Instability of Truncated Symmetric Powers of sheaves

0.31 MB·English

by  Lingguang Li

Instability of Truncated symmetric powers
2
1
0 Lingguang Li , Fei Yu
2

n Abstract
a
J Let X be a smooth n-dimensional projective variety over an al-

1 gebraic closed field k with chark > 0, F : X → X1 be the rela-
r
tive Frobenius morphism and E a vector bundle on X. X.Sun has
]
G
proved that the instability of Fr∗E is bounded by the instability of
A E ⊗Tl(Ω )(0 ≤ l ≤ n(p−1)). It is also known that there is an esti-
X
h. mate about the instability of tensor products(and symmetric powers
t etc.). In this paper we are interested in estimating the instability of
a
m the truncated symmetric powers Tl(E) (Theorem 5.6).
[

3 1 Introduction
v
8
2
Let X be a smooth n-dimensional projective variety over an algebraic closed
2
4 field k with chark > 0, F : X → X be absolute Frobenius morphisms of
. X, F : X → X1 be the relative Frobenius morphism and E a vector bundle
0 r
1 on X. Mehta and Pauly have showed that if E is semistable then F E is
r∗
0
also semistable when X is a curve of genus g ≥ 2 in ([6]). In the higher-
1
: dimensional case Sun has proved that the instability of F E is bounded by
v r∗
i the instabilities of E⊗Tl(Ω )(0 ≤ l ≤ n(p−1)) in [8]. There are also some
X X
further refined discussion in algebraic surface case in [3][9].
r
a It is also known that there is an estimation about the instability of tensor
products (and symmetric powers etc.)[5]. In this paper we are interested in
giving an estimation about the truncated symmetric powers Tl(E).
WecollectsomewellknownresultsaboutHarder-Narasimhanpolygon([7]),
strongly semistability [4][5] and direct image of torsion free sheaves under
Frobenius morphism ([8][10]).
We describe the main idea of the proof: Firstly, As we know that when
k is larger enough, the torsion free sheaf Fk0∗E has a Harder-Narasimhan
0
filtration such that all quotients are strongly semistable, then we can use it
1
to construct a flat family Tl(E) whose generic fiber is Tl(Fk0∗E) and cen-
ter fiber is Tl(⊕gri(Fk0∗E)). Secondly, the instability of Tl(Fk0∗E) can be
bounded by the instability of Tl(⊕gri(Fk0∗E)) by the upper semi continuity
of Harder-Narasimhan polygon. In fact, the bound of I(Tl(⊕gri(Fk0∗E))) is
computable, since we can use the direct sum decomposition of the long exact
sequence to look for the maximum and the minimum slope strong semistable
part of it and the slope is computed as symmetry product case. Finally, we
can use it to give a bound of instability of Tl(E).
2 Harder-Narasimhan polygon
Letk beanalgebraicallyclosedfieldofanycharacteristic. LetX beasmooth
n-dimensional projective variety over k with an ample divisor H. If E be a
rank r torsion-free sheaf on X then one can define its slope:
c (E)Hn−1
1
µ(E) =
r
Then E is slope H-semistable if for any nonzero subsheaf E ⊂ F we have
µ(E) ≤ µ(F). There exists a unique Harder-Narasimhan filtration:
0 = E ⊂ E ⊂ ... ⊂ E = E,
0 1 m
such that
µ (E) = µ(E ) > µ(E /E ) > ... > µ(E /E ) = µ (E)
max 1 2 1 m m−1 min
and E /E ,(1 ≤ i ≤ m) are semistable torsion free sheaves.
i i−1
Define the instability of E: I(E) = µ (E)−µ (E).
max min
ForanytorsionfreesheafGwemayassociatethepointp(G) = (rkG,degG)
in the coordinate plane. Now, we consider the points p(E ),...,p(E ) and
0 m
connect them successively by line segments and connecting the last point
with the first one. The resulting polygon HNP(E) is called the Harder-
Narasimhan polygon of E.
Proposition 2.1. [7] Let E be an flat family of torsion-free sheaf on X,
parameterized by the scheme S. Let s,s ∈ S with s a specialization of s.
0 0
Then HNP(E ) ≥ HNP(E )(in the partial ordering of convex polygons); i.e,
