Cohomological study on variants of the Mumford system, and integrability of the Noumi-Yamada system

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by  Rei Inoue

Cohomological study on variants of the Mumford system,
and integrability of the Noumi-Yamada system
Rei Inoue † and Takao Yamazaki ‡
Abstract: The purpose of this paper is twofold. The first is to apply the method introduced in
6 the works of Nakayashiki and Smirnov [11, 12] on the Mumford system to its variants. The other is
0 to establish a relation between the Mumford system and the isospectral limit Q(I) and Q(II) of the
0 g g
2 Noumi-Yamada system [15]. As a consequence, we prove the algebraically completely integrability of
n the systems Q(I) and Q(II), and get explicit descriptions of their solutions.
g g
a
J

7 1 Introduction
1

3 Let g be a natural number. The Mumford system [10] is an integrable system with the Lax
v
8 matrix
4
0 v(x) w(x)
l(x)= ∈ M (C[x]), (1.1)
1 u(x) −v(x) 2
0 (cid:18) (cid:19)
5
where u(x) and w(x) are monic of degree g and g +1, and v(x) is of degree ≤ g −1. The
0
/ coefficients of u(x),v(x),w(x) constitute the phase space M ≃ C3g+1 equipped with the g
h g
p dimensionalvector fieldgeneratedbythecommutingoperatorsD ,...,D (seeTheorem3.2).
1 g
-
h Thecoefficients of −detl(x) = u(x)w(x)+v(x)2 are invariants of Di. For a monic polynomial
t
a f(x) of degree 2g+1, the level set is given by
m
v: Mg,f = {l(x) ∈ Mg | u(x)w(x)+v(x)2 = f(x)} ⊂ Mg. (1.2)
i
X It is a classical fact [10] that, when f(x) has no multiple zero, the level set M is isomorphic
g,f
r
toJ(X)\Θ,whereJ(X)andΘaretheJacobivarietyandthethetadivisorofthehyperelliptic
a
curve X defined by y2 = f(x).
We write A for the affine ring of M . Nakayashiki and Smirnov [11] pointed out the
f g,f
importance of the space A / D A of the classical observables modulo the action of D ’s.
f i i f i
They defined a complex C∗ such that Hg(C∗) is isomorphic to this space, and obtained the
f P f
following results on the cohomology groups Hk(C∗).
f
(1) When f(x) = x2g+1, the q-Euler characteristic of H∗(C∗) is determined ([11] eq. (17),
f
see also Theorem 3.3).
(2) Whenf(x)hasnomultiplezero,Hk(C∗)isisomorphictothesingularcohomologygroup
f
Hk(J(X)\Θ,C). The dimension of this cohomology is determined later by Nakayashiki
([13], see also Theorem 2.1).
†Research Institute of Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. E-mail address:
[email protected]
‡Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan. E-mail address:
[email protected]
1
The q → 1 limit in (1) coincides with the (usual) Euler characteristic in (2). This supports
their conjecture that dimHk(C∗) is independent of f(x).
f
The even Mumford system and the hyperelliptic Prym system are introduced in [17, 8]
in connection with the periodic Toda lattice and the periodic Lotka-Volterra lattice. They
both have similar properties to the Mumford system. In particular, the level set is generically
isomorphic to the Jacobi variety minus some translations of the theta divisor. For these
systems,wedefineacomplexanalogoustoC∗,andestablishthecounterpartsto(1),(2)above.
f
Actually, part (1) turns out to be a routine, but for part (2) we need a detailed analysis of
the theta divisors. Our main results are Theorems 2.2 and 3.5 for the even Mumford system,
and Theorems 2.3 and 3.8 for the hyperelliptic Prym system. These results show that the
q → 1 limit in (1) coincides with the Euler characteristic in (2) in this situation as well.
