Table Of ContentSkoda’s Ideal Generation from Vanishing Theorem
for Semipositive Nakano Curvature
and Cauchy-Riemann Inequality for Tensors
7 Dedicated to Lawrence Ein
1
0
2 Yum-Tong Siu
n
a Since the nineteen sixties the Bochner-Kodaira technique of completion
J of squares, for manifolds either compact or with pseudoconvex or pseudocon-
0
cave boundary condition, has been a most important tool in algebraic and
1
complex geometry. Sometimes ingenious ad hoc adaptations are required in
]
its use. A remarkable example is the 1972 result of Skoda on ideal gener-
V
ation [Skoda1972, pp.555-556, Th´eor`eme 1], which is a crucial ingredient in
C
. the analytic approach to the finite generation of the canonical ring and the
h
t abundance conjecture. Up to this point, special analytic techniques devel-
a
m oped by Skoda, other than applications of the usual vanishing theorems and
L2 estimates for the ∂¯ equation, are required for its proof.
[
1 In recent years analytic results from vanishing theorems and solvability of
v ∂¯equationhave contributed to thesolution ofa number of longstanding open
4
0 problems in algebraic geometry. The interaction between algebraic geometry
4 and analytic methods in several complex variables and partial differential
2
equations has been a very active and productive area of investigation. For
0
. such an interaction it is advantageous to minimize the use of ad hoc analytic
1
0 methods in favor of approaches which are more amenable to adaptation to
7
formulations in algebraic geometry.
1
:
v Inthisnotewegiveasimpler, morestraightforwardproofofSkoda’sresult
i
X on ideal generation which makes it a natural consequence of the standard
r techniques in vanishing theorems and solving ∂¯-equation with L2 estimates.
a
Our proof readily gives other similar results on ideal generation. In 6 we
§
present a number of such similar results. Our proof uses the following three
ingredients.
(I) The Cauchy-Schwarz inequality for tensors
S, T 2 min(r,n) S, T 2
h iCr⊗Cn ≤ kh iCnkCr⊗Cr
for elements S an(cid:12)d T of Cr (cid:12)Cn. Detailed statement and proof will be given
(cid:12) ⊗(cid:12)
in 2. The usual Cauchy-Schwarz inequality is the special case of n = 1.
§
1
Our Cauchy-Schwarz inequality for tensors is needed in order to handle the
curvatureoperatorcondition fornonnegative Nakano curvature. Just like the
usual Cauchy-Schwarz inequality, it is derived by reduction to certain special
orthonormal situation. Our Cauchy-Schwarz inequality for tensors should
be a simple case of a more general theory of inequalities of Cauchy-Schwarz
type for general tensors (obtained by reduction to orthonormal situations
in Young tableau). In 7 where the manipulations in computation between
§
our proof and Skoda’s original proof are compared, one such inequality of
Cauchy-Schwarz type for more general tensors is given.
(II) Nonnegativity of the Nakano curvature of the metric for the kernel bun-
dle of (g , ,g ) : ( )⊕p which is induced from the standard flat
1 p Ω Ω
··· O → O
−q
metric of ( )⊕p and twisted by the weight function q g 2 , where
OΩ j=1| j|
Ω is a domain in Cn and q = min(n,p 1). This i(cid:16)s verified b(cid:17)y a simple
− P
straightforward computation in normal final coordinates at the point under
considerationandtheuseoftheaboveCauchy-Schwarz inequalityfortensors.
Detailed statement and proof will be given in 3.