s0 s
the Harder-Narasimahan Polygon rises under specialization. In particular
I(E ) ≥ I(E ).
s0 s
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3 Strongly semistable
If p = char k > 0, let F : X → X be absolute Frobenius morphisms
of X,F : X → X1 be the relative k-linear Frobenius morphism, where
r
X1 := X × k is the base change of X/k under the Frobenius morphism of
k
Spec(k). The natural morphism X1 → X is an isomorphism. We say that
E is slope strongly H-semistable if for all m ≥ 0 the pull back (Fm)∗E are
slope (Fm)∗H-semistable.
For any torsion free sheaf E, there exists an integer k such that for any
0
k ≥ k and all quotients in the Harder-Narasimhan filtration of Fk∗E are
0
strongly semistable.
Let
µ (Fk∗(E)) µ (Fk∗(E))
max min
L = lim ,L = lim
max k→∞ pk min k→∞ pk
Clearly, L (E) = −L (E∗). By definition, E is strongly semistable if
min max
and only if L (E) = µ(E) = L (E).
min max
Let α(E) = max(L (E)−µ (E),µ (E)−L (E)). We denote
max max min min
Lmax(ΩX) ifµ (Ω ) > 0
L := p max X
X 0 ifµ (Ω ) ≤ 0
(cid:26) max X
This give a uniform bound of the difference of instability after Frobenius pull
back.
Corollary 3.1. [4] Let E be a torsion-free sheaf E of rank r, then
α(E) ≤ (r −1)L .
X
We also know that for any torsion free sheaves E (1 ≤ i ≤ l),
i
l
L (⊗l E ) = L (E )
max i=1 i max i
i=1
X
and⊗,F∗,F, ,Symofstronglysemistablebundlesarealsostronglysemistable.
r
Corollary 3V.2. Let E ,(1 ≤ i ≤ l) be torsion free sheaves on X, then
i
I(⊗l E ) ≤ l I(E )+2(−l + l rk(E ))L
i=1 i i=1 i i=1 i X
P P
3
Proof. We have L ( l E ) = l L (E ) by 2.3.3 in [4]. Since
max i=1 i i=1 max i
L (E) = −L (E∨) so we can get
min max
N P
l l l l
∨ ∨
L ( E ) = −L ( E ) = −L ( (E ) ) = L (E )
min i max i max i min i
i=1 i=1 i=1 i=1
O O O X
Hence,
l l l l l
I( E ) ≤ L ( E )−L ( E ) = L (E )− L (E )
i max i min i max i min i
i=1 i=1 i=1 i=1 i=1
O O O X X
l l
= (L (E )−µ (E )−L (E )+µ (E ))+ (µ (E )−µ (E ))
max i max i min i min i max i min i
i=1 i=1
X X
l
≤ 2( r −l)I(X)+ I(E ).
i i
i=1
X X
4 Direct images under Frobenius morphism
Let S be the symmetric group of l elements with the action on V⊗l by
l
(v ⊗···⊗v )·σ = v ⊗···⊗v for v ∈ V and σ ∈ S . Let e ,··· ,e be a
1 l σ1 σl i l 1 n
basis of V, for k ≥ 0 with k +···+k = l define
i 1 n
v(k ⊗···⊗k ) = (e ⊗k1 ⊗···⊗e ⊗kn)·σ.
1 l 1 n
σX∈Sl
Definition 4.1. [8] Let Tl(V) ⊂ V⊗l be the linear subspace generated by
all vectors v(k ,...,k ) for all k ≥ 0 satisfying k + ··· + k = l. It is a
1 n i 1 n
representation of GL(V). If V is a vector bundle of rank n, the subbundle
Tl(V) ⊂ V⊗l is defined to be the associated bundle of the frame bundle of
V⊗l(which is a principal Gl(n)-bundle )through the representation Tl(V).
By sending any ek1ek2...ekn ∈ Syml(V) to v(k ,...,k ), we have
1 2 n 1 n
Syml(V) → Tl(V) → 0
Which is an isomorphism in char 0. When char k = p, Tl(V) is isomorphic
to the quotient of Syml(V) by the relations ep = 0, 1 ≤ i ≤ n. Tl(V) is
called “truncated symmetric powers”[1].
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Proposition 4.2. [8]Let l(p) ≥ 0 be the unique integer such that 0 ≤ l −
l(p)p < p. Then in the category of GL(n)-representations, we have long exact
sequence
l(p) l(p)−1
0 −→ Syml−l(p)p(V)⊗ (F∗V) −→ Syml−(l(p)−1)p(V)⊗ (F∗V) −→ ...
k r k r
^ ^
−→ Syml−p(V)⊗ (F∗V) −→ Syml(V) −→ Tl(V) −→ 0 (1)
k r
For any vector bundle W^on X, there exists a canonical filtration
∗
0 = V ⊂ V ⊂ ... ⊂ V ⊂ V = V = F (F W)
n(p−1)+1 n(p−1) 1 0 r r∗
Such that the caonical connection ∇ : V → V ⊗ Ω1 induces injective
X
morphisms V /V →∇ (V /V ) ⊗ Ω1 and the isomorphisms V /V ∼=
l l+1 l−1 l X l l+1
W ⊗Tl(Ω1 ). Let
X
I(W,X) = Max{I(W ⊗Tl(Ω ))|0 ≤ l ≤ n(p−1)}
