Osaka University Knowledge Archive

シンメイ, アサコ

神明, あさ子

ショウカイ　ミハマチョウシ　ヘンサン　イインカイ　ヘン　ワカサ　ミハマチョウシ　ミハマ　ノ　ブンカ　ダイ4カン　マウ　オドル

Ishimura, Koshiro

Ito, Shunji

Characterization of periodic points of the negative slope algorithm

Tsutsumi, Yukihiro

Casson-Walker invariants of cyclic covers branched along satellite knots

Gasser, Ingenuin

Marcati, Pierangelo

Osaka University
Title On a generalization of the ''div-curl Lemma''
Author(s)Ga ser, Ingenuin; Marcati, Pierangelo
CitationOsaka Journal of Mathematics. 45(1) P.211-P.214
Issue Date2008-03
Text Versionpublisher
URL http://hdl.handle.net/11094/7516
DOI
Rights
Gasser, I. and Marcati, P.
Osaka J. Math.
45 (2008), 211–214
ON A GENERALIZATION OF THE “DIV-CURL LEMMA”
INGENUIN GASSER and PIERANGELO MARCATI
(Received July 22, 2005, revised March 13, 2007)
Abstract
We present a generalization of the div-curl lemma to a Banach space framework
which is not included in the almost existing generalizations. An example is shown
where this generalization is needed.
1. The div-curl lemma and its generalization
In this note we present a generalization of the famous “div-curl lemma”, which
was first formulated by [1] in a Hilbert space setting. This lemma is widely used in
the analysis of nonlinear partial differential equations. In [5] the result was generalized
to an Banach space framework.
Here we present a further generalization to a setting, where on allows every com-
ponent vki , wki of the vectors vk , wk to lie in different L pi spaces (for details see be-
low). This is of special interest in problems arising from limiting procedures in the
hydrodynamic equations for plasmas and semiconductors [3], where both the original
version and the version presented in [5] are not sufficient for the analysis.
We denote by
(1) (curl w)i , j = wi ,x j − w j ,xi
the curl (matrix) of a vector field. The superscript ( )′ indicates the conjugate index
with 1 = 1=p + 1=p′. Then, the generalized version of the “div-curl lemma” reads
Theorem 1.1. Let U ∈ Rn be a bounded, open, smooth domain. Let 1 < pi <∞
for i = 1, : : : , n and let us denote pmin = min1≤i≤n pi and pmax = max1≤i≤n pi . Let
v
k(x), wk(x) ∈ Rn for k ∈ N satisfiing
• {vki }∞k=1 and {wki }∞k=1 are bounded sequences in L p
′
i (U ) and L pi (U ), respectively,
with 1=pmin − 1=n < 1=pmax.
• {div vk}∞k=1 lies in a compact set of W−1,t (U ), where t ≥ max1≤i≤n(p′i ) = (pmin)′.
• {(curl wk)i , j }∞k=1 lies in a compact set of W−1,si , j (U ) for 1 ≤ i , j ≤ n, where
min1≤ j≤n s j ,i ≥ pi for 1 ≤ i ≤ n.
2000 Mathematics Subject Classification. 35L65.
212 I. GASSER AND P. MARCATI
Then, the following convergence holds in D′(U )
(2) vk · wk * v · w,
where v, w denote the weak limits (of subsequences) of {vk}∞k=1 and {wk}∞k=1, respectively.
REMARK 1.1. Note that in the case n ≤ pmin the condition 1=pmin−1=n < 1=pmax
is always satisfied.
REMARK 1.2. The classical “div-curl lemma” is obtained by setting pi = p′i = 2
for 1 ≤ i ≤ n. Then {div vk}∞k=1 and {(curl wk)i , j }∞k=1 have to be in a compact set of
H−1(U ).
REMARK 1.3. As already mentioned in [5] a Banach space framework of the
“div-curl” lemma is given. There, the case of pi = p, 1 ≤ i ≤ n is covered.