§
(III) Vanishing theorem (from solving a ∂¯-equation with L2 estimate) for a
holomorphic vector bundle V with nonnegative Nakano curvature Θ on a
strictly pseudoconvex domain Ω in an ambient complex manifold for smooth
∂¯-closed V-valued (0,1)-form F with finite (Θ−1F, F) , where Θ−1 is de-
L2(Ω)
fined as the limit, as the positive number ε approaches 0, of the inverse of
Θ+εI with I being the identity operator. Detailed statement and proof will
be given in 4. Usually application of the technique of vanishing theorems
§
to produce holomorphic sections indispensably requires strict positivity of
curvature at least at some point. The special feature of the application here
of the vanishing theorem from solving a ∂¯-equation with L2 estimate requires
only semipositive Nakano curvature and no strict positivity at any point. The
difficulty of producing holomorphic sections by solving ∂¯-equation with L2
estimate without positivity of curvature at any point is that it is not possible
to get a useable right-hand side for the ∂¯equation by applying ∂¯to the prod-
uct of a cut-off function and a local holomorphic section. For the problem
of ideal generation, the nonnegative curvature condition is sufficient for the
application, because there is a natural useable choice for the right-hand side
of the ∂¯ equation. Another known result requiring only nonnegative cur-
vature condition is the extension result of Ohsawa-Takegoshi. Applications
2
to produce holomorphic sections from only nonnegative curvature condition
remain mostly an area yet to be explored.
Notations. The set of all positive integers is denoted by N. We will use
the notation to denote the structure sheaf of X. To minimize the use
X
O
of notations we will loosely refer to the locally free sheaf associated to a
holomorphic vector bundle simply as a holomorphic vector bundle and vice
versa. For example, we use the notation ( Cn)⊕p to denote both the globally
O
trivial vector bundle of rank p over Cn and the direct sum of p copies of the
structure sheaf Cn of Cn. We use Γ(Y,W) (or Γ(Y, )) to denote the space
O F
of holomorphic sections of a bundle W (or a coherent sheaf ) over Y.
F
For a holomorphic vector bundle V of rank r with a metric H over a
jk¯
complex manifold X of complex dimension n with a Ka¨hler metric g , we
λν¯
will usetheLatinletters j,k,i,ℓetc. fortheindices ofthefiber coordinatesof
V and use the Greek letters λ,ν,µ,ρ etc. for the local coordinates z1, ,zn
of X. We use gν¯λ to mean the inverse of g and use Hk¯j to mean the i·n·v·erse
λν¯
of g . Unless there is a possible confusion, without explicit mention we will
λν¯
use Hk¯j and gν¯λ to raise the subscripts of tensors and use H and g to
jk¯ λν¯
lower thesuperscripts oftensors. ThenotationT denotesthe(holomorphic)
X
tangent bundle of X. The notation K denotes the canonical line bundle of
X
X.
For a holomorphic subbundle W of V, when we use the metric of W
which is induced from the metric H of V, we simply say the metric H of
jk¯ jk¯
W instead of introducing a new notation.
When we refer to a positive (1,1)-form √ 1 η dzλ dzν, we
− 1≤λ,ν≤n λν¯ ∧
will drop the factor √ 1 and simply say that the (1,1)-form η is positive.
− P λν¯
The notation ∂ means ∂ and the notation ∂ means ∂ . The notation
λ ∂zλ λ¯ ∂zλ
( , ) or , means the inner product and the notation means the norm,
· · h· ·i k·k
sometimes with a subscript to indicate in which space the inner product or
the norm is taken.
The study of vanishing theorems and L2 estimates for the ∂¯ equation
has a very long history and a very extensive literature. To avoid a long
bibliography which nowadays can easily be fetched from readily available
online searchable databases, we keep to a minimum the listing of references
at the end of this note.
3
§1. Algebraic Formulation and Application of Skoda’s Ideal
Generation
Though this is not part of our proof of Skoda’s result on ideal generation,
to draw attention to its use in algebraic geometric problems, we give here
the algebraic geometric formulation of Skoda’s result and how it is used in
the analytic approach to the finite generation of the canonical ring before we
present our proof.
We state in the following way a trivially more general form of Skoda’s
result than is given in [Skoda1972, pp.555-556, Th´eor`eme 1] by using a Stein
domain spread over Cn instead of a Stein domain in Cn, in order to be able
to formulate it in an algebraic geometric setting.