X
Using this, Sun has proved that instability of F W is bounded by instability
∗
of I(W,X):
Corollary 4.3. [8] Let X be a smooth projective variety of dim(X) = n,
whose canonical divisor K satisfies K ·Hn−1 ≥ 0. Then
X X
I(F W) ≤ pn−1rk(W)I(W,X).
r∗
5 Instability of truncated symmetric powers
Remark 5.1. Any torsion free sheaf F defines a rational vector bundle V
(vector bundle over a big open set). A rational vector bundle V can be extend
to many torsion free sheaves, a canonical one is jV. In our following dis-
!
cussion, since codimension 2 part doesn’t affect our slope, for convenience,
we can choose a common big open set, such that all torsion sheaves restrict
on it are vector bundles. Thus we can only consider the vector bundle case.
Lemma 5.2. In the short exact sequence:
0 −→ E −→ F −→ G −→ 0
Any two of E,F,G are semistable(strongly semistable) with the same slope,
then theotheris trival(codim(support) ≥ 2)orsemistable(stronglysemistable)
with the same slope.
5
Proof. Assume E,F are semistable with the same slope. If G is nontrival,
let G be last item in the Harder-Narasimhan filtration
m−1
0 = G ⊂ G ⊂ ... ⊂ G ⊂ G = G.
0 1 m−1 m
By surjective homomorphism F −→ G/G , we know that
m−1
µ(G/G ) ≥ µ(F) = µ(G) ≥ µ(G/G )
k−1 k−1
So G is semistable. For any integer k, we consider the following exact se-
quence
0 −→ Fk∗E −→ Fk∗F −→ Fk∗G −→ 0.
We get strongly semistable case by a similar discussion.
Remark 5.3. In fact, for the long exact sequence:
0 −→ H −→ ... −→ H −→ H −→ 0
1 j−1 j
As we have showed above, if H is strongly semistable, µ(H ) = u or H
j−1 i i
trival (0 ≤ i ≤ j−1). Then H is trival or strongly semistable with µ(H ) =
j j
u. This is enough for our following application.
Proposition 5.4. Assume E = E ⊕ ... ⊕ E with E (1 ≤ i ≤ m) are
1 m i
strongly semistable torsion free sheaves and µ(E ) ≥ µ(E ) ≥ ··· ≥ µ(E ),
1 2 m
r = rk(E). If l ≥ r(p−1), then I(TlE) = 0. If l < r(p−1), then we have
I(TlE) ≤ Min{l,[r/2](p−1)}I(E).
Proof. Step1: ConstructionofHarder-Narasimhanfiltration. Since⊗,F∗, ,Sym
r
of strongly semistable vector bundles are also strongly semistable, so
V
b1 bi bm
E := Ea1⊗···⊗Eai ···⊗Eam⊗ F∗E ⊗···⊗ F∗E ⊗···⊗ F∗E
a1...am,b1...bm 1 i m r 1 r i r m
^ ^ ^
isatrivalorstronglysemistabledirectsumpartofSyma1+...+amE⊗ b1+...+bmE,
for any non-negative integers a ,b ,(1 ≤ i ≤ m). If E 6= ∅, then
i i a1...am,b1...bm V
µ(E ) = (a +pb )µ(E )+...+(a +pb )µ(E ).
a1...am,b1...bm 1 1 1 m m m
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Using the natural homomorphism F∗E −→ Symp(E ), we can decom-
r i i
pose the long exact sequence (4.2) into strongly semistable direct sum part
according to µ = c µ(E )+...+c µ(E ),c +...+c = l,
1 1 m m 1 m
l(p) l(p)−1
0 −→ (Syml−l(p)p(E)⊗ (F∗E)) −→ (Syml−(l(p)−1)p(E)⊗ (F∗E)) −→
k r µ k r µ
^ ^
... −→ (Syml−p(E)⊗ (F∗E)) −→ Syml(E) −→ Tl(E) −→ 0
k r µ µ µ
So by Remark 5.3, Tl(E)^decompose into strongly semistable part Tl(E)
µ
according to µ = c µ(E )+...+c µ(E ),c +...+c = l.
1 1 m m 1 m
Step 2: The computation of I(Tl(E)).
Letr = rk(E),r = rk(E ),wechoosealocalbase{x ,...,x }
i i r1+..+ri−1+1 r1+..+ri
of E . For l ≤ r(p−1),l = t(p−1)+s,0 ≤ s < p−1, write:
i
(d ,...,d ,d ,...,d ) = (p−1,..,p−1,s,...,0)
1 t t+1 r
Then r xdi ( r xdi ) belong to the maximum (minimum) slop stong
i=1 i i=1 r−i+1
semistable part of Tl(E). Denote µ(x ) = µ(E ), if x ∈ E , Then
i j i j
Q Q
r r
µ (TlE) = d µ(x ),µ (TlE) = d µ(x )
max i i min r−i+1 i
i=1 i=1
X X
r [r/2]
µ (TlE)−µ (TlE) = (d −d )µ(x ) = (d −d )(µ(x )−µ(x ))
max min i r−i+1 i i r−i+1 i r−i+1
i=1 i=1
X X
[r/2]
≤ (d −d )(µ (E)−µ (E)) =
i r−i+1 max min
i=1
X
lI(E) l ≤ [r/2](p−1)
(r(p−1)−l)I(E) r = 2k,r(p−1) > l > [r/2](p−1)