(1)
Our second aim is to study the integrability of the Noumi-Yamada system of A type
N−1
by making use of a relation with the Mumford system. Recall that the Noumi-Yamada
system was introduced as a higher order Painlev´e equation [15], and given by the following
autonomous equations:
g
∂q
k
= q q −q +e −e +αδ for N = 2g+1, (1.3)
k k+2i−1 k+2i k k+1 k,2g+1
∂t
i=1
X(cid:0) (cid:1)
∂q
k
= q q q −q q
k k+2i−1 k+2j k+2i k+2j+1
∂t
1≤i≤j≤g
X (cid:0) (cid:1) for N = 2g+2, (1.4)
g+1 g+1
α
+ (e −e )− q +(e −e ) q
k+2i−1 k+2i k k k+1 k+2i−1
2
i=1 i=1
(cid:0)X (cid:1) X
for k = 1,...,N. Here e and α are parameters, and we set the periodicity q = q and
k k+N k
e = e . The original form of this system was obtained in [19, 1]. The case of g = 1
k+N k
corresponds to the fourth and the fifth Painlev´e equations.
We consider the isospectral limit α = 0 of (1.3) and (1.4), and denote them by Q(I) and
g
Q(II) respectively. We prove their algebraically completely integrability by relating them to
g
the Mumford system in Theorem 4.6. Especially, we show that the level sets of Q(I) and
g
Q(II) are generically isomorphic to that of the Mumford system (1.2) and the disjoint union
g
of two copies of (1.2) respectively. We further obtain an explicit description of the solutions
of Q(I) and Q(II) in terms of the theta functions. While the original Painlev´e property of the
g g
Noumi-Yamada systems is lost at α = 0, we hope that one may obtain some information of
the solutions for the Noumi-Yamada systems ( of α 6= 0) by studying the perturbation theory
on the Mumford system around α = 0.
Before closing Introduction, we state the definition of the complete integrability of a finite
dimensional dynamical system to clarify our position. We follow [2] and [18] Chapter V:
Definition 1.1 Let M = {u = (u ,...,u )} ≃ Cm be the phase space equipped with the m
1 m H
dimensional commuting Hamiltonian vector field on M. Let F ,...,F ∈ C[u ,...,u ] be
1 mI 1 m
the integrals of motion of the vector field. For f = (f ,··· ,f )∈ CmI, the level set of M is
1 mI
defined as {u ∈M | F (u) = f , i = 1,...,m }.
i i I
2
(i) M is completely integrable if the dimension of the level set is m for generic f.
H
(ii) M is algebraically completely integrable if M is completely integrable and satisfies the
following conditions: the level set over generic f is isomorphic to an affine part of an abelian
variety of dimension m . On this abelian variety, the flows of the above vector fields are
H
linearized.
This paper is organized as follows: §2 is devoted to the computation of the cohomology of
affine Jacobi varieties. This part is technically independent of the rest of the paper. In §3 we
first recall the results of Nakayashiki-Smirnov on the Mumford system, then we explain how
their results are generalized to the even Mumford system and the hyperelliptic Prym system.
In §4 we study the integrability and solution of Q(I) and Q(II).
g g
Acknowledgement
The authors thank Atsushi Nakayashiki for his kind advice on approach to this subject. We
express our gratitude to Kiyoshi Takeuchi who explained the proof of Theorem 2.9 to us.
Finally, we also thank Yoshihiro Takeyama for a discussion.
2 Cohomology of affine Jacobi varieties
2.1 Summary of results
Let X be a hyperelliptic curve of genus g. Let J and Θ be the Jacobi variety and the theta
divisor associated with X. The following theorem is due to Nakayashiki.
