Higher cohomology of the pluricanonical bundle is not deformation invariant

0.14 MB·English

by  Ning Hao

HIGHER COHOMOLOGY OF THE PLURICANONICAL BUNDLE IS NOT
DEFORMATION INVARIANT
NINGHAO,LI LI
8
0
0
2
Abstract. In this paper, we compute the dimensions of the 1st and 2nd cohomology groups

n of all the pluricanonical bundles for Hirzebruch surfaces, and the dimensions of the 1st and 2nd
a cohomology groups of the second pluricanonical bundles for a blow-up of a projective plane along
J
finitemanydistinctpoints. Bothofthemgiveexamplesshowingthatthehighercohomologyofthe

6 pluricanonical bundleis not deformation invariant.
1

]
G Contents
A
1. Introduction 1
h. 2. Deformation of a Hirzebruch surface: Proof of Theorem 2 2
t 3. Deformation of a blow-up of a projective space 4
a
m References 6
[

2
v
1. Introduction
6
0
Let ∆ = {t ∈C : |t|< 1} bethe unitdisc in the complex plane. AssumeY is a compact complex
0
2 manifold. Denote by KY the canonical line bundle of Y. Denote by hi(Y,L) = dimCHi(Y,L) for
1 any linebundleLon Y. ThedimensionofH0(Y,mK )iscalled them-thplurigenusof Y. In[S02],
Y
6
Siu proved the following remarkable theorem on the deformation invariance of the plurigenera.
0
/
h Theorem 1. [S02] Let π : X → ∆ be a holomorphic family of compact complex projective algebraic
t manifolds with fiber X . Then for any positive integer m, the complex dimension of H0(X ,mK )
a t t Xt
m is independent of t for t ∈ ∆.
:
v It is then natural to ask the following question:
i
X Question 1.1. Is the dimension of Hq(X ,mK ) independent of t, for q > 0?
t Xt
r
a
The main point of the present article is to give a negative answer to Question 1.1. In fact, the
following two theorems are our main results.
Theorem 2. For m ≥ 0, Let F = P(O⊕O(m)) be the m-th Hirzebruch surface. Then
m
1 2k 2k
dimH0(−kK )= 4k+(k− )m+2 k+ +1 .
Fm 2 m m
(cid:16) h i (cid:17)(cid:16) h i (cid:17)
Theorem 3. Let M → P2 be the blow-up of P2 along v distinct points. The dimension of H0(−K )
M
is 10 −v for v ≤ 4; for v > 4, the dimension depends on the position of the v points: for each
integer i such that max(10−v,0) ≤ i ≤ 6, there exist v distinct points such that the corresponding
M satisfies dimH0(−K ) = i.
M
2000 Mathematics Subject Classification. 32L10 14F10.
1
Returning to Question 1.1, Theorem 2 shows that for each k the dimensions h0(−kK ) depend
Fm
on m in a non-constant way. On the other hand, it is shown in [K86, Example 2.16] that for any
positive integer ℓ ≤ m/2 there is a holomorphic family M → ∆ whose fiber M is isomorphic to
t
F for t 6= 0, and whose central fiber M is isomorphic to F . By Serre duality, we find that
m−2ℓ 0 m
h2((k+1)K ) = h0(−kK )
Mt Mt
depends on t in a non-constant fashion.
Similarly, consider the situation of Theorem 3. Let B = {(q ,··· ,q ) ∈ (P2)v : q 6= q if i 6= j}
1 v i j