(1.1) Skoda’s Theorem on Ideal Generation. Let Ω be a domain spread over
Cn which is Stein. Let ψ be a plurisubharmonic function on Ω, g ,...,g be
1 p
holomorphicfunctionsonΩ(notallidenticallyzero), α > 1,q = min(n,p 1),
and g 2 = p g 2. Then for any holomorphic function f on Ω with −
| | j=1| j|
P f 2e−ψ
| | < ,
g 2(αq+1) ∞
ZΩ | |
there exist holomorphic functions h ,...,h on Ω with f = p h g on Ω
1 p j=1 j j
such that
p h 2e−ψ α f 2e−ψ P
k=1| k| | |
g 2αq ≤ α 1 g 2(αq+1)
ZΩ P | | − ZΩ | |
holds.
The following algebraic geometric formulation of Skoda’s result is easily
obtained by representing the compact complex algebraic manifold X of com-
plex dimension n minus an some complex hypersurface as a Stein domain
over C and considering a holomorphic line bundle over X minus some com-
plex hypersurface as globally trivial by division by a meromorphic section of
it.
(1.2) Theorem (Algebraic Geometric Formulation of on Ideal Generation).
Let X be a compact complex algebraic manifold of complex dimension n
and L and E be respectively holomorphic line bundles on X with (possibly
4
singular) Hermitianmetrics e−ϕL ande−ϕE such that ϕ andϕ areplurisub-
L E
harmonic. Let s , ,s Γ(X,L) and 0 < γ 1 and q = max(n,p 1).
1 p
··· ∈ ≤ −
Then for any s Γ(X,(q+2)L+E +K ) with
X
∈
s 2e−(1−γ)ϕL−ϕE
C := | | < ,
s ZX pj=1|gj|2 q+1+γ ∞
(cid:16) (cid:17)
P
then there exist h , ,h Γ(X,((q + 1)L + E + K ) such that s =
1 p X
p h s with ··· ∈
j=1 j j
P p s 2e−(1−γ)ϕL−ϕE q
k=1| k| 1+ C.
ZX P pj=1|gj|2 q+γ ≤ (cid:18) γ(cid:19)
(cid:16) (cid:17)
P
(1.3) Theorem (Finite Generation of Section Module). Let L be a holomor-
phic line bundle over a compact complex algebraic manifold X of complex
dimension n. Let s(m), ,s(m) be a C-basis of Γ(X,mL). Let N and m be
1 ··· qm 0 0
positive integers such that N m (n+2) and Γ(X,m L) = 0. Let ε (for
0 0 0 m
≥ 6
m N) be a sequence of positive numbers which decrease rapidly enough to
∈
yield the convergence of the infinite series
∞ qm 2
Φ := ε s(m) m
m j
mX=1 Xj=1 (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
on X. Suppose there exist elements s(m0), ,s(m0) of Γ(X,m L) such that
1 ··· qm0 0
for any m N and for any s(m) Γ(X,mL + K ) the (locally defined)
0 X
≥ ∈
function
s(m) 2
(1.3.1)
n+2
Φm−(n+2)m0 (cid:12)(cid:12) qm(cid:12)(cid:12)0 s(m0) 2
j=1 j
(cid:18) (cid:19)
(cid:12) (cid:12)
P (cid:12) (cid:12)
is locally bounded on X. Then for m N(cid:12) , (cid:12)
0
≥
Γ(X,mL+K ) = s(m0) ν1 s(m0) νqm0 Γ((m m ℓ )L+K ),
X 1 ··· qm0 − 0 m X
ν1+···+Xνqm0=ℓm(cid:16) (cid:17) (cid:16) (cid:17)
where ℓ the largest integer with m ℓ m N .
m m 0 0
− ≥
5
In particular, if one denotes by R(X,F) the ring ∞ Γ(X,kF) of all
k=1
global holomorphic sections of positive tensor powers of a line bundle F
over X, then the module ∞ Γ(X,mL + K ) is gLenerated by a finite
m=1 X
number of elements over the ring R(X,m L) and the finite number of ele-
0
ments can be taken to be aLC-basis of the finite-dimensional C-vector space
N0 Γ(X,mL+K ). In other words,
m=1 X
L ∞ N0
Γ(X,mL+K ) = R(X,m L) Γ(X,mL+K ) .