Theorem 2.1 [13]
2g 2g
dimHk(J \Θ,C)= − for k = 0,1,...,g.
k k−2
(cid:18) (cid:19) (cid:18) (cid:19)
2g 2g
In particular, the Euler characteristic χ(J \Θ) is given by (−1)g( − ).
g g−1
(cid:18) (cid:19) (cid:18) (cid:19)
Let∞ and∞ betwo(distinct)pointsonX whichareconjugateunderthehyperelliptic
+ −
involution. Let O be a Weierstrass point on X. The main results in this section are the
following:
Theorem 2.2 Let Θ′ = Θ∪(Θ+[∞ −∞ ]). Then we have
− +
2g+1 2g+1
dimHk(J \Θ′,C) = − for k = 0,1,...,g.
k k−2
(cid:18) (cid:19) (cid:18) (cid:19)
2g+1 2g+1
In particular, the Euler characteristic χ(J \Θ′) is given by (−1)g( − ).
g g−1
(cid:18) (cid:19) (cid:18) (cid:19)
3
Theorem 2.3 Let Θ′′ = Θ∪(Θ+[∞ −O])∪(Θ+[∞ −O]). Then we have
+ −
2g+2
dimHk(J \Θ′′,C) = , for k = 0,1,...,g.
k
(cid:18) (cid:19)
2g+1
In particular, the Euler characteristic χ(J \Θ′′) is given by (−1)g .
g
(cid:18) (cid:19)
Therestofthissectionisdevotedtotheproofofthem. Thereaderwhoismainlyinterested
in an integrable system is advised to skip it in the first reading.
2.2 Notations and reformulation
We shall prove the theorems stated above in a slightly general form. After introducing some
notations, we formulate the general result in this subsection.
For a variety S, we let Hk(S) = Hk(S,C) and hk(S) = dimHk(S). We also use the
compact supportcohomology Hk(S) = Hk(S,C), and writehk(S) for its dimension. For each
c c c
0 ≤ r ≤ g, we let X(r) = Xr/S be the r-th symmetric product of X, which is identified
r
with the space of effective divisors of degree r. We regard X(r) as a subvariety of X(g) via
the map r P 7→ r P +(g−r)O. We let ϕ : X(g) → J be the Abel-Jacobi map with
i=1 i i=1 i
respect to O.
P P
For each 0 ≤r ≤ g−1, we define
W = W0 = im[X(r) ֒→ X(g) →ϕ J],
r r
W+ = W +[∞ −O], W− = W +[∞ −O],
r r + r r −
W0+ = W ∪W+, W0− = W ∪W−,
r r r r r r
W± = W+∪W−, W0± = W ∪W+∪W−.
r r r r r r r
We also let W = J. The following relations are deduced from [8] Lemma 2.4. (Note that
g
∞ +∞ is linearly equivalent to 2O.)
+ −
W ∩W+ = W0+ , W ∩W− = W0− , W+∩W− = W . (2.1)
r r r−1 r r r−1 r r r−1
Since Θ is the translate of W by Riemann’s constant, Theorems 2.1, 2.2 and 2.3 are
g−1
obtained as the special case r = g of the following theorem.
Theorem 2.4 For k = 0,1,...,r, we have
2g 2g
hk(W \W ) = − , (2.2)
r r−1 k k−2
(cid:18) (cid:19) (cid:18) (cid:19)
2g+1 2g+1
hk(W \W± ) = − , (2.3)
r r−1 k k−2
(cid:18) (cid:19) (cid:18) (cid:19)
2g+2
hk(W \W0± ) = . (2.4)
r r−1 k
(cid:18) (cid:19)
(For other values of k, they are zero.)
The proof of this theorem occupies the rest of this section.
4
2.3 Review of known results
We recall some results in a literature. The following propositions are due to Mumford and
Nakayashiki respectively.
Proposition 2.5 ([10] Proposition 1.2) The subvariety W \W of W is affine.
r r−1 r
Proposition 2.6 ([13] Corollary 2) The Euler characteristic of W is given by
r
2g−2 2g−2
χ(W )= (−1)r − .
r r r−2
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17)
Thefollowing Proposition is aconsequence of Macdonald’s explicit description [9](3.2), (6.3),
(14.1) and (14.3) of the basis of the cohomology groups Hk(J),Hk(X(r)) and their image
under the natural maps.