be the configuration space of v points in P2. Denote by Γ ⊂ B ×P2 the graph of the morphism
πi
π : B → P2 induced from the projection (P2)v → P2 to the i-th factor. Let M be the blow-up
i
of B ×P2 along v disjoint submanifolds Γ for 1 ≤ i ≤ v. The natural morphism M → B is a
πi f
holomorphic family with each fiber M over t = (q ,··· ,q ) being isomorphic to the blow-up of P2
t 1 v
f
along v points q ,··· ,q . Theorem 3 says that
1 v
h2(2K ) = h0(−K )
Mt Mt
is not constant for v > 5. Since B is obviously smooth and connected, we can find a small disc
∆ ⊂ B such that h2(2K )) is not constant for t ∈ ∆.
Mt
Remark. Sofarwehaveshown,modulotheproofsofTheorems2and3,thatthereareholomorphic
families M → ∆ such that h2(kK ) depends on t. On the other hand, we know from Theorem 1
Mt
(and in fact from earlier results using the classification of surfaces) that h0(kK ) is constant in t.
Mt
Now, suppose X is a complex projective algebraic surface. Let K be the canonical line bundle of
X. Applying the Riemann-Roch Theorem for surfaces, we have
1 1
h0(kK)−h1(kK)+h2(kK) = kK(kK −K)+χ(O ) = k(k−1)K2+χ(O ).
X X
2 2
Moreover, by Noether’s formula,
K2+χ (X)
top
χ(O ) = ,
X
12
and therefore
1
h1(kK) = h0(kK)+h2(kK)−(6k2−6k+1)χ(O )+ k(k−1)χ (X).
X top
2
Now, a deformation does not change the differentiable structure, and χ (X) is a topological
top
invariant. Therefore, χ (X) is a deformation invariant. Next, χ(O ) = h0,0 −h1,0 +h2,0 is also
top X
invariant for a family of Ka¨hler manifolds, because Hodge numbers are invariant for such a family.
By Siu’s result of invariance of plurigenera, h0(kK) is also a deformation invariant. It follows that,
since h2(kK) is not a deformation invariant, neither is h1(kK).
Acknowledgements. WearegratefultoDrorVarolin,whobroughtthisproblemtoourattention and
provided considerable advice and encouragement throughout our investigations, to Blaine Lawson
for his valuable comments and encouragement, to Yujen Shu for useful discussions.
2. Deformation of a Hirzebruch surface: Proof of Theorem 2
Recall that the Hirzebruch surfaces F (m ≥ 0) is the P1-bundle over P1 associated to the sheaf
m
OP1 ⊕OP1(m). For example, F0 is isomorphic to P1×P1, and F1 is the blow-up of P2 at a point.
P1 can be covered by two affine line U ,U ∼= C. Let z and z be affine coordinates for U and
1 2 1 2 1
U2 respectively. Then on U1 and U2 we have z1z2 = 1. The sheaf OP1(m) can be constructed by
gluing two trivial bundles U ×C and U ×C together as follows: a section of s of U ×C can be
1 2 1 1
glued with a section s of U ×C if s (z ) = zms (z ) on U ∩U where z = 1/z (that is to say,
2 2 1 1 1 2 2 1 2 2 1
2
the transition function of the corresponding vector bundle is zm). Therefore, F can be obtained
1 m
by gluing U ×P1 with U ×P1 where (z ,ζ ) ∈ U ×P1 is identified with (z ,ζ ) ∈ U ×P1 if
1 2 1 1 1 2 2 2
z z = 1, and ζ = zmζ .
1 2 1 2 2
