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УДК 515:17+517:545
The Spaces of Meromorphic Prym Differentials on Finite Tori
Olga A. Chuesheva*
Kemerovo State University, Krasnaya, 6, Kemerovo, 650043,
Russia
Received 12.01.2014, received in revised form 24.02.2014, accepted 30.03.2014 In this article we construct all kinds of elementary Prym differentials for arbitrary characters on a variable torus with a finite numbers of punctures and find the dimensions of two important quotient spaces. As a consequence, this yields the dimension of the first holomorphic de Rham cohomology group of Prym differentials for arbitrary characters on torus. Also, we construct explicit bases in these quotient spaces.
Keywords: Prym differentials for arbitrary characters, the Gunning cohomological bundle over the Te-ichmuller space torus with a finite numbers of punctures.
Introduction
The theory of multiplicative functions and Prym differentials in the case of special characters on a compact Riemann surface has found applications in the geometric function theory of complex variable, the analytic number theory and in the mathematical physics [1-7]. In [1-3] the development of the general theory of multiplicative functions and Prym differentials on a compact Riemann surface of genus g > 2 for arbitrary characters has been started. The function theory on compact Riemann surfaces differs substantially from that on finite Riemann surfaces even for the class of single-valued meromorphic functions and abelian differentials. A number of basic spaces of functions and differentials on a finite Riemann surface F' of type (g, m), with g ^ 1,m > 0, are infinite-dimensional.
In this article we start constructing the general function theory on a variable torus with a finite numbers of punctures for multiplicative meromorphic functions and differentials with arbitrary characters. We construct all kinds of elementary Prym differentials for arbitrary characters on such surfaces and find the dimensions of two important quotient spaces. As a consequence, this yields the dimension of the first holomorphic de Rham cohomology group of Prym differentials for arbitrary characters on torus. Also, we construct explicit bases in these quotient spaces.
1. Preliminaries
Fix a smooth compact oriented surface F of genus g =1, with a marking {a, b}, which is an ordered tuple of generators for n1(F). Let F0 be a fixed complex analytic structure on F. From now on, for brevity, the Riemann surface (F; F0) will be denoted by F0. Fix distinct points P1,...,Pm G F. Suppose that F' = F\{P1,...,Pm} is a surface of type (1,m), with m > 1. Denote by r' the Fuchsian group of the first kind that acts invariantly on the disk U = {z G C : |z| < 1} and uniformizes the surface F0, i.e. F0 is conformally equivalent to U/r'. This group has the representation r' = {A, B, y1, ..., Ym : [A,B]j1 ...jm = I), where [A, B] = ABA-1B-1 for A, B G r', and I is the identity mapping [6].
* [email protected] © Siberian Federal University. All rights reserved
Every complex analytic structure on F' is determined by some Beltrami differential ^ on F0; i. e., an expression of the form ^,(z)dz/dz independent of the choice of a local parameter on F0, where ^(z) is a complex function on F0 and IImIIl^f;) < 1- Denote this structure on F' by F^. It is clear that ^ = 0 corresponds to F0. Let M (F') be the set of all complex analytic structures on F' endowed with the topology of Cœ convergence on F0, Di// + (F') be the group of all orientation-preserving smooth diffeomorphisms of F' leaving all punctures fixed, and Di//0(F') be the normal subgroup of Di// +(F') consisting of all diffeomorphisms homotopic to the identity diffeomorphism of F0. The group Di// +(F ') acts on M (F ') as ^ ^ / where / G Di// + (F '), ^ G M (F '). Then the Teichmuller space Tim(F ') = Ti m(F0) is the quotient space M(F')/Di//o(F') [6].
Since the mapping U ^ FO = U/r' is a local diffeomorphism, every Beltrami differential ^ on FO lifts to a Beltrami r'-differential ^ on U; thus, ^ G LTO(U), IImIIl^,^) = esssup |^(z)| < 1,
zGU
and m(T(z))T'(z)/T (z) = m(z), z G U, T G r'.
Extend the r'-differential m on U to C\U by putting m = 0. Then there exists a unique quasiconformal homeomorphism wM : C ^ C with fixed points +1, — 1,i, which is a solution to the Beltrami equation wz = m(z)wz. The mapping T ^ TM = wMT(wM)-1 determines an isomorphism of r ' onto the quasi-Fuchsian group = wMr '(wM)-1 = (AM,BM,7f, ...,7^ :
[A^]7f ...7& = I >.
