Vector Bundle of Prym Differentials over Teichmu ̈ller Spaces of Surfaces with Punctures Текст научной статьи по специальности «Математика»

УДК 515.17+517.545

Vector Bundle of Prym Differentials over Teichmiiller Spaces of Surfaces with Punctures

Alexander V. Chueshev Victor V. Chueshev*

Institute of Fundamental Sciences Kemerovo State University Krasnaya, 6, Kemerovo, 650043 Russia

Received 12.10.2018, received in revised form 18.01.2019, accepted 04.03.2019 In this paper we study multiplicative meromorphic functions and differentials on Riemann surfaces of finite type. We prove an analog of P. Appell's formula on decomposition of multiplicative functions with poles of arbitrary multiplicity into a sum of elementary Prym integrals. We construct explicit bases for some important quotient spaces and prove a theorem on a fiber isomorphism of vector bundles and n!-sheeted mappings over Teichmuller spaces. This theorem gives an important relation between spaces of Prym differentials (Abelian differentials) on a compact Riemann surfaces and on a Riemann surfaces of finite type.

Keywords: Teichmiiller spaces for Riemann surfaces of finite type, Prym differentials, vector bundles, group of characters, Jacobi manifolds. DOI: 10.17516/1997-1397-2019-12-3-263-275.

Introduction

In the paper we study multiplicative meromorphic functions and differentials on Riemann surfaces of type (g, n). Recently the interest in this subject has increased in relation with applications in theoretical physics, in particular, in description of vortex-like patterns in ferromagnetics [1].

Theory of functions on compact Riemann surfaces differs significantly from that on Riemann surfaces of finite type even for the class of single-valued meromorphic functions and Abelian differentials, since some of basic spaces of functions and differentials on Riemann surfaces F' of type (g,n), g ^ 1, n > 0 are infinite-dimensional.

In this paper we continue constructing the general theory of functions on Riemann surfaces of type (g, n) for multiplicative meromorphic function and differentials. We prove an analog of P. Appell's formula about the expansion of a multiplicative function with poles of arbitrary multiplicity into a sum of elementary Prym integrals. Also we construct explicit bases for some important quotient spaces and prove a theorem about fiber isomorphism of vector bundles and n!-sheeted mappings over Teichmiiller spaces. This theorem gives an important relation between spaces of Prym differentials (abelian differentials) on compact Riemann surfaces and Riemann surfaces of finite type.

* [email protected] © Siberian Federal University. All rights reserved

1. Preliminaries

Let F be a smooth compact oriented surface of genus g > 2, with the marking {ak, bk}gk=1, i.e. an ordered collection of standard generators of n1(F), and F0 be a compact Riemann surface with the fixed complex-analytic structure on F. Fix different points P1,... ,Pn € F. We assign type (g,n) to a surface F' = F\{P1,... ,Pn}. By r' we denote the Fuchsian group of genus 1 acting invariantly in the disk U = {z € C : \z\ < 1} and uniformizing the surface F'. Thus, F' = U/r', where r' has the representation [2,3]

g

r' = (Au ...,Ag ,B1 ,...,Bg ,Ci,...,Cn : n A Bo A-1B-1C1 ---Cn = I).

j=1

Any other complex analytic structure on F' is given by a Beltrami differential a on F', i. e. by an expression of the form ¡(z)dz/dz, invariant with respect to the choice of the local parameter on F', where ¡(z) is a complex-valued function on F' and WaWl^(f') < 1. We denote this structure on F' by F^.

Let M(F') be the set of all complex analytic structures on F' with the topology of C convergence on F', Diff + (F') be the group of all orientation preserving smooth diffeomorphisms of F' onto itself, which leave all punctures fixed, and Diff0(F') be the normal subgroup of Diff+(F') of diffeomorphisms homotopic to the identity diffeomorphism on F'. The group Diff +(F') acts on M(F') by a ^ f *¡a,, where f € Diff +(F'), ¡j € M(F'). Then the Teichmiiller space Tgn(F') is the quotient space M(F')/Diff0(F') [2].