Proposition 2.7 The dimension of the image of the composition of the natural maps
2g 2g
Hk(J) → Hk(X(g)) → Hk(X(r)) is if k ≤ r and is if r ≤ k ≤ 2r.
k 2r−k
(cid:18) (cid:19) (cid:18) (cid:19)
Lastly, we recall a result of Bressler and Brylinski in the following form:
Proposition 2.8 ([5] Proposition 3.2.1.) For any r and k, we have a canonical isomorphism
Hk(W ,Q) ≃ IHk(W ,Q).
r r
Here the right hand side is the intersection cohomology with the middle perversity. In par-
ticular, Hk(W ) has a Hodge structure of (pure) weight k, and satisfies the Poincar´e duality
r
(see, for example, [7] §5.4).
In [5], this proposition is proved for r = g − 1 (and was used by Nakayashiki in [13]
Theorem 4). The proof for general r is identical, but we include a brief account here for the
reader’s convenience. We recall that the singular locus of W coincides with W if r < g
r r−2
([3] Chapter IV Corollary 4.5, Theorem 5.1). We fix 0 ≤ r < g and consider W . We have
r
a stratification W = X ∪X ∪··· where X = W \W . Note that the codimension
r 0 2 l r−l r−l−2
of X in W is l. Proposition 2.8 is a direct consequence of the following theorem applied to
l r
l = 0:
Theorem 2.9 Let j :X ֒→ W be the immersion. Then we have an isomorphism
Xl l r
(j ) Q ≃ Q for l = 0,2,4,....
Xl !∗ Xl Wr−l
We briefly recall the proof of Bressler and Brylinski (loc. cit.) The key idea is to use the
method of Borho and MacPerson [4]. We write π : X(r) → W for the map r P 7→
r i=1 i
[ r P −rO]. Then, π restricted to X is a fiber bundle with fiber Pl/2. This implies three
i=1 i l P
consequences: (i) π is semi-small in the sense of [4] §1.1. (ii) all the relevant pairs for π are
P
the constant sheaves Q on X (with multiplicity one) for each l = 0,2,4,··· . (See [4] §1.2
Xl l
5
for the definition of relevant pairs.) (iii) the stalk of Hk(Rπ Q ) at x∈ X is of rank one if
∗ X(r) l
k is even and k ≤ l, or is trivial otherwise. By the decomposition theorem due to Beilinson,
Bernstein and Deligne (see [4] §1.7), (i) and (ii) imply
Rπ Q ≃ (j ) Q [−l].
∗ X(r) Xl !∗ Xl
l=0,2,4,···
M
In view of (iii), each direct summand (j ) Q must be isomorphic to Q without any
Xl !∗ Xl Wr−l
higher cohomology sheaf.
2.4 Lemmas on W
r
We introduce two auxiliary lemmas concerning the cohomology of W . Let ak : Hk(W ) →
r r r
Hk(W ) and bk : Hk(J) → Hk(W ) be the maps induced by the inclusions W → W
r−1 r r r−1 r
and W → J.
r
Lemma 2.10 If 0≤ k ≤ r, the map bk :Hk(J) → Hk(W ) is an isomorphism. If r ≤ k ≤ 2r,
r r
there exists a canonical isomorphism ˇbk : Hk+2(g−r)(J)(g − r) −∼→ Hk(W ). (Here (g − r)
r r
indicates the Tate twist of the Hodge structure [6].) In particular, the dimension of Hk(W )
r
2g 2g
is or according to 0 ≤ k ≤ r or r ≤ k ≤ 2r.
k 2r−k
(cid:18) (cid:19) (cid:18) (cid:19)
Proof. Whenr = g,theassertionistrivial. Weshowtheassertion bythedecreasinginduction
on r. We assume the assertion for r. Thanks to Proposition 2.5, we have Hk(W \W ) = 0
c r r−1
if k < r. By the long exact sequence
··· → Hk(W \W ) → Hk(W ) →akr Hk(W ) → Hk+1(W \W )→ ··· ,
c r r−1 r r−1 c r r−1
we see that ak is an isomorphism if k ≤ r−2 and an injection if k = r−1. By the Poincar´e
r
duality assured by Proposition 2.8, we obtain a map aˇk : Hk(W ) → Hk+2(W )(1) which
r r−1 r
is an isomorphism if k ≥ r and a surjection if k = r−1. It remains to show the bijectivity
of ar−1 and aˇr−1. To show this, we compare the dimensions. By the inductive hypothesis we
r r
2g
have hr−1(W ) = hr+1(W ) = . Since we have proved the lemma for bk and ˇbk
r r r−1 r−1 r−1
(cid:18) (cid:19)
with k 6=r−1, we can compute hr−1(W ) as
r−1
2g
hr−1(W ) =(−1)r−1 χ(W )− (−1)khk(W ) = .