The classical results of Ahlfors and Bers [6], and other authors assert that 1) T1 ,m(F') is a complex manifold of dimension m for m > 1; 2) T1 ,m(F') carries a unique complex analytic structure such that the natural mapping ^ : M(F ') ^ T1 ,m(F ') is holomorphic; furthermore, ^ has only local holomorphic sections; 3) the elements of depend holomorphically on the moduli [m] of finite Riemann surfaces F^.
Two Beltrami r '-differentials m and v are conformally equivalent if and only if wMT(wM)-1 = wvT(wv)-1, with T G r '. It is natural that the choice of generators (a,b} U {71,...,7m} in n1(F') is equivalent to the choice of a systems of generators {aM,bM} U (7f,...,7m} in n1(F^), and (AM, BM} U {7^,..., 7^} in for every [m] in T1jm. This implies the identifications M(F')/Di//0(F') = T1jm(F') = T1jm(r'). Furthermore, there is a bijective correspondence between the classes of Beltrami differentials [m], the classes of conformally equivalent marked finite Riemann surfaces [F^; (aM, bM} U {7^,..., 7^,}] and marked quasi-Fuchsian groups r^ [5, 6]. In [6, p. 99] Bers constructed a holomorphic abelian differential Z[m] = dz on FM for ev-
r
ery [m] G T1, with the condition / Z([M],w)dw = 1, C G C, and it depends holomorphically on the moduli [m] for FM. Moreover, the b-period on FM is the complex number r b»(h)
M = Z(M,w)dw, C G C, and it depends holomorphically on the moduli [m].
Define for arbitrary fixed [m] G T1 and £0 G C the classical Jacobi mapping ^ : C ^ C
r5
by ¥>(£) = Z([M],w)dw. The quotient space J(FM) = C/L(FM) is called the marked Jacobian
■ho
variety for FM, where L(FM) is the lattice over Z generated by the elements 1 and m. The universal Jacobian variety for torus is a fibration over T1, whose fiber over [m] G T1 is the Jacobian J(FM) of the surface FM [5,7].
Next, given an integer n > 1, there exists a fibration over T1 whose fiber over [m] G T1 is the space of all degree n integer divisors on FM. The holomorphic sections of this bundle determine on every FM a degree n integer divisor which holomorphically depends on [m]. Also there exists a holomorphic mapping from this bundle into the universal Jacobian bundle, n > 1, whose restriction to the fibres extends the Jacobi mapping ^ : FM ^ J(FM) = FM. We can obtain local holomorphic sections of these bundles over a neighborhood U([m0]) C T1 (for every n > 1) from local holomorphic Earle sections s of ^ : M(F) ^ T1 over U([m0]) [7].
A character p for F; is every homomorphism p : (n1(F^), •) ^ (C*, •), C* = C \ {0}. Each character is uniquely determined by an ordered tuple (p(a;), p(b;), p(y;),..., p(Ym)) G (C*)2+m.
Definition 1.1. A multiplicative function f on F'; for a character p is a meromorphic function f on w;(U) such that f (Tz) = p(T)f (z), z G w;(U), T G r;.
Definition 1.2. A Prym q-differential with respect to a Fuchsian group r' for p, or (p, q)-differential, is a differential w(z)dzq such that w(Tz)(T'z)q = p(T)w(z), z G U, T G r', p : r' ^ C*, q G N.
If f0 is a multiplicative function on F; for p without zeros and poles then f0(P) =
fo(Po)exp / 2nic(M,p)C([m]), where Po[m] = fs[;](Po) G F;, c([M],p) G C, c depend holo-
Jp0 [;]
morphically on [m] and p. Furthermore, integration is performed from a fixed point P0 [m] to the current point P on the variable surface F;, and s[m] is the Earle section [7] over U([mo]) C T1. We deduce that the character p for f0 is of the form p(a;) = exp2nic([M],p), p(b;) = exp(2nic([M], p)m). Refer to these characters p as unessential, while to f0 with this character, as a unit. The characters which are not unessential we call essential on n1(FM). Denote by Hom(r, C*) the group of all characters on r with the natural multiplication. The unessential characters constitute a subgroup L1 of Hom(r, C*) [1].
Definition 1.3. ^ on F' = U/r' for p is called multiplicatively exact, whenever ^ = df (z) and f (Tz) = p(T)f (z), T G r', z G U; thus, f is a multiplicative function on F' for p.
Given p G Hom(r;, C*), denote by Z 1(r^, p) the set of all mappings ^ : r; ^ C such that ^(ST) = + p(s)^(T), s,t G r; [1].