Proof. The proof follows the ideas of the proof of the “div-curl lemma” given
in [2].
In a frist step we define functions uki , 1 ≤ i ≤ n as (unique) solutions of the problem
(3) −1uki = wki in U , uki = 0 on U .
The uki are uniformly bounded in W 2, pi (U ) for 1 ≤ i ≤ n.
In the second step we define the functions
(4) zk = −div uk , yk = wk −∇zk
with
(5) yki = wki − zkxi =
(
uk( j ,xi ) − uki ,x j
)
x j
= ((curl uk) j ,i )x j .
Therefore, {zk}∞k=1 is bounded in W 1, pmin (U ) and compact in Lr (U ) with 1=pmin−1=n <
1=r ≤ 1. The sequence {yki }∞k=1 is on one hand bounded in L pmin (U ) (second derivatives
of uk), on the other hand compact in Lmin1≤ j≤n (s j ,i )(U ). Indeed, according to the as-
sumptions (curlwk) j ,i is compact in W−1,s j ,i (U ). Therefore, (curl uk) j ,i lies compactely
in W 1,s j ,i (U ) according to the results in [4] (at this point the smooth boundary (C∞)
is required).
Then, the limits z, y, u of zk , yk , uk satisfy z = −div u, y = w −∇z and
(6) −1ui = wi in U , ui = 0 on U .
Finally, using a testfunction 8 ∈ C∞0 (U ) we obtain for min1≤ j≤n(s j ,i ) ≥ pi , 1 ≤
i ≤ n
(7)
∫
U
v
k · yk8 dx →
∫
U
v · y8 dx .
DIV-CURL LEMMA 213
Similarly, for t ≥ (pmin)′ = max1≤i≤n(p′i ) we have
(8)
∫
U
div vk zk8 dx →
∫
U
div vz8 dx .
Due to the assumption 1=pmin − 1=n < 1=pmax there exist always values of r such that
(where r is the parameter used above)
(9) 1
pmin
− 1
n
<
1
r
<
1
pmax
.
Then, the second inequality (in (9)) garantuees the convergence
(10)
∫
U
v
k · ∇8zk dx →
∫
U
v · ∇8z dx .
Combining (7)–(10) we obtain
(11)
∫
U
v
k · wk8 dx =
∫
U
v
k · yk8 dx −
∫
U
div vk zk8 dx −
∫
U
v
k · ∇8zk dx
→
∫
U
v · y8 dx −
∫
U
div vz8 dx −
∫
U
v · ∇8z dx
=
∫
U
v · w8 dx .
This ends the proof.
EXAMPLE. Suppose n = 2 and p1 < 2 (p′1 > 2), p2 > 2 (p′2 < 2). The curl
matrix has two (nonvanishing) elements (curl wk)1,2 = −(curl wk)2,1. Then t > p′1 and
s1,2 > p2 are required in order to apply the theorem. This is an examples where the
results in [5] do not apply.
References
[1] L. Tartar: Compensated compactness and applications to partial differential equations; in Non-
linear Analysis and Mechanics: Heriot-Watt Symposium IV, Pitman, Boston, Mass, 1979,
136–212.
[2] L.C. Evans: Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS
Regional Conference Series in Mathematics 74, Published for the Conference Board of the
Mathematical Sciences, Washington, DC, 1990.
[3] I. Gasser and P. Marcati: A quasi-neutral limit in the hydrodynamic model for charged fluids,
Monatsh. Math. 138 (2003), 189–208.
214 I. GASSER AND P. MARCATI
[4] C.E. Kenig: Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Prob-
lems, CBMS Regional Conference Series in Mathematics 83, Published for the Conference
Board of the Mathematical Sciences, Washington, DC, 1994.
[5] J.W. Robbin, R.C. Rogers and B. Temple: On weak continuity and the Hodge decomposition,
Trans. Amer. Math. Soc. 303 (1987), 609–618.
Ingenuin Gasser
Fachbereich Mathematik
Universität Hamburg
Bundesstraße 55, D-20146 Hamburg
Germany
e-mail: gasser@math.uni-hamburg.de
Pierangelo Marcati
Dipartimento di Matematica Pura e Applicata
Universitá degli Studi dell’Aquila
I-67100 L’Aquila
Italy
e-mail: marcati@univaq.it


On a generalization of the ''div-curl Lemma''

