u) where dσ(u) is given as in (b). (Of course, one has a similar formula inCn withz=ru and dλ(z) =r2n−1dr∧dσ(u)).
Hint. Notice thatdu1∧. . .∧dun= 0 sincedimSn−1< n.
(d) LetS2n−1⊂Cn be the unit sphere, and consider the projection
S2n−1→Cn−1, z= (z1, . . . , zn)7→z0= (z1, . . . , zn−1).
Writez0=ρu,ρ∈R+,u∈S2n−3⊂Cn−1, and zn =rneiθn in polar coordinates on Cn−1 (resp. C).
Ifλ0 is the Lebesgue measure ofCn−1and σ0 the area measure ofS2n−3, show that
dσ(z) =dλ0(z0)∧dθn =ρ2n−3dρ∧dσ0(u)∧dθn
in the complement ofS2n−1∩ {zn= 0} 'S2n−3(which is dσ-negligible) inS2n−1. Derive from there the Fubini type formula
Z
S2n−1
f(z)dσ(z) = Z
ρ∈[0,1]
Z
θn∈[0,2π]
Z
u∈S2n−3
f ρu,(1−ρ2)1/2eiθn
ρ2n−3dρ dθndσ0(u).
Hint. Usedθn= Im(dlogzn) = 2i1(dzzn
n −dzzn
n ) to compute dr∧dλ(z0)∧dθn.
(e) Show by induction on n that the area of S2n−1 is σ2n−1 = (n−1)!2πn . The area of S(0, R) ⊂ Cn is
2πn
(n−1)!R2n−1and the volume ofB(0, R)⊂Cn is πn!nR2n. 2. LetΩ⊂Cn be an open set such that
∀z∈Ω, ∀λ∈C, |λ| ≤1 ⇒ λz ∈Ω.
Show that Ω is a union of polydisks of center 0 (with respect to coordinates z0 = u(z) associated with arbitrary unitary matricesu∈U(n)) and infer that the space of polynomialsC[z1, . . . , zn]is dense inO(Ω)
1
for the topology of uniform convergence on compact subsets. IfΩ is bounded, show thatC[z1, . . . , zn]is dense inO(Ω)∩C0(Ω)for the topology of uniform convergence on Ω.
Hint: consider the Taylor expansion of a function f ∈ O(Ω) at the origin, writing it as a series of homogeneous polynomials. To deal with the case of O(Ω)∩C0(Ω), first apply a dilation tof.
3. The goal of this exercise is to prove the Cauchy formula for the unit ball inCn. Let B ⊂Cn be the unit Hermitian ball,S=∂B andf ∈O(B)∩C0(B). Our goal is to check the following Cauchy formula:
f(w) = 1 σ2n−1
Z
S
f(z)
(1− hw, zi)n dσ(z).
(a) By means of a unitary transformation and exercise 2, reduce the question to the case when w = (0, . . . ,0, wn)and f(z) is a monomialzα=z1α1. . . znαn.
(b) Show that the integralR
Szαzkndσ(z) vanishes unlessα= (0, . . . ,0, k). Compute the value of the non zero integral by means of suitable integration by parts.
Hint. Use formula 1 (d), invariance ofσ by rotation, and/or Fourier series arguments.
(c) Prove the formula by means of a suitable power series expansion of(1− hw, zi)−n. 4. Montel spaces
Let E be a topological vector space E over K =R or C. A subset S ⊂ E is said to be bounded if for every neighborhoodU of zero inE, there existsλ∈K∗ such thatA⊂λU (or equivalently, λ−1A⊂U).
(a) Show that ifA is compact, then A is bounded.
Hint. Use the continuity of the operations to prove the fact that for any given open neighborhoodU of0, one can find a neighborhoodV of0 andδ >0such thatV +µV ⊂U for allµ∈K,|µ|< δ, and coverA by translatesxj+V.
(b) If (E,k k) is a normed vector space, show that A is bounded if and only if supx∈Akxk < +∞.
More generally, ifE is locally convex and the topology ofE is defined by a collection of semi-norms (pα)α∈I, thenA is bounded if and only if for everyα∈I one hassupx∈Apα(x)<+∞.
(c) A Fréchet spaceEwill be said to be aMontel spaceif the compact subsets ofE are exactly the closed bounded subsets of E. Show that for every open set Ω ⊂ Cn, the space O(Ω) is a Montel space;
likewise, forΩ⊂Rn,C∞(Ω)is a Montel space.
(d) Show that an infinite dimensional Banach space is never a Montel space. Derive from there that the topology of O(Ω)orC∞(Ω)cannot be defined by a single norm.
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