Let ^ be a closed Prym differential on F0 for p. Integrating this differential, we obtain f(Tz) - f(Tzo) = p(T)(f(z) - f(zo)), where $ = df(z), z G U, f(z) is a Prym integral on the disk U for which is determined up to an additive term. Hence, T G r' satisfies f (Tz) = p(T)f(z) + f o(T), where f(T) = f (Tzo) - p(T)f (zo). Therefore, the period mapping : r' ^ C for ^ is defined. It depends on the choice of a Prym integral f(z)
on U and a base point z0. Given another Prym integral f1(z) = f(z) + c for we have f ,z0 (T) = $f,z0 (T) + ca(T), a(T) = 1 - p(T), T G r'. It is easy to verify that both mappings f Z0 and &flJ0 satisfy the cocycle relation ^(ST) = ^(S) + p(S)^(T), S,T G r'. They belong to the space Z 1(r', p) and represent the same class of periods in H 1(r',p) = Z1(r', p)/B1(r', p) for the Prym differential ^ for p on F', where B1 (r', p) is generated by a.
For a closed Prym differential ^ we can determine the classical periods. For T G r' the
j- TZ0
corresponding classical period ^Z0 (T) = / ^ and we have the equality ^Z0 (T) = fZ0 (T) -
Z0
f (zo)a(T).
Consequently, the mappings of the form T ^ fZ0 (T) (periods in the sense of Gunning) and of the form T ^ ^Z0 (T) (the classical periods) determine the same class of periods G H 1(r', p) for a Prym differential ^ on F' for p. Thus, we have a well-defined C-linear mapping p : ^ ^ from the vector space of closed Prym differentials ^ on F' for p in the vector space H 1(r' ,p).
Denote by Q2jp(F;) the space of Prym differentials of the second kind with finitely many poles on F; for character p [5,2]. The space A1(p) consists of Prym differentials for p on F' that have finitely many poles on F' and extend meromorphically on F.
Lemma 1.1. If the differential w G Q2jP(F;)nA1(p) has the class of periods [w] = 0 in H 1(r;,p), then w is a multiplicatively exact differential on F'; for p.
Proof. It suffices to verify this for a fixed surface and a fixed character. We obtain the classical periods 71,..., Yfc while going separately around the poles Q1,..., Qk of the differential w. They
all vanish, being equal to the residues at the poles of second or higher order for the branches of our multivalued differential w the second kind.
If the class of periods [w] = 0, then the classical period wZo (T) = ca(T), c = 0 for every T, where wZo (T) = /(Tz0) — /(z0) = c(1 — p(T)), while / is some Prym integral for w. Then / = (/ — c) is a multiplicative function for p and w = d/ = d(/ — c). Thus, the periods Wzo jo (a), Wzo jo (b), Wzo jo (71),..., wzo jo (7m) in the sense of Gunning all vanish for some representative of class [w]. Consequently, w is a multiplicatively exact differential for p on F;. The proof of Lemma 1.1 is complete. □
A divisor on F; is a formal product D = P^1 ... pmk, pj G F;, m^ G Z, j = 1,..., k.
Theorem (Riemann-Roch's theorem for characters [5,2]). Let F be a compact Riemann surface of genus one. Then for every divisor D on F and every character p the equality rp(D-1) = degD + ip-i (D) holds.
Theorem (Abel's theorem for characters [5, 2]). Let D be a divisor on a marked variable compact Riemann surface [F;, {a;, b;}] of genus one and p a character on n1(FM). Then D is a divisor of a multiplicative function / on F; for p ^ degD = 0 and
y(D) = ^ logp(bM) — ¿Mlogp(aM)(= V(p, [m]))
in C modulo the integer lattice L(F;) generated by the complex numbers 1 and m.
Observe that by a theorem of Bers [6, p. 99], the mapping ^ depends locally holomorphically on p and [m].
Every element ^ G Z 1(r ', p) is unique determined by the ordered tuple of complex numbers ¿(A), ¿(B), ¿(71),...,^(7m).
Lemma 1.2 ( [8]). For every ¿ G Z 1(r ', p) there holds the equality
m— 1
a(B^(A) — a(A^(B) + ¿(71) + £ p(71 ... y^-+1) = 0. (1)
j=1
Lemma 1.3 ( [8]). The holomorphic principal Hom(r', C*)-bundle E = |J Hom(r' , C*) is analytically equivalent to the trivial bundle T1 ,m(F') x Hom(r', C*) over the base T1 ,m(F').