r−1 r−1 r−1 r−1
(cid:16) k6=Xr−1 (cid:17) (cid:18) (cid:19)
Here we used Proposition 2.6. This completes the proof. (cid:3)
Lemma 2.11 The map ak : Hk(W ) → Hk(W ) is surjective for any k and r. (When
r r r−1
k ≤ r−1, this is an isomorphism as proved in the above lemma.)
Proof. We fix k. We consider the following commutative diagram:
bk ak
Hk(J) →r Hk(W ) →r Hk(W )
r r−1
↓ ↓
c
Hk(X(g)) →dr Hk(X(r)).
6
It is enough to show the surjectivity of ak ◦ bk = bk for all r. Thus we shall show the
r r r−1
surjectivity of bk instead. The diagram shows the inequality
r
dimIm(d ◦c) ≤ dimImbk ≤ hk(W ).
r r r
ByProposition2.7andLemma2.10,weseethedimensionofIm(d ◦c)coincides withhk(W ).
r r
Therefore the equality holds in the inequality above, and the proof is done. (cid:3)
Remark 2.12 Assumek ≥ r.Viatheisomorphismsˇbk inLemma2.10, the mapak inLemma
r r
2.11 can be rewritten as (a Tate twist of)
Hk+2(g−r)(J) → Hk+2(g−r)+2(J)(1).
This map seems to coincide with the cup product with a hyperplane section, up to a multiplica-
tion by a non-zero constant. (This would imply Lemma 2.11 by the Hard Lefschetz Theorem.)
When r =g, this was shown by Nakayashiki [13].
2.5 Proof of (2.2)
We introduce the following notations:
ak
Hk = Hk(W ), Kk = ker[Hk ։r Hk ].
r r r r r−1
(If k ≤ r − 1 we have Kk = 0.) Recall that ak is surjective by Lemma 2.11. By Lemma
r r
2g 2g
2.10 we see the dimension of Kk is zero if 0 ≤ k ≤ r−1, and is − if
r 2r−k 2r−k−2
(cid:18) (cid:19) (cid:18) (cid:19)
r ≤ k ≤ 2r. The long exact sequence
··· → Hk(W \W )→ Hk(W ) →akr Hk(W ) → Hk+1(W \W )→ ···
c r r−1 r r−1 c r r−1
provides an isomorphism
Hk(W \W )∼= Kk.
c r r−1 r
BythePoincar´eduality(fortheusualcohomologytheory),weseehk(W \W )= h2r−k(W \
r r−1 c r
W ). This proves (2.2). We remark that our proof in the case of r = g is basically same
r−1
as Nakayashiki’s proof, except that the use of the Hard Lefschetz theorem was avoided in
Lemma 2.11 (see Remark 2.12).
2.6 Proof of (2.3)
According to (2.1), we have an exact sequence of sheaves on W± :
r
0 −→ Q −→ Q ⊕Q −→ Q → 0.