X 0 X
!
m=1 m=1
M M
In the special case of L = K , under the condition of the local bounded-
X
ness of (1.3.1) the canonical ring R(X,K ) is finitely generated.
X
Proof. Lete−ϕL = 1. Form N andanys Γ(X,mL+K ), byTheorem
Φ ≥ 0 ∈ X
(1.2) from
s 2e−(m−(n+2)m0)ϕL
| | < ,
2 n+2 ∞
ZX qm0 s(m0)
j=1 j
(cid:18) (cid:19)
(cid:12) (cid:12)
it follows that there exist P (cid:12) (cid:12)
(cid:12) (cid:12)
h , ,h Γ(X,(m m )L)
1 ··· qm0 ∈ − 0
such that s = qm0 h s(m0). If m (n+2)m is still not less than N , we
j=1 j j − 0 0
can apply the argument to each h instead of s until by induction on ν we
j
P
get
h(j1,···,jν), ,h(j1,···,jν) Γ(X,(m m (ν +1))L+K )
1 ··· qm0 ∈ − 0 X
for 1 j , ,j q with 1 ν ℓ such that
≤ 1 ··· ν ≤ m0 ≤ ≤ m
ν
s = h(j1,···,jν−1) s(m0)
jν jλ
1≤j1,·X··,jν≤qm0 λY=1
with h(0) = h for 1 j q . Q.E.D.
j j ≤ ≤ m0
We now introduce the three ingredients for our proof of Skoda’s ideal
generation.
§2. Cauchy-Schwarz Inequality for Tensors
6
Thefollowing Cauchy-Schwarz inequality fortensorsisintroducedtohan-
dle inequalities for curvature operator conditions. Its proof will be done by
applying the usual Cauchy-Schwarz inequality for vectors and a reduction to
the special case of an orthonormal set of vectors by linear transformations.
(2.1) Proposition (Cauchy-Schwarz Inequality for Tensors). Let S and T be
elements of Cr Cn. Then
⊗
S, T 2 min(r,n) S, T 2 .
h iCr⊗Cn ≤ kh iCnkCr⊗Cr
(cid:12) (cid:12)
(2.2) Remark. When the tensors S and T are represented by r n matrices
(cid:12) (cid:12)
×
S = (s ) and T = (t ) ,
ℓ,k 1≤ℓ≤r,1≤k≤n ℓ,k 1≤ℓ≤r,1≤k≤n
the inequality in Proposition(2.1) becomes
2 n 2
s t min(r,n) s t .
ℓ,k ℓ,k m,k ℓ,k
≤
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)1≤ℓ≤Xr,1≤k≤n (cid:12) 1≤Xm,ℓ≤r(cid:12)Xk=1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
The special(cid:12) case of n = 1 i(cid:12)s simply the followin(cid:12)g usual Ca(cid:12)uchy-Schwarz
inequality for vectors
2
s t s 2 t 2
ℓ ℓ ℓ ℓ
≤ | | | |
(cid:12) (cid:12) ! !
(cid:12)1X≤ℓ≤r (cid:12) 1X≤ℓ≤r 1X≤ℓ≤r
(cid:12) (cid:12)
(cid:12) (cid:12)
where sℓ = sℓ,1 an(cid:12)d tℓ = tℓ,1(cid:12).
(2.3)Proof ofProposition(2.1). LetS = (s ) andT = (t ) .