(Here and below we adopt the standard abuse of notation, using an affine coordinate ζ to denote
1
a point in the fiber P1.)
We recall the holomorphic family {M : t ∈ C} defined in [K86, Example 2.16]: Fix a positive
t
integer ℓ ≤ m/2, define M as
t
M = U ×P1∪U ×P1,
t 1 2
where (z ,ζ )∈ U ×P1 and (z ,ζ ) ∈ U ×P1 are the same point in M if
1 1 1 2 2 2 t
z z = 1 and ζ = zmζ +tzℓ.
1 2 1 2 2 2
It is proved in [K86] that M is isomorphic to Hirzebruch surface F if t = 0 and isomorphic to
t m
F if t 6= 0. Thus F is a deformation of F .
m−2ℓ m m−2ℓ
Now we calculate dimH0(−kK ) = dimΓ 2T Nk for k > 0. First, we assert that a
Fm Fm
section in Γ U1×P1, 2TU1×P1 ⊗k is of the (cid:0)fo(cid:0)rVm (cid:1) (cid:1)
(cid:0) (cid:0)V (cid:1) (cid:1)
∂ ∂ ⊗k
v (z ,ζ ) = a (z )ζ2k +a (z )ζ2k−1+...+a (z ) ∧ ,
1 1 1 2k 1 1 2k−1 1 1 0 1 ∂z ∂ζ
(cid:16) (cid:17)(cid:16) 1 1(cid:17)
i.e., it the expansion contains no term of power of ζi for i > 2k. Indeed, expand v (z ,ζ ) into a
1 1 1
power series ∞ a (z )ζi ( ∂ ∧ ∂ )⊗k. Now cover U ×P1 with two open subset U ×V and
i=0 i 1 1 ∂z1 ∂ζ1 1 1 1
U ×V wher(cid:0)ePV and V ar(cid:1)e isomorphic to C and cover P1. Assume the coordinates of V and V
1 2 1 2 1 2
are ζ and η respectively. Then ζ η = 1 on V ∩V , and therefore ∂ = −η2 ∂ . In U ×V , the
1 1 1 1 1 2 ∂ζ1 1∂η1 1 2
section v (z ,ζ ) is of the form
1 1 1
∞ ∞
1 ∂ ∂ ⊗k ∂ ∂ ⊗k
a (z ) ∧(−η2) = (−1)k a (z )η2k−i ∧ .
i 1 ηi ∂z 1 ∂η i 1 1 ∂z ∂η
(cid:16)Xi=0 1(cid:17)(cid:16) 1 1(cid:17) (cid:16)Xi=0 (cid:17)(cid:16) 1 1(cid:17)
For this section to be holomorphic, all the terms a (z ) for i> 2k should vanish.
i 1
Similarly, a section in Γ U2×P1, 2TU2×P1 ⊗k is of the form
(cid:0) (cid:0)V (cid:1) (cid:1)
∂ ∂ ⊗k
v (z ,ζ )= b (z )ζ2k +b (z )ζ2k−1+...+b (z ) ∧ .
2 2 2 2k 2 2 2k−1 2 2 0 2 ∂z ∂ζ
(cid:16) (cid:17)(cid:16) 2 2(cid:17)
To glue v and v into a section of Γ F , 2T ⊗k , we need v (z ,ζ ) = v (z ,ζ ) over
1 2 m Fm 1 1 1 2 2 2
(U U )×P1. Since (cid:0) (cid:0)V (cid:1) (cid:1)
1 2
T ∂ ∂ ∂
= −z2 +mzm−1ζ
∂z 1∂z 2 2∂ζ
2 1 1
and
∂ ∂
= z−m ,
∂ζ 1 ∂ζ
2 1
we have
∂ ∂ ∂ ∂
∧ = −z2−m ∧ .
∂z ∂ζ 1 ∂z ∂ζ
2 2 1 1
3
Therefore on (U ∩U )×P1,
1 2
v (z ,ζ )
2 2 2
∂ ∂ ⊗k
= b (z )ζ2k +b (z )ζ2k−1+...+b (z ) ∧
2k 2 2 2k−1 2 2 0 2 ∂z ∂ζ
(cid:16) (cid:17)(cid:16) 2 2(cid:17)
1 1 1 ∂ ∂ ⊗k
= b ( )z2kmζ2k +b ( )z(2k−1)mζ2k−1+...+b ( ) (−z2−m)k ∧
(cid:20) 2k z 1 1 2k−1 z 1 1 0 z (cid:21) 1 ∂z ∂ζ
1 1 1 (cid:16) 1 1(cid:17)
1 1 1 ∂ ∂ ⊗k
=(−1)k b ( )z(m+2)kζ2k +b ( )z(m+2)k−mζ2k−1+...+b ( )z(2−m)k ∧
(cid:20) 2k z 1 1 2k−1 z 1 1 0 z 1 (cid:21) ∂z ∂ζ
1 1 1 (cid:16) 1 1(cid:17)
Comparing the coefficients above with the coefficients of
∂ ∂ ⊗k
v (z ,ζ ) = a (z )ζ2k +a (z )ζ2k−1+...+a (z ) ∧ ,
1 1 1 2k 1 1 2k−1 1 1 0 1 ∂z ∂ζ
(cid:16) (cid:17)(cid:16) 1 1(cid:17)
we get
(−1)kb ( 1 )z2k+km = a (z ),
2k z1 1 2k 1
 (−1)kb ( 1 )z2k+(k−1)m = a (z ),
 2k−1 z1 1 2k−1 1
 ···