Epstein, Neil

Vraciu, Adela

A length characterization of *-spread

Vermeire, Peter

Osaka University
Title The Hilbert scheme of elliptic curves and reflexive sheaves onFano-3folds
Author(s)Vermeire, Peter
CitationOsaka Journal of Mathematics. 45(1) P.85-P.89
Issue Date2008-03
Text Versionpublisher
URL http://hdl.handle.net/11094/9548
DOI
Rights
Vermeire, P.
Osaka J. Math.
45 (2008), 85–89
THE HILBERT SCHEME OF ELLIPTIC CURVES
AND REFLEXIVE SHEAVES ON FANO 3-FOLDS
PETER VERMEIRE
(Received November 7, 2006, revised January 10, 2007)
Abstract
We study the cohomological properties of reflexive rank 2 sheaves on smooth
projective threefolds. Applications are given to the relationship between moduli
of reflexive sheaves and Hilbert schemes of associated elliptic curves on Fano
threefolds.
1. Introduction
We work over an algebraically closed field of characteristic 0.
In this paper we continue the study of reflexive sheaves on projective threefolds
[5], [6], [7]. Recall that a coherent sheaf F is torsion-free if the natural map of F to
its double-dual h: F → F∗∗ is injective, and that F is reflexive if h is an isomorphism.
Recall the following Serre correspondence for reflexive sheaves:
Theorem 1 ([4, 4.1]). Let X be a smooth projective threefold, M an invertible
sheaf with H1(X , M∗) = H2(X , M∗) = 0. There is a one-to-one correspondence between
(1) pairs (F , s) where F is a rank 2 reflexive sheaf on X with detF = M and s ∈ 0(F )
is a section whose zero scheme has codimension 2
(2) pairs (Y ,  ) where Y is a closed Cohen-Macaulay curve in X , generically a local
complete intersection, and  ∈ 0(Y , !◦Y ⊗ !∗X ⊗ M∗) is a section which generates the
sheaf !◦Y ⊗ !∗X ⊗ M∗ except at finitely many points.
Furthermore, c3(F ) = 2pa(Y )− 2 + c1(X )c2(F )− c1(F )c2(F ).
Note that if F is locally free, then the corresponding curve Y is subcanonical.
In Section 2 we give examples of the relationship between the structure and the
cohomology of a reflexive sheaf F . In particular, we give some cohomological criteria
to determine when a reflexive sheaf is actually locally free—this is equivalent to the
vanishing of the third Chern class and is needed in the last section. In Section 3, we
investigate the influence of global sections on the cohomology of F and in our main
result (Theorem 7) we give an application of these results to the case of elliptic curves
on Fano threefolds. The main result is another example of the relationship between the
moduli space of vector bundles and the Hilbert scheme of curves on a threefold [6].
1991 Mathematics Subject Classification. 14F05, 14D20, 14J60.
86 P. VERMEIRE
2. Local freeness via cohomology
As a simple example of the connection between the cohomology and the struc-
ture of F , recall that for a locally free sheaf F and a very ample line bundle L , if
hi (X , F ⊗ Ln) = 0 for i = 1, 2, n ∈ Z, we say F is L-aCM (arithmetically Cohen-
Macaulay), as in this case the associated curve Y is arithmetically Cohen-Macaulay (in
the embedding by L) if and only if F is L-aCM ([1], [2]). If F is reflexive we have:
Proposition 2. Let F be a rank 2 reflexive sheaf on a smooth projective 3-fold
X , L an ample invertible sheaf. If H2(X , F ⊗ Ln) = 0 for all n ≪ 0, then F is locally