ℓ,k 1≤ℓ≤r,1≤k≤n ℓ,k 1≤ℓ≤r,1≤k≤n
The inequality
2
S, T r S, T
h iCr⊗Cn ≤ kh iCnkCr⊗Cr
is the same as (cid:12) (cid:12)
(cid:12) (cid:12)
2 n 2
s t r s t ,
ℓ,k ℓ,k m,k ℓ,k
≤
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)1≤ℓ≤Xr,1≤k≤n (cid:12) 1≤Xm,ℓ≤r(cid:12)Xk=1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
which simply fo(cid:12)llows from the fo(cid:12)llowing applic(cid:12)ation of the(cid:12) Cauchy-Schwarz
inequality with focus on the summation over the double index (ℓ,m) (where
7
δ for 1 ℓ,m r is the Kronecker delta)
ℓm
≤ ≤
2
s t
ℓ,k ℓ,k
(cid:12) (cid:12)
(cid:12)1≤ℓ≤Xr,1≤k≤n (cid:12)
(cid:12) (cid:12)
2
(cid:12) (cid:12)
(cid:12) (cid:12)
= δ s t
ℓm m,k ℓ,k
(cid:12) (cid:12)
(cid:12)1≤ℓ,mX≤r,1≤k≤n (cid:12)
(cid:12) (cid:12)
(cid:12) n (cid:12)2
(cid:12) (cid:12)
= δ s t
ℓm m,k ℓ,k
(cid:12) !(cid:12)
(cid:12)1≤Xℓ,m≤r Xk=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) n 2
(cid:12) (cid:12)
(δ )2 s t
mℓ m,k ℓ,k
≤ ! (cid:12) (cid:12)
1≤Xm,ℓ≤r 1≤Xm,ℓ≤r(cid:12)Xk=1 (cid:12)
(cid:12) (cid:12)
n 2 (cid:12) (cid:12)
(cid:12) (cid:12)
= r s t .
m,k ℓ,k
(cid:12) (cid:12)
1≤Xm,ℓ≤r(cid:12)Xk=1 (cid:12)
(cid:12) (cid:12)
The inequality (cid:12) (cid:12)
(cid:12) (cid:12)
2
S, T n S, T
h iCr⊗Cn ≤ kh iCnkCr⊗Cr
is the same as (cid:12) (cid:12)
(cid:12) (cid:12)
2 n 2
s t n s t .
ℓ,k ℓ,k m,k ℓ,k
≤
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)1≤ℓ≤Xr,1≤k≤n (cid:12) 1≤Xm,ℓ≤r(cid:12)Xk=1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Insteadofprovi(cid:12)ng thisinequality(cid:12)we provethefo(cid:12)llowing equ(cid:12)ivalent inequality
(cid:12) (cid:12) (cid:12) (cid:12)
which is obtained by replacing t by its complex conjugate t .
ℓ,k ℓ,k
2 n 2
(2.3.1) s t n s t .
ℓ,k ℓ,k m,k ℓ,k
≤
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)1≤ℓ≤Xr,1≤k≤n (cid:12) 1≤Xm,ℓ≤r(cid:12)Xk=1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Denote the n co(cid:12)lumn r-vectors of(cid:12)the r n ma(cid:12)trix (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
×
S = (s )
ℓ,k 1≤ℓ≤r,1≤k≤n
by~s , ,~s . Likewise we denote the n column r-vectors of the r n matrix
1 n
··· ×
T = (t )
ℓ,k 1≤ℓ≤r,1≤k≤n
by ~t , ,~t .
1 n
···
8
Toprove(2.3.1)itsuffices toassumethatn r andbyslightlyperturbing
≤
~s , ,~s andthentakingthelimittogobacktotheoriginalsetof~s , ,~s ,
1 n 1 n
··· ···
we can assume without loss of generality that the vectors ~s , ,~s of Cr
1 n
···
are C-linearly independent. We apply a C-linear transformation given by
an n n matrix R to the n elements ~t , ,~t of Cr and at the same time
1 n
× ···
the C-linear transformation given by the n n matrix (Rt)−1, which is the
×
complex-conjugate transpose inverse of R, to the n elements ~t , ,~t of Cr
1 n
···
in order to preserve the inner product
n
s t
j,k ℓ,k
k=1
X
for any fixed 1 j,ℓ r. So without loss of generality we can assume
≤ ≤
that the set of n elements ~s , ,~s of Cr are orthonormal with respect to
1 n
···
, and explicitly given by s being equal to the Kronecker delta δ for
Cr j,k jk
h· ·i
1 j,k n and equal to 0 for n < j r. In particular, the expression
≤ ≤ ≤
n 2
s t
m,k ℓ,k
(cid:12) (cid:12)
1≤Xm,ℓ≤r(cid:12)Xk=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
is equal to (cid:12) (cid:12)
r n n 2 r n n
δ t = t 2 = ~t 2 .