The set G' = |J H 1(r^,p) is called the Gunning cohomological bundle over the base
[;],p=1
T1,m x Hom(r', C*)\{1} [1]. For G' and p =1 we use Gunning's isomorphism [1] between the complex vector space H 1(r^,p) and the vector space Homp([r;, r;], C), that consists of homomorphisms ¿0 : [r;,r;] ^ (C, +) with the condition ¿0(STS—1) = p(S^0(T), T G [r;, r;], S G r;. Here [r;, r;] is the commutant group r;. Thus, the bundle G' is isomorphic to the vector bundle |J Homp([r;, r;], C). [;],p=1
Moreover, we can determine the transition matrices of this bundle in terms of two coordinate neighborhoods U1 = {p : p(A) = 1}, U2 = {p : p(B) = 1}, which cover the base Hom(r ', C*)\{1} provided that p(7j-) = 1, j = 1,...,m. For the neighborhood U1 we have a(A;) = 0. Every element ¿0 G Homp([r;, r;], C) for p G U1 can be defined as ¿0 = ¿;|[r ] for ¿; G Z 1(r;,p) such that ¿1(A;) = 0 and ¿1(T) = ct(A;) —^([T, A;]), T G r; [1].
Theorem ( [1,8]). The Gunning cohomological bundle G' over T1jm(F') x (Hom(r', C*)\{1}) is a holomorphic vector bundle of rank m for m ^ 1.
2. Elementary Prym differentials
In the construction of a general theory of single-valued and multiplicative differentials, an important role is played by the elementary differentials [3,5] of arbitrary order with the minimal number of poles, either one pole of order > 1, or two simple poles, depending holomorphically on the character p and the moduli [m] of Riemann surfaces. In this section we find the general form of elementary (p, q)-differentials on F».
Proposition 2.1 ( [2]). A degree 0 divisor D is a divisor of a meromorphic (p, q)-differential w on a compact Riemann surface F of genus g =1 for a character p with q ^ 1 if and only if V(D) = ^(p) in J(F).
The proof proceeds as in the case q =1 considered in [3,5], taking into account that —2K = 0 in J(F) for the torus F.
Theorem 2.1. Given a point Q, a character p on F» of type (1,m), m ^ 1, and natural numbers n ^ 2, q ^ 1 there exists an elementary (p, q)-differential t^. q of the class Ai(p) with a unique pole Q of order exactly n on F». The general form of its divisor is (t^.q) = RlQnRN Pfcl *km , where y>(Ri) = v(Qn) — v№ ... Rn) + V^1 ) + • • • + V^™ ) + V>(p), kj > 0,
^ p1 ... pr
kj G N, j = 1,..., m. The points R2,...,RN are chosen as a local holomorphic section of divisors of degree N — 1 on F»\{Q}, and N = n + k1 + • • • + km. Moreover, these differentials
depend locally holomorphically on [m] and p, and for an essential character we have t^. q = (£ + O(1)ldzq, z(Q) = 0. "
Proof. Given q ^ 1, find the general form of (p, q)-differentials of the second kind with a unique pole at the point Q of order exactly n > 1 on F».
The Riemann-Roch theorem for (p, q)-differentials on F; [3] yields the dimension
V.g^npki^.prr) = dimc np(Qnpt1...p!r), where kj ^ 0,j = 1,...,m. We have v,? (D) =
— deg D + r^, where D = npk 1 pkr , f [m] is an arbitrary multiplicative function for p
i
on F;, which locally holomorphically depends on [m] and p [3]. Hence
ij,? \ ^ nk 1 nfc r
n + k1 + ••• + km > 1. Here r^= 0, since deg ^> 0 under our assumptions. Indeed, deg(f [m]) = 0 and deg (D) ^ m > 0. We can prove this fact in a different way. If there exists a function g = 0 for p on F; satisfying (g) > QnP'-f 1 .. .P^r (f [m]), then 0 = deg(g) > deg(QnPf 1 .. .P^™(f [m])) > 2; this is a contradiction.
It is clear that V,g(Qnpk1 pfc™ ) = V,g(Qn-1 ^i Pfc™ ) + 1. Consequently there exists a (p, q)-differential tJ^q with a pole of order exactly n at the point Q on F;. Therefore,
(tÎÏq) = ^^pkr' on F». Thus (tJJ.q) = ^Q^ on f;.
These (p, q)-differentials w = t-Jt;)q from A'(p) onto F» are determined non uniquely on F»
because of their zeros and poles: (w) = R Q'Rn k 1 1 k m , kj > 0, j = 1,..., m. Fix k',..., km,
Q p1 ... pm
as the orders of possible poles at the points P',... ,Pm. Furthermore, deg(w) =0 on F;. This implies that N = n + k' + • • • + km.