free. In particular, F reflexive and L-aCM implies that F is locally free.
Proof. This is essentially [4, 2.5.1], where it is shown that H2(X ,F⊗ Ln) = c3(F )
for all n ≪ 0, and that c3(F ) = 0 if and only if F is locally free.
REMARK 3. It is shown in [8] that there are aCM curves on the general sextic
threefold which are not subcanonical. If C is such a curve, then by Proposition 2 no
reflexive sheaf F associated to C by the Serre correspondence is aCM.
As a second example, we look at a case where the Riemann-Roch formula be-
comes especially simple.
Proposition 4. Let F be a rank 2 reflexive sheaf on a smooth projective 3-fold
X with c1(F ) = c1(!X ). Then c3(F ) = 2 h2(X , F ) − 2 h1(X , F ). Hence the following
are equivalent:
(1) h2(X , F ) ≤ h1(X , F ).
(2) h2(X , F ) = h1(X , F ).
(3) (X , F ) = 0.
(4) F is locally free.
Proof. From the usual spectral sequence, we have the exact 5-term sequence
0 → H1(X , F∗ ⊗ !X ) → H2(X , F )∗ → H0(X , Ext1OX (F , !X ))
→ H2(X , F∗ ⊗ !X ) → H1(X , F )∗ → 0.
Further, by [4, 2.6] we have c3(F ) = h0(X , Ext1(F , !X )). Our hypotheses imply that
F∗ ⊗ !X = F and the statements immediately follow.
In general, we have the following:
DEFINITION 5. Let X be a projective threefold. A rank 2 reflexive sheaf F has
canonical parity if det F ⊗ !∗X = L⊗2 for some invertible sheaf L .
ELLIPTIC CURVES AND REFLEXIVE SHEAVES 87
By Proposition 4 we have
Corollary 6. Let F be a rank 2 reflexive sheaf with canonical parity on a smooth
projective 3-fold X . Then F is locally free if and only if
(X , F ⊗ (det F∗ ⊗ !X )1=2) = 0.
3. Sheaves associated to elliptic curves
Our main application is to the relation between the moduli space of torsion free
sheaves and the Hilbert scheme of curves.
Theorem 7. Let C ⊂ X be an irreducible local complete intersection curve with
pa(C) = 1 on a smooth projective threefold with Hi (X , OX ) = 0 for i ≥ 1, H2(X , !∗X ) =
0, and h0(C , !X ⊗ OC ) = 0 (e.g. X is Fano). Via the Serre correspondence one can
associate to C a rank 2 vector bundle F with det F = !∗X .
If F is simple then h1(C , NC=X ) ≥ ext2OX (F , F ), and h0(C , NC=X )− ext1OX (F , F ) ≥
h0(X , IC ⊗ !∗X )− h1(X , IC ⊗ !∗X ).
If we also have h1(X , IC ⊗ !∗X ) = 0, then h1(C , NC=X ) = ext2OX (F , F ) and
h0(C , NC=X )− ext1OX (F , F ) = h0(X , IC ⊗ !∗X ).
Proof (Cf. [6, Proposition 23]). Because detF = !∗X , F∗ has canonical parity and
so h1(X , F∗) = h2(X , F∗) by Proposition 4. We have the sequence
0 → !X → F∗ → IC → 0.
Now, H1(X , !X ) = H2(X , !X ) = 0 by hypothesis, hence H1(X , F∗) = H1(X , IC ) = 0.
By hypothesis, Hi (X , !X ) = 0 for 0 ≤ i ≤ 2 and h0(C , !X ⊗ OC ) = 0, therefore
hi (X , IC ⊗ !X ) = 0 for i = 0, 1. From the sequence
0 → !⊗2X → F∗ ⊗ !X → IC ⊗ !X → 0
we see H1(X ,F∗⊗!X ) = H1(X , !⊗2X ). However, H2(X ,F ) = H1(X ,F∗⊗!X )∗ by Serre
duality and H1(X , !⊗2X ) = H2(X , !∗X ) = 0 by hypothesis, hence H2(X , F ) = 0.
The short exact sequence in the first paragraph gives H3(X ,F∗) = H0(X ,!X ⊗F ) =
H0(X ,F∗) = 0. Then tensoring that sequence with F yields Exti
OX
(F ,F ) = Hi (X ,IC⊗F ).
88 P. VERMEIRE
Tensoring the standard sequence
0 → IC → OX → OC → 0
with F yields