mk ℓ,k ℓ,m k Cr
(cid:12) (cid:12)
Xℓ=1 mX=1(cid:12)Xk=1 (cid:12) Xℓ=1 mX=1(cid:12) (cid:12) Xk=1(cid:13) (cid:13)
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:13) (cid:13)
(cid:12) (cid:12)
The left-hand of the(cid:12)inequality(cid:12)(2.3.1) can be rewritten as
2 n 2
s t = ~s , ~t ,
ℓ,k ℓ,k k k Cr
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:12)1≤ℓ≤Xr,1≤k≤n (cid:12)(cid:12) (cid:12)(cid:12)Xk=1(cid:10) (cid:11) (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
which by the Cauc(cid:12)hy-Schwarz inequa(cid:12)lity (cid:12)is (cid:12)
n n
n ~s , ~t 2 n ~t 2
≤ k k Cr ≤ k Cr
k=1 k=1
X(cid:12)(cid:10) (cid:11) (cid:12) X(cid:13) (cid:13)
(cid:12) n (cid:12) 2 (cid:13) (cid:13)
n s t .
m,k ℓ,k
≤
(cid:12) (cid:12)
1≤Xm,ℓ≤r(cid:12)Xk=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) 9 (cid:12)
Q.E.D.
§3. Nakano Curvature of Bundle and Twisting by Metric of
Trivial Line Bundle
We now straightforwardly compute the Nakano curvature in the case of a
special subbundle of rank p 1 in the trivial vector bundle of rank p whose
−
induced metric is twisted by a special weight function (i.e., a metric of the
trivial line bundle) to conclude the nonnegativity of the Nakano curvature.
(3.1) Nakano Curvature of Vector Bundles and Subbundles. Let U be a
domain in Cn with coordinates z1, ,zn. Let V be a holomorphic vector
···
bundle of rank r on U with smooth Hermitian metric H whose components
are H (for 1 h,k r). We use the notation ΘV (for 1 j,k r and
jk¯ ≤ ≤ jk¯λν¯ ≤ ≤
1 λ,ν n) to denote the components of the curvature tensor ΘV of V
≤ ≤
which is defined as
ΘV = ∂ ∂ H + Hℓ¯i(∂ H ) ∂ H .
jk¯λν¯ − λ ν¯ jk¯ λ kℓ¯ ν i¯j
1≤i,ℓ≤r
X (cid:0) (cid:1)
When there is no possible confusion, we will simply use Θ to denote ΘV
jk¯λν¯ jk¯λν¯
and use Θ to denote ΘV.
The sign convention is chosen here so that when r = 1, the nonnegativity
of the curvature of the metric e−ϕ means the plurisubharmonicity of the
function ϕ.
When we have a holomorphic subbundle W of V with rank 1 s < r and
≤
local frame e , ,e so that e , ,e is orthonormal at the point P under
1 s 1 r
··· ···
consideration (after the application of an P-dependent element of GL(r,C)),
the curvature ΘW for W (for 1 j,k s and 1 λ,ν n) is given by
jk¯λν¯ ≤ ≤ ≤ ≤
s
(3.1.1) ΘW = ∂ ∂ H + ∂ H (∂ H )
jk¯λν¯ − λ ν¯ jk¯ λ jℓ¯ ν ℓk¯
ℓ=1
X(cid:0) (cid:1)
r
= ΘV ∂ H (∂ H )
jk¯λν¯ − λ jℓ¯ ν ℓk¯
ℓ=s+1
X (cid:0) (cid:1)
when W is given by the metric induced from the metric of V.
10