Since the mapping U ^ F' = U/r' is a local diffeomorphism, any Beltrami differential a on F' lifts to a Beltrami r'-differential a on U, i. e. a € L^(U), ||m||to = esssupzeu \a(z)\ < 1, and a(T(z))T'(z)/T'(z) = a(z), z € U, T € r', see [2].

If the r'-differential a on U is continued on C\U, setting a = 0, then there is a single quasiconformal homeomorphism wM : C ^ C with fixed points +1, -1,i, which is a solution of the Beltrami equation wz = a(z)wz. The map T ^ T^ = w^T(wM)-1 defines an isomorphism of the group r' onto the quasi-Fuchsian group P'u = wMr'(w^)-1.

In the work [2, p. 99] there were constructed abelian differentials (1[a], ..., Zg[a] on F^], that form a canonical base dual to a canonical homotopy base {a%, b%}gk=1 on F^, which depends holomorphically on moduli [a] for a class of conformal equivalency of a marked Riemann surface F %. Further on, for brevity we shall write simply F % for the class of equivalence ]. Here we assume that the class [a] has Bers coordinates h1 ,h2,..., h3g-3 when embedding the Teichmiiller space Tg(F0) of compact Riemann surfaces into C3g-3. Moreover, the matrix of b-periods

q(a) = (njk [A])9jk=1 on F% consists of complex numbers njk [¡] = f Zj ([¡],w)dw,£ € w% (U),

«

and depends holomorphically on [¡].

For any fixed [a] € Tg and £0 € w%(U) define the classical Jacobi mapping ^ : w%(U) ^ Cg

«

by the rule: pj (£) = f Zj ([a],w)dw, j = 1,... ,g. The quotient space J (F) = Cg/L(F) is called

«0

the marked Jacobi manifold for F = F0, where L(F) is a lattice over Z, generated by the columns

of the matrix (Ig, il), where Ig is an identity matrix of order g. The universal Jacobi manifold of order g is a fibered space over Tg, with a fiber over [¡] € Tg being a marked Jacobi manifold J(F%) for a marked Riemann surface F% [4].

A character p for F" is any homomorphic p : (n1(F'), ■) ^ (C*, ■), C* = C \ {0}. Further on we shall assume that p(y%) = 1, where is a simple loop around only one puncture Pj on F^,

j = 1,...,n.

Definition 1. A multiplicative function f on Fp for the character p is a mermorphic function f on wp(U) such that f (Tz) = p(T)f (z), z £ wp(U), T £ r'p.

Definition 2. A Prym q-differential with respect to a Fuchsian group r' for p, or a (p,q)-differential, is a differential w(z)dzq such that w(Tz)(T'z)q = p(T)v(z), z £ U, T £ r',p : r' ^ C*.

If a multipicative function f0 on Fp for p does not have zeroes or poles, then the character p is called non-essential and f0 is called a unit. The characters which are not non-essential are called essential on ni(Fp). The set Lg of non-essential characters form a subgroup in the group Hom(rp, C*) of all characters on rp. A divisor on Fp is a formal product D = Pf1 .. .Pfk, Pj £ Fp, nj £ Z, j =1,..., k.

Theorem (Abel's theorem for characters, [3,5]). Let D be a divisor on a marked variable compact Riemann surface [Fp, {ap,..., ap, bp,..., bp}] of genus g ^ 1, and p be a character on ni(Fp).

Then D is a divisor of a multiplicative function f on F p for p if and only if degD = 0 and 1 g 1 g

*(D) = loSp(b>U) - ^ EloSpJU)M( = Hp, M)),

j=i j=i

where ^[p] : Fp ^ J(Fp) is the Jacobi mapping.

The class Mi(p) consists of those Prym differentials for p on F", which have finitely many poles on F^ and admit meromorphic continuation to Fp.