Wr± Wr+ Wr− Wr−1
(For simplicity we write Q instead of ι Q for a closed immersion ι : S ֒→ T.) The long
S ∗ S
exact sequence deduced from this implies (again by Lemma 2.11)
Hk(W±) ∼= Hk ⊕Kk.
r r r
7
We then consider the long exact sequence
··· → Hk(W \W± ) → Hk(W ) →frk Hk(W± ) → Hk+1(W \W± )→ ··· .
c r r−1 r r−1 c r r−1
With respect to the isomorphisms Hk(W )∼= Hk and Hk(W± )∼= Hk ⊕Kk , the map fk
r r r−1 r−1 r−1 r
reads as fk(x) = (ak(x),0). Therefore we obtain an exact sequence
r r
0 → Kk−1 → Hk(W \W± )→ Kk → 0.
r−1 c r r−1 r
2g+1 2g+1
In particular, hk(W \W± ) is − if r ≤ k ≤ 2r, and is zero otherwise.
c r r−1 2r−k 2r−k−2
(cid:18) (cid:19) (cid:18) (cid:19)
The Poincar´e duality shows hk(W \W± )= h2r−k(W \W± ). This proves (2.3).
r r−1 c r r−1
2.7 Proof of (2.4)
The first step is to study the cohomology of W0+. The relation (2.1) gives a resolution of the
r
sheaf Q :
Wr0+
0→ Q → Q ⊕Q → Q ⊕Q → Q ⊕Q → ··· .
Wr0+ Wr Wr+ Wr−1 Wr+−1 Wr−2 Wr+−2
We consider the deduced spectral sequence
Ei,j = Hj(W )⊕Hj(W+ ) ⇒ Hi+j(W0+).
1 r−i r−i r
i,j j⊕2 i,j
If we identify E with H , the boundary map d reads as
1 r−i 1
j⊕2 j⊕2 j j
H → H ; (x,y) 7→ (a (x−y),a (x−y)).
r−i r−i−1 r−i r−i
Therefore we obtain
j j
H ⊕K (i = 0)
i,j r r
E =
2 Kj (i > 0)
( r−i
i,j
Since the weight of the Hodge structure on E are different for different j’s by Proposition
2
2.8, we have the degeneration of the spectral sequence at E -terms and the decomposition [6]
2
min(k,r)
Hk(W0+)∼= Hk ⊕ Kk−i.
r r r−i
i=0
M
The same formula holds also for Hk(W0−).
r
The rest of the proof is similar to the previous subsection. By (2.1), we have a long exact
sequence
··· → Hk(W0±) → Hk(W0+)⊕Hk(W−) → Hk(W0− ) → Hk+1(W0±) → ··· ,
r r r r−1 r
and hence an exact sequence
min(k,r) min(k,r)
0 → Kk−i → Hk(W0±)→ Hk ⊕Kk⊕2⊕ Kk−i → 0.
r−i r r r r−i
i=1 i=1
M M
8
We put this description in the long exact sequence
··· → Hk(W \W0± ) → Hk(W ) →grk Hk(W0± ) → Hk+1(W \W0± )→ ··· .
c r r−1 r r−1 c r r−1
Then we get
min(k,r)
dimkergk = dimKk, dimcokergk = 2 dimKk−i ,
r r r r−i−1
i=0
X
so hk(W \ W0± ) = dimkergk + dimcokergk−1 can be determined by a straight forward
c r r−1 r r
computation. Now the Poincar´e duality hk(W \W0± ) = h2r−k(W \W0± ) completes the
r r−1 c r r−1
proof of (2.4). (cid:3)
3 Mumford system and its variants
3.1 Mumford system
The Mumford system [10] is described by the Lax matrix (1.1) with
u(x) = xg +u xg−1+···+u
1 g
v(x) = v3xg−1+v5xg−2+···+vg+1 (3.1)
2 2 2
w(x) = xg+1+w xg +···+w .
1 g+1
ThecoefficientsofthesepolynomialsconstitutethephasespaceM ≃ C3g+1. Wealsoconsider
g
the space of monic polynomials C[x]monic ∼= C2g+1 of degree 2g+1. We define a map
deg=2g+1
ψ :M → C[x]monic ; l(x) 7→ u(x)w(x)+v(x)2. (3.2)
g deg=2g+1
Recall (1.2) that for f(x) = x2g+1 +f x2g +···+f ∈ C[x]monic , the fiber ψ−1(f) is
1 2g+1 deg=2g+1
denoted by M . Assume f(x) has no multiple zero, and let X be the hyperelliptic curve X
g,f
of genus g defined by y2 = f(x). Let J(X) and Θ be the Jacobi variety of X and its theta
divisor. As mentioned in the introduction, we have the following theorem.