 (−1)kbi(z11)z12k+(i−k)m = ai(z1),
···

 (−1)kb0(z11)z12k−mk = a0(z1).

Thuswe seethat fora and b tobeholomorphic, a (z )is apolynomial of degree≤ 2k+(i−k)m
i i i 1
(thus it has 2k +(i−k)m+1 degrees of freedom for i ≥ k− 2k and 0 otherwise), and then b is
m i
uniquely determined by a . Therefore,
i
dimH0(−kK ) =dimΓ 2T Nk
Fm Fm
(cid:0)(cid:0)^ (cid:1) (cid:1)
2k
= 2k+km+1 + 2k+(k−1)m+1 +...+ 2k− m+1
m
h i
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
1 2k 2k
= 4k+(k− )m+2 k+ +1 .
2 m m
(cid:16) h i (cid:17)(cid:16) h i (cid:17)
Thus Theorem 2 is proved.
It is then a direct calculation to get the following
Corollary 2.1. The dimension of the first cohomology of mK is
Fm
1 2(k−1) 2(k−1)
h1(kK ) = 4k−2+(k−1)m− m k+ −4k2 +4k−1.
Fm 2 m m
(cid:16) h i (cid:17)(cid:16) h i(cid:17)
3. Deformation of a blow-up of a projective space
Take v distinct points q ,··· ,q on Pn, let M → Pn be the blow-up of Pn along these v points,
1 v
and let E ,··· ,E be the corresponding exceptional divisors.
1 v
Let Z be the set of nonnegative integers. For α = (α ,··· ,α ) ∈ Zn, define |α| = n α .
+ 1 n + i=1 i
Let f(z1,...,zn) be a polynomial of n variables. Denote P
∂|α|f ∂|α|f
= .
∂zα ∂zα1···∂zαn
1 n
We need the following lemma:
4
Lemma 3.1. For an integer k > 0, the vector space H0(−kK ) is isomorphic to the vector space
M
consisting of polynomials f(z ,...,z ) of degree ≤ (n+1)k satisfying
1 n
∂|α|f
(q )= 0, ∀1≤ i ≤ v,∀|α| < (n−1)k.
∂zα i
Proof. The line bundle −kKM (resp. −kKPn) is isomorphic to ( nTM)⊗k (resp. ( nTPn)⊗k).
Thus the natural push-forward of vector fields V V
π∗ :Γ(TM) → Γ(TPn)
induces a natural push-forward
n ⊗k n ⊗k
π∗ :Γ TM → Γ TPn
(cid:0)(cid:0)^ (cid:1) (cid:1) (cid:0)(cid:0)^ (cid:1) (cid:1)
hence a natural push-forward
e
π∗ :Γ(−kKM)→ Γ(−kKPn).
Since π : M → Pn is an isomorphism outside the exceptional divisors, π must be injective.
∗
e
Indeed, if a holomorphic section of Γ(−kK ) is zero on a open subset of M, then it must be
M
identically zero. Therefore the dimension of Γ(−kK ) is equal to the imageeof π .
M ∗
Now we examine which elements of Γ(−kKPn) are in the image of π∗ : Γ(−kKM) → Γ(−kKPn).
Locally, we consider an open polydisc U = {z = (z ,··· ,z ) : |z | < ǫ,∀i} of Cen where ǫ > 0 is
1 n i
e
sufficiently small. Let U be the blow-up of U along the center (0,··· ,0). Then
Ue = {(z,ζ) ∈ U ×Pn−1 : ziζj = zjζi,∀1 ≤ i< j ≤ n},
where (ζ1,··· ,ζn) is thee homogeneous coordinates of ζ ∈ Pn−1. Let
U = {(z,ζ) ∈U :ζ 6= 0}.
i i
Without loss of generality, we consideer i= 1. e
U = {(w ,··· ,w ): |w |,|w w |,··· ,|w w |< ǫ}
1 1 n 1 1 2 1 n
with the embedding U1 ֒→e U given by (w1,w2,··· ,wn) 7→ (w1,w1w2,··· ,w1wn).
An element of ( nT )⊗k is of the form
U
e e
V ∂ ∂
⊗k
f(z ,z ,··· ,z ) ∧···∧ .
1 2 n
∂z ∂z
(cid:0) 1 n(cid:1)
Since w = z ,w = z /z ,··· ,w = z /z , we have ∂ = ∂ − w2 ∂ − w3 ∂ −···− wn ∂ ,
1 1 2 2 1 n n 1 ∂z1 ∂w1 w1∂w2 w1∂w3 w1 ∂wn
∂ = 1 ∂ , ···, ∂ = 1 ∂ . Therefore it induces a meromorphic section of ( nT )⊗k:
∂z2 w1∂w2 ∂zn w1∂wn Ue1
V
1 ∂ ∂
⊗k
f(w ,w w ,··· ,w w ) ∧···∧ .
1 1 2 1 n w1(n−1)k(cid:0)∂w1 ∂wn(cid:1)
Expanding f in a power series, we see that the smallest degree of nonzero terms of the power series
should be no less than (n−1)k to ensure the above section being holomorphic.
Thus, the local calculation shows Γ(−kKM) is isomorphic to the subspace of Γ(−kKPn) where
the Taylor expansion of each section at q (1 ≤ i≤ v) contains no term of degree < (n−1)k.
i
We can always find a hyperplane H ⊂ Pn that does not contain any q (1 ≤ i ≤ v). The
i
complementof H,denotedbyV, isisomorphictoCn. Itiswell knownthatsections ofΓ(−kKPn)=
OPn((n+1)k) are in one-to-one correspondence with polynomials on V ∼= Cn of degree ≤ (n+1)k.
Thus the dimension of Γ(−kKPn) is (n+1n)k+n . Therefore the subspace Γ(−kKM) = H0(−kKM)
can be identified with the set of poly(cid:0)nomials f(cid:1)of degree ≤ (n+1)k satisfying
∂|α|f
(q )= 0, ∀1≤ i ≤ v,∀|α| < (n−1)k.
∂zα i
5
(cid:3)
Now we examine the case when n = 2 and k = 1, i.e. H0(−K ) where M is the blow-up of P2
M
at v points q ,··· ,q . The above lemma says that
1 v
H0(−K )= {polynomials f of degree ≤ 3 such that f(q )= 0,∀1 ≤ i ≤ v.}
M i
The dimension of H0(−K ) depends on the position of q ’s. Indeed, let f = a zizj be
M i i+j≤3 ij 1 2
the polynomial corresponding to the vector v = (a ) ∈ C10. Consider thPe map C2 → C10
f ij i+j≤3
sending a point q = (x,y) to qˆ= (xiyj) = (1,x,y,x2,xy,y2,x3,x2y,xy2,y3) (the map q 7→ qˆ
i+j≤3
can be thought of as the affine version of Veronese embedding) and denote the image of this map
by S. Then the condition f(q) = 0 means exactly that fˆ·qˆ = 0. As there is no proper linear
subspace of C10 containing S, generic v points qˆ ,··· ,qˆ in S are linearly independent for v ≤ 10.
1 v
Therefore, H0(−K ) = 10−v for generic v(≤ 10) points p ,··· ,p .
M 1 v
Nowweassertthatforanyv ≤ 4pointsq = (x ,y ),thecorrespondingv points{qˆ}v inS span
i i i i i=1
a linear subspace of dimension v in C10 (in another word, they are in general position). Indeed,
after changing coordinates if necessary, we can assume all x ’s are different. Then the matrix with
i
the i-th row being the coordinate of qˆ is
i
1 x y x2 x y y2 x3 x2y x y2 y3
1 1 1 1 1 1 1 1 1 1 1 1
 ... ... ... ... ... ... ... ... ... ... ,
 1 xv yv x2v xvyv yv2 x3v x2vyv xvyv2 yv3 
 