0 → H0(X , IC ⊗ F ) → H0(X , F ) → H0(C , NC=X )
→ H1(X , IC ⊗ F ) → H1(X , F ) → H1(C , NC=X )
→ H2(X , IC ⊗ F ) → 0.
Noting that h0(X , IC ⊗ F ) = ext0OX (F , F ) = 1 by simplicity, we have
h1(F )− h0(F ) + 1 = h1(NC=X )− h0(NC=X ) + h1(IC ⊗ F )− h2(IC ⊗ F ).
By Serre duality, Hi (X , F ) = H3−i (X , F∗ ⊗ !X )∗ and so again from the sequence
0 → !⊗2X → F∗ ⊗ !X → IC ⊗ !X → 0
we see
h1(F )− h0(F ) + 1 = − h0(!∗X ) + h0(!∗X ⊗ OC ) + h1(!∗X ).
The long exact cohomology sequence above immediately implies that h1(C , NC=X ) ≥
ext2
OX
(F , F ). Combining this with the two equalities above gives
h0(NC=X )− ext1OX (F , F ) ≥ h0(!∗X )− h0(!∗X ⊗ OC )− h1(!∗X )
= h0(X , IC ⊗ !∗X )− h1(X , IC ⊗ !∗X ).
For the last statement, note that if h1(X , IC ⊗ !∗X ) = 0 then the sequence
0 → OX → F → IC ⊗ !∗X → 0
implies that H1(X , F ) = 0. Now the long exact cohomology sequence above gives us
h1(C , NC=X ) = ext2OX (F , F ), and h0(C , NC=X ) − ext1OX (F , F ) = h0(X , IC ⊗ !∗X ) since
we no longer have an inequality above.
It is easy to see that the hypotheses of Theorem 7 are not vacuous:
EXAMPLE 8. If C ⊂ X ⊂ P4 is a nondegenerate elliptic normal curve on a smooth
hypersurface X of degree 2 ≤ d ≤ 4, then the associated rank two vector bundle is
stable (hence simple).
Similarly, if C ⊂ X ∩ P3 ⊂ P4 is a degenerate elliptic normal (in P3) curve and
X is a smooth hypersurface of degree 4, then the associated rank two vector bundle is
stable.
ELLIPTIC CURVES AND REFLEXIVE SHEAVES 89
ACKNOWLEDGMENTS. I would like to thank the referee for a careful reading
of this manuscript and for providing useful criticism; in particular a simple argument
provided by the referee replaced a technical result used in an earlier version.
References
[1] A. Beauville: Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64.
[2] L. Chiantini and C. Madonna: ACM bundles on a general quintic threefold, Matematiche (Cata-
nia) 55 (2000), 239–258 (2002).
[3] R. Hartshorne: Algebraic Geometry, Springer-Verlag, New York, 1977.
[4] R. Hartshorne: Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176.
[5] P. Vermeire: Stable reflexive sheaves on smooth projective 3-folds, Pacific J. Math. 219 (2005),
391–398.
[6] P. Vermeire: Moduli of reflexive sheaves on smooth projective 3-folds, J. Pure Appl. Algebra
211 (2007), 622–632.
[7] P. Vermeire: An effective bound for reflexive sheaves on canonically trivial 3-folds, to appear.
[8] C. Voisin: Sur une conjecture de Griffiths et Harris; in Algebraic Curves and Projective Ge-
ometry (Trento, 1988), Lecture Notes in Math. 1389, Springer, Berlin, 1989, 270–275.
Department of Mathematics
214 Pearce
Central Michigan University
Mount Pleasant MI 48859
USA
e-mail: verme1pj@cmich.edu


The Hilbert scheme of elliptic curves and reflexive sheaves on Fano 3-folds

Lescot, Paul

Reduction theorems for characteristic functors on finite p-groups and applications to p-nilpotence criteria

Tomassini, Adriano

Vezzoni, Luigi

Contact Calabi-Yau manifolds and special Legendrian submanifolds

特集「大学におけるeラーニングプラットフォーム : システムの概要、将来展望と導入例

梶田, 将司

コース　カンリ　システム　ノ　オープンカ　ト　ダイガク　ノ　ジョウホウ　センリャク

山崎, 淳

カンノンミョウオウシュウ　ノ　セイカク　ト　ケンキュウ　ノ　カダイ

Osaka University

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