In [6] it was proved that for any essential character p, a point Qi £ Fp, and natural q > 1 or a non-essential character p, a point Qi £ Fp, and natural q > 1 there exists an elementary (p, q)-differential Tpq;Q1 of the third kind with a unique simple pole Qi[/j] on Fp. For any nonessential character p, a point Qi £ F 'p if q = 1 there is no elementary (p, 1)-differential tpq1 . Also it is proved there that on a variable surface Fp of genus g > 2 for any natural q > 1 there exists an elementary (p, q)-differential rp,q-Q1Q2 of the third kind with simple poles Qi,Q2 £ Fp, and

TPqQ\ = ("m + ^(1))dzq, z(Qi) = 0, of the second kind with the pole Qi[p] of order m > 2. These differentials depend locally holomorphically on [p] and p.

Let p : E ^ B be a locally trivial holomorphic vector bundle of rank m, i.e. E, B are complex analytic manifolds, the base B is covered by a system of open simply-connected sets {Ua} such that there exists a system of holomorphic fiber coordinate homeomorphisms a : Ua x Cm ^ P-i(Ua) for all a. On intersections UanUp = 0 there are given ypa = : (UanUp) x Cm ^

(UanUp) xCm, holomorphic matrix transition functions, which satisfy on (UanUpnUY) xCm the relations pa = id, where id is the identity homeomorphism for all indices. The transition

functions <^pa(x,z) = (x,ippa(x)z) define holomorphic mappings ippa : Ua n Up ^ GL(m, C), where x £ B, z £ Cm h <p>aYipYp<p>pa = 1. These conditions on B, Cm, {Ua} and such ^pa,<p>pa are sufficient to define a locally trivial holomorphic vector bundle E of rank m over B [7].

Any holomorphic section s : B ^ E, i.e. ps(x) = x,x £ B, may locally be described as -is : Ua ^ Ua x Cm, which define holomorphic vector-valued functions sa : Ua ^ Cm by the formula (^-is)(x) = (x, sa(x)),x £ Ua. On intersections Ua n Up = 0 these functions satisfy the compatibility conditions sp(x) = <p>pa(sa(x)).

Conversely, given a set of holomorphic vector-valued functions sa : Ua ^ Cm with the compatibility conditions satisfied, then the formula s(x) = a(x, sa(x)) uniquely, i.e. independently of the choice of the covering {Ua}, defines a holomorphic section s : B ^ E.

If E is a locally trivial holomorphic vector bundle of rank m over B, then there exists a base of locally holomorphic sections for {Ua} given by sko = <a(x, ek), k = 1,... ,m, x G Ua, where e1,... ,em is the standard base in Cm.

Conversely, given a base of locally holomorphic sections ska, k = 1,... ,m, x G Ua of E, the

m m

coordinate homeomorphisms can be defined by <a(x,z) = ^ ZjSja, where z = ^ Zjej, which

j=i _ j=i are holomorphic in x G B and z G Cm. Besides, from (s1a(x),..., sma(x))t = <paß(x)(s1ß(x),... ...,smß(x))t it follows that the transition functions <aß(x) are holomorphic on intersections Ua n Uß = In this manner (E,p,B) is endowed with the structure of a holomorphic vector bundle of rank m over B.

2. An analog of Appel's decomposition formula

for a multiplicative function on a variable Riemann surface of finite type

p

Denote by T^Q = — f t(2Q an elementary Prym integral of second kind on F^ for an essential

Qo

character p with only simple pole at Q and with residue +1 in Q that depends holomorphically

(2)

on [a] and p, where t^.Q has zero residue at Q [5,6,8].