Theorem 3.1 [10] M is isomorphic to J(X)\Θ.
g,f
The space M is equipped with the structure of a dynamical system by the following
g
theorem.
Theorem 3.2 ([10] Theorem 3.1) There are the independent and commuting invariant vector
fields D ,...,D on M given by
1 g g
u(x )v(x )−v(x )u(x )
1 2 1 2
D(x )u(x ) =
2 1
x −x
1 2
1 w(x )u(x )−u(x )w(x )
D(x )v(x )= 1 2 1 2 −α(x +x )u(x )u(x ) (3.3)
2 1 1 2 1 2
2 x −x
1 2
(cid:16) (cid:17)
v(x )w(x )−w(x )v(x )
1 2 1 2
D(x )w(x )= +α(x +x )v(x )u(x )
2 1 1 2 1 2
x −x
1 2
where D(x)= g xg−iD , and α(x) = 1.
i=1 i
P
9
To make the notion of the dynamical system clearer, we introduce g times t ,...t given by
1 g
∂ = D . The coefficients of u(x)w(x)+v(x)2 are the invariants of D . Thus the vector fields
∂ti i i
are well-defined on the level set M . It is shown in [10] §5 that if f(x) has no multiple zero,
g,f
then the flow of D is linearized on J(X)\Θ.
i
The map ψ :M → C[x]monic induces an inclusion between their affine rings
g deg=2g+1
A= C[u ,...,u ,v ,...,v ,w ,...,w ] ←֓F = C[f ,...,f ].
1 g 3 g+1 1 g+1 1 2g+1
2 2
The actions of D given by (3.3) are naturally extended to A. Let C1 be the free A-module
i
with the basis dt ,...,dt , and let Ck = ∧kC1. We define the complex
1 g
0 −→ C0 −d→ C1 −d→ ··· −d→ Cg−1 −d→ Cg −→ 0, (3.4)
where the differential d is given by
g
d= dt ∧D : Ck → Ck+1. (3.5)
i i
i=1
X
Let f(x) be a monic polynomial of degree 2g+1. We write A for the affine ring of M .
f g,f
ThenA isaquotient ringofAdividedby therelation u(x)w(x)+v(x)2 = f(x).By tensoring
f
A over A with (3.4), we get the complex
f
0 −→ C0 −d→ C1 −d→ ··· −d→ Cg−1 −d→ Cg −→ 0.
f f f f
We note that the highest cohomology group of this complex is isomorphic to the space men-
tioned in the introduction
g
Hg(C∗)≃ A / D A .
f f i f
i=1
X
Furthermore, when f(x) has no multiple zero we have an isomorphism
Hk(J(X)\Θ,C)≃ Hk(C∗)
f
due to Theorem 3.1 and the algebraic de Rham theorem. This cohomology was computed in
Theorem 2.1.
We define a grading on A by setting deg(∗ ) = i for ∗ ∈ {u,v,w}. Then F ⊂ A is a
i
graded subring and deg(fi) = i. We set deg(Di) = i− 12 (note that DiA(j) ⊂ A(i+j−21)), and
deg(dt ) = −i+ 1 so that the degree of d is zero. We write A for A = A/ 2g+1f A,
i 2 0 x2g+1 i=1 i
which becomes a graded ring. Similarly, we write C∗ for the complex C∗ . The q-Euler
0 x2g+1 P
characteristic of C∗ is defined to be χ (C∗) = g (−1)kch(Ck), where we write ch(O) =
0 q 0 k=0 0
∞ qkdimO(k) for a graded space O= ⊕ O(k). The notations
k=0 k≥0 P
P [k] = (1−qk),
q
[k] [k−1] ···[1] for k ∈ Z ,
q q q >0
[k] !=
q ([k]q[k−1]q···[21]q for k ∈ 21 +Z≥0,
are used to state the following result of Nakayashiki-Smirnov.
10