LOGARITHMIC GROWTH

LOGARITHMIC GROWTH

5(a) Logarithmic growth

I explain the proof of a version of the Dolbeault lemma ( ¯ ∂-Poincar´e lemma) with logarithmic singularities, published in the Comptes Rendus in a joint paper with D. H. Phong, and refined in a paper with Zucker. In what follows, let ∆ denote the disk of radius ¹ ₂ around the origin in C, ∆ ^∗ = ∆ \ {0} the punctured disk. The variable is denoted z = re ^iθ for z 6= 0.

Lemma. Let N ∈ Z , N 6= 1, and let g ∈ C ^∞ (∆ ^∗ ) be a function of logarithmic growth of order N :

|g(z)| < C | log r| ^N

r , C ∈ R ⁺ .

Then there is a solution f ∈ C ^∞ (∆ ^∗ ) to the equation ∂ f ¯ = g that satisfies

|f (z)| < C ^′ | log r| ^N+1 , C ^′ ∈ R ⁺ .

Proof. We begin as usual by writing the formula for the fundamental solution of the ¯ ∂ equation:

f(z) = 1 2πi

Z Z

∆

g(w)

w − z dwd w, z ¯ ∈ ∆ ^∗ .

We first show that this integral converges to a C ^∞ solution to ¯ ∂ f = g. For any w ∈ ∆ and a > 0, let B(w, a) be the disk of radius a centered on w. Let z ₀ ∈ ∆ ^∗ , a > 0 such that the closure of B(z 0 , 2a) is contained in ∆ ^∗ . Then we can write g = g ₁ + g ₂ as a sum of two functions in C ^∞ (∆ ^∗ ), where supp(g ₁ ) ⊂ B(z ₀ , 2a) and g 2 | _B(z

,a) ≡ 0. In particular, the singularity at 0 is contained in the support of g 2 .

Now setting w = ρe ^iθ we find

| 1 2πi

Z Z

∆

g ₂ (w)

w − z dwd w| ¯ < 1 2πa |

Z Z

∆

g 2 (w)dwd w| ¯

< C πa

Z 2π 0

Z 1/2 0

| log ρ| ^N

ρ ρdρdθ = O(

Z 1/2 0

| log ρ| ^N dρ) and for any N , | log ρ| ^N is integrable on [0, ¹ ₂ ], so this integral converges. (There is a terrible misprint in my article with Phong! The integrability is by integration by parts: we have

Z t 0

|log ρ| ^N dρ = ρ · |log ρ| ^N | ^t ₀ − N Z t

0

|log ρ| ^N−1 |dρ

and we have it by induction for N > 0, and for N < 0 there is nothing to prove.)

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1

2 LOGARITHMIC GROWTH

Thus

f ₂ (z) = 1 2πi

Z Z

∆

g ₂ (w)

w − z dwd w ¯ ∈ C ^∞ (B(z ₀ , a)).

Moreover, we can differentiate under the integral sign and since the integrand is holomorphic in z, ¯ ∂ f ₂ = 0 on B(z ₀ , a).

Now g ₁ ∈ C ^∞ _c (∆) so the usual arguments apply and we find that f ₁ (z) = 1

2πi Z Z

∆

g 1 (w)

w − z dwd w ¯ ∈ C ^∞ (B(z ₀ , 2a))

and that ¯ ∂ f 1 = g 1 on B(z 0 , 2a). Thus f = f 1 + f 2 is C ^∞ in a neighborhood of z 0

and is a solution to ¯ ∂ f = g on B(z ₀ , a).

Now z 0 is arbitrary, so it remains to show that f has logarithmic growth. Choose z ∈ ∆ ^∗ , r = |z|. We decompose ∆ ^∗ in three regions: ∆ ^∗ = D ₁ ∪ D ₂ ∪ D ₃ where

D ₁ = B(0, r

2 ) \ {0}, D ₂ = B(z, r

2 ) ∩ ∆, D ₃ = ∆ ^∗ \ (D ₁ ∪ D ₂ ).

We bound the integral on each of these three regions.

On D ₁ we have _|w−z| ¹ ≤ ² _r . Thus 1

2π | Z Z

D

g(w)

w − z dwd w| ¯ < 4C 2πr

Z Z

D

| log ρ| ^N dρdθ = C 1 1 r

Z r/2 0

| log ρ| ^N dρ.

Integration by parts as above shows that this is O(|log ^r ₂ | ^N ) = O(|log ^r ₂ | ^N ). (Again a terrible misprint!)

On D ₂ we have the inequality |g(w)| < C ^{∗ | log} _r

^|

. where C ^∗ is independent of r. Thus

1 2π |

Z Z

D

g(w)

w − z dwd w| ¯ < C ^∗ 2π

|log ^r ₂ | ^N r · |

Z Z

D

dwd w ¯

|w − z|

≤ C ^∗ 2π

|log ^r ₂ | ^N r · |

Z Z

B(0,r/2)

dud ¯

|u| (u = w − z)

= C ^∗ |log r 2 | ^N by polar coordinates.