(r(p−1)−l+(d −(p−1)))I(E) r = 2k +1,r(p−1) > l > [r/2](p−1)
 k+1

 0 l ≥ r(p−1)



Let E ,E be torsion free sheaves on X, The projective space
1 2
P := P(Ext1 (E ,E )∨ ⊕k)
X 2 1
7
parameterizes all extensions of E ,E , including the trivial one E ⊕E . And
1 2 1 2
there is a tautological family
∗ ∗ ∗
0 −→ q E ⊗p O (1) −→ E −→ q E −→ 0
1 P 2
on P × X. Since any extension have the same Hilbert polynomial, so E is
P-flat.([2],p.186)
For any non-split extension
0 −→ E −→ E −→ E −→ 0
1 2
WecanconstructatorsionfreesheafE onX×P1,suchthatE| = E(s 6= 0)
X×s
and E| = E ⊕E , we denote this E E ⊕E .
X×0 1 2 1 2
Proposition 5.5. Let E be a torsion free sheaf on X,
0 ⊂ E ⊂ ... ⊂ E = E
1 m
be a filtration such that E /E (1 ≤ i ≤ m) be torsion free sheaves, then
i i−1
I(Tl(E)) ≤ I(Tl(⊕m (E /E ))).
i=1 i i−1
Proof. We get a series of specialization as following
E E /E ⊕E ... E /E ⊕E /E ⊕...⊕E
m m m−1 m−1 m m−1 m−1 m−2 1
Thus construct a family E of torsion free sheaf, then TlE. There is a long
exact sequence of TlE:
l(p) l(p)−1
0 −→ Syml−l(p)p(E)⊗ (F∗E) −→ Syml−(l(p)−1)p(E)⊗ (F∗E) −→ ...
k r k r
^ ^
−→ Syml−p(E)⊗ (F∗E) −→ Syml(E) −→ Tl(E) −→ 0
k r
Tl(E) is also a flat family. ^
Sincerestrictfunctorisexactandcommuteswithtensor(symmetric,wedge,F∗)
r
operations. By the upper semicontinuity ofthe Harder-Narasimhan polygons
(Proposition 2.1), We have:
I(Tl(E)) = I((TlE) ) ≤ I((TlE) ) = I(Tl(⊕i=m(E /E ))).
s s0 i=1 i i−1
8
Theorem 5.6. For any torsion free sheaf E of rank r, If l ≥ r(p−1), then
I(TlE) = 0. If l < r(p−1), then
I(TlE) ≤ Min{l,[r/2](p−1)}(I(E)+2(r −1)L ).
X
Proof. There is an integer k , such that if
0
0 ⊂ E ⊂ ... ⊂ E = Fk0∗E
1 m
is the Harder-Narasimhan filtration of Fk0∗E, then E /E (1 ≤ i ≤ m) are
i i−1
strongly semistable torsion free sheaves. Therefore,
I(TlE) ≤ I(Fk0∗(TlE))/pk0
= I(Tl(Fk0∗E))/pk0
≤ I(Tl(⊕m (E /E )))/pk0
i=1 i i−1
≤ Min{l,[r/2](p−1)}I(⊕m (E /E ))/pk0
i=1 i i−1
= Min{l,[r/2](p−1)}(L (E)−L (E))
max min
≤ Min{l,[r/2](p−1)}(I(E)+2(r−1)L ).
X
ThesecondandthirdinequalitiesfollowfromProposition5.5andProposition
5.4, respectively. This completes the proof.
By Theorem 5.6 and Corolary 3.2, we have the following corollary.
Corollary 5.7. Let W be a torsion free sheaf on X, then
I(W,X) ≤ I(W)+[n/2](p−1)I(Ω )+(2(rk(E)−1)+2(n−1)[n/2](p−1))L
X X
Acknowledgments: We would like to express our thanks to Prof. Xi-
aotao Sun. The second author also express his thank to Prof. Kang Zuo for
the invatation to visit Johannes Gutenberg University.
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Academy of Mathematics and Systems Science,
Chinese Academy of Science
Beijing, P.R.China
E-mail: [email protected], [email protected]
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