Proposition 2.1 yields the equation vp № ... RN) — Vp (Qn) — Vp (Pk 1 ... Pm™) = ^(p) in the Jacobian variety J(F;). Consequently, v(R1 ... RN) = v(Qn) + k1v(P>1) + • • • + kmy(Pm) + ^(p) = M or v(R1) = M — v(R2 ... RN). Therefore, to specify the zeros of the differential we have N — 1 = n — 1 + k1 +-----+ km > 1, for k1 > 1, free parameters which can choose arbitrarily
on F^. Solving the Jacobi inversion problem, we find Ri, which is the unique holomorphic solution to the equation.
Show now that the point Q is a pole of order exactly n on F^. The proof is by contradiction. Assume that only first points R1,..., Rk, k < n, coincide with Q on FM. Then we have the equality
^(Rfc+i) = ^(Qn-fc) + V(p) + ki^(Pi) + • • • + km^(Pm) - ^(Rfc+2...RW) = M. (*)
Choosing the point Rk+1 G FM such that y(Rk+1) is not equal to the constant M and the point Rk+1 is not Q, we have the contradiction. Thus, the point Q is indeed a pole of order exactly n on FM.
Consequently, the divisor (t^.q) = Qn has the most general form for the (p, q)-
differentials t^. q of class A1(p) with a unique pole Q G F^ exactly of order n > 2 on F^ = FM\{P1,..., Pm}. By induction on n, taking into account the case n =1 and an essential character p [8], we have t^. q = (zn + O(1))dzJ.
The proof of Theorem 2.1 is complete. □
We can prove the next statement in the same fashion.
Theorem 2.2. Given distinct points Q1,Q2 on a surface F^ of type (1,m),m ^ 1, a character p on F^ and a natural number q ^ 1, there exists an elementary (p, q) —differential tpj9 . q1q2 of the third kind and class A1(p), with exactly simple poles Q1 and Q2 on F^, with the general form of divisor (t„,, . QiQi ) = rqiqn Pki ^ , where = ^(Q^) - ...Rn ) + ^(P'1 ) +
• • • + ^(Pm™) + ^(p), kj ^ 0, j = 1,..., m. Furthermore, we choose the points R2,..., RN as locally holomorphic section of divisors of degree N — 1 on F/^\{Q1, Q2}, N = 2 + k1 + • • • + km. Also, these differentials depend locally holomorphically on [p] and p.
3. Prym differentials for an unessential characters
Given a character p, denote by Q2 p(F^) the space of meromorphic differentials of the second kind with finitely many poles on F^, and by ne p(F^), the subspace of all multiplicatively exact
Prym differentials for p on F^. Let t~ be an abelian differential of the second kind on FM with
a unique pole of order exactly n > 2 at the point jP1, and with zero a-periods [5]. The point P1 is chosen using the condition rp( -J- ) = 1.
For every character p =1 define the mapping from Q2jp(F^) into H 1(r^, p), associating to a differential w its class of periods [w].
Suppose that w G Q2jp(F^) lifts to U, where F^ = U/r, and r is the Fuchsian group of the
/•Tzo /•
first kind uniformizing F^ on U [2, 6]. Find the classical periods wzo (T) = / w + nm+1 / w +
zo
Ym + l
• • • + nm+k j w, where nj G Z, j = m + 1,..., m + k. Here Ym+1,..., Ym+fc stand for loops
Ym+k
r Tzo
enclosing only the poles Q1,..., Qk of w on F^ respectively. We take the integral / w along
z0
some fixed particular path in the disk U avoiding the poles of w.
Since w is a differential of the second kind, all residues at its poles vanish. Thus, there exists a global primitive, a meromorphic function f (z) on U satisfying w = df on F^\{Q1,..., Qk}.
s r Tzo ~ ~
Lifting now w = df to U, relative to r, we obtain wzo (T) = / df (z) for every T G r, where r
is the Fuchsian group of the first kind on U uniformizing the surface FM, which results from F^ by removing all k poles Qi,..., Qk of the differentials w. Define the mapping
9 w ^ [w] = |wZ0 (A), (B), (yi), ...,
Wzo (Ym-i), Wzo (Ym+i), . . . , Wzo (Ym+k)} G H 1(r, p'),
where p'(ys) = 1, s = m + 1,... ,m + k and p' = p on = T. Since all wzo(ys), for s = m+1,..., m+k, vanish, [w] is expressed in terms of only wzo (A), wzo (B), wzo (y1), ..., wzo (Ym-1), satisfying (1), while for p =1 we have wzo(A) = 0 for p(A) = 1, and wzo(B) = 0 for p(B) = 1. Hence, the mapping Q2 p(F^) 9 w ^ [w] G H 1(r^,p) is well-defined.