which contains a v×v square
1 x ··· xv−1
1 1
 .. .. .. 
. . .
 1 xv ··· xvv−1 
 
Its determinant is (x −x ) which is nonzero since we assume that the x ’s are different.
1≤i<j≤v j i i
Therefore the abovQe v ×10 matrix is of full rank, which implies the v points {qˆi}vi=1 in S span a
linear subspace of dimension v in C10.
On the other hand, we can easily find as finitely many points as we want in S which lie in
a 4 dimensional subspace of C10. For example, we can take {q = (i,0)}v , the corresponding
i i=1
{qˆ}v ∈ C10 satisfy the requirement. for v ≥ 5, start from v points of S in a 4 dimensional
i i=1
subspace, move them point by pointto a general position, we get v points in S spanninga subspace
of dimension 4,5,··· ,min(v,10), respectively. Therefore we have proved the following theorem:
Theorem 4. Let M → P2 be the blow-up of P2 along v distinct points. The dimension of H0(−K )
M
is 10 −v for v ≤ 4; for v > 4, the dimension depends on the position of the v points: for each
integer i such that max(10−v,0) ≤ i ≤ 6, there exist v distinct points such that the corresponding
M satisfies dimH0(−K ) = i.
M
It is then a direct calculation to get the following
Corollary 3.2. The dimension of the first cohomology of 2K is
0, if v ≤ 4;
h1(2K) =
(cid:26) integer between max(0,v−10) and v−4, if v ≥ 5.
References
[K86] K.Kodaira,Complexmanifoldsanddeformationofcomplexstructures.TranslatedfromtheJapanesebyKazuo
Akao.With an appendix byDaisukeFujiwara. Grundlehren der Mathematischen Wissenschaften [Fundamental
Principles of Mathematical Sciences], 283. Springer-Verlag, NewYork,1986. x+465 pp.
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[S02] Siu,Y.-T.,Extensionoftwistedpluricanonicalsectionswithplurisubharmonicweightandinvarianceofsemipos-
itively twisted plurigenera for manifolds not necessarily of general type. Complex geometry. Collection of papers
dedicated to Hans Grauert. Springer-Verlag, Berlin, 2002. (223–277)
Ning Hao
Mathematics Department, SUNY at Stony Brook
Stony Brook, NY 11794, US
Email: [email protected]
Li Li
Korean Institute for Advanced Study
207-43 Cheongryangri-dong, Seoul 130-722, Korea
Email: [email protected]
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