On D ₃ , finally, we have |w − z| ≥ |w/3| = ρ/3. So 1

2π | Z Z

D

g(w)

w − z dwd w| ¯ < 6C 2π

Z Z

D

| log ρ| ^N

ρ ² ρdρdθ

< 3C π

Z Z

D

∪D

| log ρ| ^N ρ dρdθ

= 6C Z 1

r/2

| log ρ| ^N

ρ dρ = 6C

N + 1 | log r/2| ^{N +1} .

This completes the proof.

LOGARITHMIC GROWTH 3

Let A ⁰ _N (∆ ^∗ ) be the space of f satisfying the growth condition of degree N , A ⁰ _si (∆ ^∗ ) = ∪ _N A ⁰ _N (∆ ^∗ ) A ⁰ _rd (∆ ^∗ ) = ∩ _N A ⁰ _N (∆ ^∗ ), A ¹ _? = A ⁰ _? · ^d¯ _{z ¯} ^z . Then the above lemma implies that ¯ ∂ : A ⁰ _? →A ¹ _? is surjective if ? = si, ? = rd. Moreover ker ¯ ∂ ∩ A ⁰ _si is the space of functions on ∆ ^∗ that are holomorphic and have logarithmic growth at 0; but then they have removable singularities, so in fact they extend holomorphically to ∆. Similarly, ker ¯ ∂ ∩ A ⁰ _rd is the space of holomorphic functions on ∆ that vanish at 0.

In order to globalize and generalize to higher dimensions, we can improve the result.

Corollary. Let θ = z∂/∂ z and θ ¯ = ¯ z ∂/∂ ¯ z ¯ In the above Lemma, suppose h = ¯ zg(z) has the property that all its derivatives of the form θ ⁱ θ ¯ ^j h are bounded by some (respectively every) power of | log ρ| Then f has the same property.

Indeed, letting I(g) = _2πi ¹ R R

∆

g(w)

w−z dwd w ¯ the main point is that ^∂I(g) _∂z = I ( ^∂g _∂z ), and then we argue by induction. This allows us to use the standard argument (see Griffiths-Harris, p. 25) to apply this lemma to complex algebraic varieties (even analytic varieties). Let M be a smooth compact complex algebraic variety of dimension n, Z ⊂ M a divisor with (simple) normal crossings. This means that every point z ∈ Z has a neighborhood D ⊂ M such that D −→ ^∼ ∆ ⁿ with z as origin and

(*) (M \ Z ) ∩ D −→ ^∼ (∆ ^∗ ) ^r × ∆ ^n−r

for some integer r. In other words, up to complex analytic change of coordinates, Z ∩ D looks like a union of some of the coordinate hyperplanes. We define

A ^0,q _si ((∆ ^∗ ) ^r × ∆ ^n−r ) = A ^0,q _si (∆ ^∗ ) ^{⊗ r} ⊗ A ^0,q (∆) ^{⊗ n−r}

and let A ^0,q _si to be the sheaf of C ^∞ (0, q) forms on M \ Z whose restriction to (M \Z )∩ D belongs to A ^0,q _si ((∆ ^∗ ) ^r ×∆ ^n−r ) for some (equivalently any) holomorphic isomorphism (*) as above. We define A ^0,q _rd similarly. Note that these define sheaves on M (not just on M \ Z (a section on an open set U is in this sheaf if and only if its restriction to U \ Z is).

Theorem (Harris-Phong). Let B be a (holomorphic) vector bundle on M and let A ^0,q _si (B) = A ^0,q _si ⊗ B, A ^0,q _rd (B) = A ^0,q _rd ⊗ B. There are natural inclusions of sheaves

B ֒ → A ^0,0 _si (B); B(−Z ) ֒ → A ^0,0 _rd (B) where B(−Z) = ker[B→B ⊗ O Z ], and the complexes

0B→A ^0,0 _si (B)→A ^0,1 _si (B)→ . . . →A ^0,n _si (B)→0;

0B(−Z )→A ^0,0 _rd (B)→A ^0,1 _si (B)→ . . . →A ^0,n _si (B)→0;

are fine resolutions of B and B(−Z ) respectively.

This follows by standard arguments from the 1-dimensional local case (and the existence of partitions of unity).

The Theorem was designed to apply to certain smooth projective compactifi- cations M of non-compact Shimura varieties S(G, X) = M \ Z (this should be

K

S(G, X ) but I omit the subscript). The idea is that automorphic vector bundles [W ] on S(G, X ) have canonical extensions [W ] ^can to M with very good properties;

in particular, the growth conditions for forms in Γ(M, A ^0,q _si ([W ] ^can ) correspond,

under the isomorphism with functions on G(Q)\G(A), to the growth conditions

defining automorphic forms.