If the class of periods [w] =0 in H 1(r^,p) for w G n2 p(F^), then the differential w is multiplicatively exact for p on FM, and so w G ne,p(F^). If w G ne,p(F^), then, as above, wzo(ys) =0 for s = m + 1,..., m + k, where Ys is a loop enclosing only the pole Qs of w. By assumption, w = df, where f is a multiplicative meromorphic function on FM, and so all periods in the sense of Gunning of w on FM vanish. So, [w] =0 in H1 (r^,p).
Thus, for every p =1 the period mapping from n2jP(F^)/nejP(F^) into H 1(r^,p) determined by the rule w+ne,p(F^) ^ [w+ne,p(F^)] = [w] is well-defined, bijective, and linear. Consequently, dime n2,p(F^)/Qe,p(F^) < m for every p =1.
Theorem 3.1. The vector bundle E1 (F^) is holomorphic of rank m over the
base T1jm x (L1\{1}) for m ^ 2. Furthermore, the tuples
f0t~i),f0TP2Pi >-.->f0TPmPi, (2)
of cosets of Prym differentials constitute bases of locally holomorphic sections of this bundle,
.
M'
where f0 is a multiplicative unit on FM for p, rp(_P1 = 1 on FM and point P1 G F'
Proof. This bundle is well-defined over this base by Lemma 1.3. Let us establish the reverse inequality dime n2iP(FM)/neiP(FM) > m and construct a basis for this quotient space.
Verify that for p =1, p(A) = 1, the differentials in the tuples (2) represent cosets in our quotient space which are linearly independent over C. For p0 = 1 on n1(FMo) there exists A G r^o satisfying p0(A) = exp2nic = 1. Thus, c = 0 for every p in a sufficiently small neighborhood U(p0) c L1\{1} and every [p] G U[p0]. Since df0 = 2nicf0Z on FM, we can express f0Z linearly via df0. Consequently, instead of the differential f0Z we can take df0, which represents the zero coset. Suppose that there exists a linear combination with nonzero coefficients
C1foT~|) + C1foTp2Pi +-----+ Tm — 1 f0 Tpm Pi = df,
where f is multiplicative function for an unessential character p on FM with p0(A) = 1.
Go around the point P2 along a small loop y2, starting from P2,0 on F^. Then the expression on the left-hand side has the residue c1 fo(P2,o)p(Y2), while this residue on the right-hand side vanishes. But f0(P2,0) = 0, p(y2) = 1, so c1 = 0. In the same fashion we calculate the residues
along small loops enclosing the points P3,..., Pm and obtain c2 = • • • = cm—1 = 0. Then we are
(2)
left with the sum c1f0r~ ' = df. Consider the coefficient c1.
1) If df has removable singularities at all punctures, then this equality on FM implies that there exists a meromorphic multiplicative function on FM with simple pole at Pc1. But this is impossible by the choice of this point and the condition = 1;
2) If the continuation of d/ to F; has at least one pole or essential singularity at the punctures, then for the combination on the left-hand side this point (puncture) is not singular, while for d/ it is singular. This is a contradiction. Therefore, c = 0.
Thus, the differentials in the tuple (2) represent cosets in our quotient space which are linearly independent over C.
The case p0(B) = 1 can be proved analogously. The proof of Theorem 3.1 is complete. □
Denote by ^p(qi 1 q ;F^) the space of the differentials for p, which are multiples of the divisor q 1 q on F;, and by ^e,P(1; F;), the subspace of the holomorphic multiplicatively exact differentials for p on F;.
Theorem 3.2. The vector bundle E2 = U^P(Qi 1 Qs; F^)(1; F^) is holomorphic of rank m + s with the base T1jm x (L1 \{1}) for m ^ 2, s ^ 1. Furthermore, the tuple
/o^Pf, f0TP2Pi >-",/oTPmPi, f0TQiPi ,...,/oTQsPi, (3)
of cosets of differentials constitutes a basis for locally holomorphic sections of this bundle, where Q1,..., Qs are distinct points on F;, depending holomorphically on [pj.
Proof. Consider the period mapping ^p(qi 1 q ;F^) 9 w ^ [wj G H 1(r'',p). The class of [wj is determined by the tuple of classical periods (w(A) = 0, w(B), w(y1), ..., w(7m-1), w(c1),..., w(cs)). Here the period w(Ym) is expressed via the remaining m + s of periods, F; = F;\{Q1,... ,Qs} = FM\{P1,... ,Pm} U {Q1,... ,QS}, and F; = U/r''.
If qi 1 q ; F;) 9 w ^ [wj = 0 in H 1(r'', p), then the differential w is multiplicatively exact on F;. The points Q1,...,Qs are removable singularities for w since 2n«(resQ3.w) =
w = 0, for j = 1,...,s. Thus, w G ne,p(1;F;). Consequently, the period mapping is well-defined, bijective, and takes ^p(qi 1 q ; F;)/Oe,p(1; F;) linearly into H 1(r'', p). Thus,
wQi.. . Qs ' x ;F;)/°e,p(1; f;) < m+s. Let us establish the reverse inequality for the dimension and construct a basis. The tuple of cosets of differentials in (3) is linearly independent over C. Indeed, if
(2) ~ ~ ' '
C1/0TK' + C1/0T P2Pi +-----+ cm_1/oTPmPi + c1/oTQiPi +-----+ cs/oTQsPi = d/,
then c1 = • • • = cm-1 = c1 = • • • = cS =0 since / is a multiplicative meromorphic function for p on F; and its residues at the points P2,..., Qs vanish.
(2)
This yields the equality C1/orp^ = d/. If the continuation / has in the punctures a unique pole P^ then (/) > on F;. But this is impossible because of the condition i) = 1, since we have the inequality (/) > 1 > . This is a contradiction. Hence, the dimension of the quotient space is at least m + s and we have constructed a basis. The proof of Theorem 3.2 is complete. □
By Grauert's theorem, since the base is simply connected, we obtain
Corollary 3.1. The holomorphic vector bundle (with fibers consisting/ of the first holomorphic
de Rh,am cohomology groups for p on f;) e2 = U HLi p(F;) = U^p(1; f; V^e,p(1; f;) is
analytically equivalent to the trivial rank m vector bundle with the base T1jm x (L1\{1}) for m > 2.
Define the period mapping x from Op(1; F') onto H 1(r',p), associating to w its class of
periods [wj, which is determined by the tuple of classical periods / w, / w, / w,..., / w .
\Ja Jb Jyi Ym — i y
Choose a representative for [wj satisfying / w = w(A) = 0.
Corollary 3.2. On every surface F; of type (1, m), m ^ 2, given an unessential character p, we have an isomorphism (1; F;) = Kerx © p(F;), where Kerx = 0e,p(1; F;) is an infinite dimensional vector space and dime p(F;) = m.
4. Prym differentials for an essential character
Lemma 4.1. On a surface F; of type (1,m), m ^ 1, given an essential character p, there exists a (p, 1)-differential t = tp;q2Pi , where Q G F;, and (t) = q2R1pkRNpkm on F;, where
kj G N, j = 2,..., m, and Rk = P1, Q, k = 1,..., N, N = 3 + k2 + • • • + km, depending locally holomorphically on [p] and p.
Proof. Proceed as in Section 3.
Theorem 4.1. The vector bundle = |J n2,p(F;)/ne,p(F;) is holomorphic of rank m with the base T1jm x Hom(r ', C*)\L1 for m ^ 2. Furthermore, the tuples : either
tp;p2pI ,...,Tp;PmPi ,tp;q2pI (4)
or
of cosets of differentials constitute a basis for locally holomorphic sections of this bundle, where
Q G f;.
Proof. Take an essential character p on F;. Define a mapping $ from Q2,p(f;) into H 1(r;, p), by associating to a differential w its class of periods [w] G H 1(r;, p).
If w G Q2jp(f;) satisfies [w] =0 in H 1(r;,p), then w is multiplicatively exact for p on F;, and hence w G ne,p(F;). It is clear also that every differential w in ne,p(F;) has the vanishing class of periods. Therefore, the kernel of $ coincides with ne,p(F;). Consequently, this mapping is well-defined on the quotient space n2,p(F;)/ne,p(F;). Furthermore, $ is bijective and linear. This implies that dime n2iP(F;)/neiP(F;) < m.
Let us establish the reverse inequality for the dimension and construct two forms of basis for our quotient space.
By Theorem 2.2, there exists a tuple tp;p2pi ,..., Tp;PmPi of elementary Prym differentials of third kind with simple poles at the points Pj and P1, j = 2,..., m, on F; respectively.
Suppose that the tuple (4) represents linearly dependent cosets in our quotient space for the essential character p; therefore, there exists a nontrivial linear combination with nonzero coefficients :
C1Tp;P2Pi +-----+ Cm— 1Tp;PmPi + CmTp;Q2Pi =
for fixed Q G F;, where f is multiplicative function on F;, (possibly, with poles of arbitrary orders and essential singularities in punctures for the branches of this function on F;.)
Consider the coefficients cj, j = 1,..., m — 1. For j = 2,..., m, take a loop Yj enclosing only
the point Pj. Then the classical period / df = ca(Yj) and, choosing instead of f the function
Yj
(f — c), we obtain / d(f — c) = 0. Consequently, all coefficients c1 = • • • = cm—1 = 0.
Yj ~
What's left to prove is the equality cmTp;Q2pi = df. Go along T1 around the point P1 and calculate residue at point, we obtain that ccm = 0.
We can prove the same statement for tuple (5) in same fashion.
The theorem 4.1 is proved. □
Theorem 4.2. The vector bundle E4 = U^p(qi 1 q ; F^)(1, F^) is holomorphic of rank m + s with the base T1jm x Hom(r', C*)\L1 for distinct points Qi,..., Qs, s ^ 1, on the surface F^ of type (1, m), m ^ 2. Furthermore, the tuple : either
tP;P2PI > . . .,Tp;PmPi ,tP;QIPI , . . . ,tP;Q=PI ,tP;P22PI , (6)
or
tP;PI , . . . ,Tp;Pm! Tp;Qi , . . . ,tP;QS , (7)
of cosets of Prym differentials constitutes a basis for locally holomorphic sections of this bundle.
Proof. It suffices to verify only the linear independence of the cosets of differentials in (6). Suppose that there exists a nontrivial linear combination :
C2Tp;P2Pi +----+ CmTp;PmPi + cm+1 Tp;QiPi +----+ Cm+sTp.Qs p + cVp.p2^ = df.
If f has essential singularities at the punctures, then we immediately obtain a contradiction since the left-hand side lacks those. Using the residues and periods, as in the proof of the previous theorem, we infer that cj = 0, for j = 2,..., m + s. It remains to consider the equality
c'Tp;P22Pi = df.
The residue at Pi is a multiple of cC, it has the form Mc', M = 0, while on the right-hand side, since df is multiplicatively exact, we can make the classical period around only point P1 vanish. Hence, c' = 0. Thus the cosets of differentials in (6) constitute a basis for our quotient space.
Analogously we can prove this statement for tuple (7).
The proof of Theorem 4.2 is complete. □
Corollary 4.1. The vector bundle E4 = |J H^oi P(F") = U^P(1; F")/Oe,p(1; F^) is holomorphic of rank m with the base T1jm x (Hom(r ', C*)\L1) for m ^ 2.
Remark 1. The vector bundles E1 and are analytically equivalent to the Gunning cohomo-logical bundle G' over their respective bases.
Remark 2. Theorems 3.2 and 4.2, taking into account theorems 2.1 and 2.2, can be generalized for vector bundles with fibers 1 ; F^), over the products T1jm x Hom(r ', C*)\L1 and
T1,m x L1\1 for s > 1, a1,. .., as G N, q > 1.
References
[1] R.C.Gunning, On the period classes of Prym differentials, J. Reine Angew. Math., (1980), no. 319, 153-171.
[2] V.V.Chueshev, Multiplicative functions and Prym differentials on a variable compact Riemann surface, Part 2, Kemerov. Gosudarstven. Universitet, Kemerovo, 2003 (in Russian).
[3] V.V.Chueshev, Multiplicative Weierstrass points and Jacobi manifold on a compact Riemann surface, Mathematical Notes, 74(2003), no. 4, 629-636.
[4] R.Dick, Holomorphic differentials on punctured Riemann surface, Differ. Geom. Math. Theor. Phys.: Phys. and Geom.; Proc. NATO Adv. Res., Workshop and 18 Int. Conf. Davis. Calif. 2-8 June. N.-Y., London, 1990, 475-483.
[5] H.M.Farkas, I.Kra, Riemann surfaces, New-York, Springer-Verlag, 1992.
[6] L.V.Ahlfors, L.Bers, Spaces of Riemann surfaces and quasi-conformal mappings, Moscow, 1961 (in Russian).
[7] C.J.Earle, Families of Riemann surfaces and Jacobi varieties, Annals of Mathematics, 107(1978), 255-286.
[8] T.S.Krepizina, V.V.Chueshev, Multiplicative functions and Prym differentials on a variable torus, Vestnik Novosibirskogo Gosuniversiteta, 12(2012), no. 1, 74-90 (in Russian).
Пространства мероморфных дифференциалов Прима на конечных торах
Ольга А. Чуешева
В данной статье построены все виды элементарных дифференциалов Прима для любых характеров на переменных торах с конечным числом проколов и найдены размерности двух важных фактор-пространств мероморфных дифференциалов Прима. Как следствие, находится размерность первой голоморфной группы когомологий де Рама дифференциалов Прима для любых характеров на торе. В этих фактор-пространствах построены явные базисы.
Ключевые слова: дифференциалы Прима для произвольных характеров, когомологическое 'расслоение Ганнинга над пространством Тейхмюллера для тора с конечным числом проколов.