Let f be a function on F^' of the class M1 for an essential character p with s simple poles Pn+1,Pn+2, ■ ■ ■, Pn+s and residues cn+1,..., cn+s at these poles respectively for some its branch. Consider an analytic continuation of this function f (denoting it by the same symbol) from F^'

(1) (1) 9-i~ P ~

to Ffj,. Consider the expression f1 = f—cn+1T(.pn i-----cn,+sT(.pn ^ — J2 cj / ~j, where Cj G C,

j=1 Qo

j = 1,... ,g — 1, and ■ ■ ■, Z9-1 is the base of Prym differentials of the first kind for an essential character p on F^ depending holomorphically on [a] end p [2]. Then f1 is a meromorphic single-valued branch of the Prym integral with an essential character p on the fundamental polygon A^,

where the surface F^ is uniformized [3], with the divisor (f1) > pqi—p^n, qj ^ 0, j = 1, ■ ■ ■ ,n,

on F ^ Here we assume P1,P2, ■ ■ ■ ,Pn+s G IntAM ■ Besides, the Prym integral f1 for p has a branch whose principal parts of Laurent series coincide with principal parts of Laurent series at Pj, j = 1, ■ ■ ■ ,n, for f and zero am-periods, m = 1, ■ ■ ■ ,g — 1, on F^ or on A^ [2]. Therefore

s (1) 9-1 P „

f = £ cn+j T(1l+j + E Cj f ~j + f1.

j=1 j=1 Qo

If Pi is a pole of order ql,ql > 2, then in the formula above one should instead of c^P, l = n + 1,... ,n + s, (for simple poles), and also for poles Pl, l = 1,... ,n, of the branch of f1 write sums of the form

A T+ A dTPP + Asd2TP1P + + A* d9!-lTiP

where Aij are coefficients of the principal part of the Laurent series for some branch of f at Pi ,j = l,..., qi(Pi), l = n + l,... ,n + s, and for a branch of fr at Pi, P2,..Pn. Indeed, in a

neighborhood of Pi we have expansions T(1l =-—— + Oil); (T(1l )' = ---r + O(l),

p'Pl z — z(Pi) ppl l (z — ai)2

z(Pi) = ai; ... ; (T% ) = 7-^r^-r + O(l), l < m < qi(Pi) — l, where qi(Pi ) is the order

(z — ai)m-1

of the pole at Pl for branches f and f1: l = 1,..., s + n. From that follows the theorem.

Theorem 1. Let f be a branch of a function of class M1 for an essential character p on a variable Riemann surface F u of type (g,n), g ^ 2, n > 0, with pairwise distinct poles at Pn+i, ..., Pn+s of multiplicities qn+1,..., qn+s with given principal parts:

Aj,qj + ... + , , „ j = n +1,...,n + s. (1)

(z - z(Pj ))qj ..... (z - z(Pj ))'

Then for an analytic continuation of f we have (f) ^ ——1 , qj ^ 0,j = 1,... ,n, on Ff

and

, dm-1 T(1

n+s qj fa dm-l T(1) ] S-1 !■ —

f ^^ (m - 1)! OP™-1 +^Cj Q j

j=1 m=1

Ajq , , Aj,2 , Aj,1

j=1 JQ

where f = --..--+ ... + --o +--+0(1) for some branch in a neighborhood

J (z - z(Pj ))qj (z - z(Pj ))2 z - z(Pj) KJJ y

of Pj, j = 1,. ..,n + s, m Fu, and all summands depend holomorphically on [p] and p.

Let now p be a non-essential character. The proof of the previous expansion formula for an essential character does not work since in this case there is no Prym integral of the second kind with only simple pole on Fu. Therefore we need a Prym differential tp;q2q2 of second kind for a non-essential character p with two poles of second order at two distinct points Q1 and Q2 on △ u

with zero residues at Q1 and Q2 [5,6]. In this case one should use as basic elements of expansion

—

the Prym integrals Fp;q1q2 = - j tp;q2q2 of second kind with two simple poles Q1 and Q2.

Qo '12

Consider one more Prym differential tp;q1q2 = f0TQ1Q2 of the third kind on Fu, where f0 is a unit for p on Fu and tq1q2 is the normalized (i.e all a-periods vanish) abelian differential with simple poles Q1 and Q2 on Fu and residues +1 and -1 at these points, respectively, which depend holomorphically on [p] and p [5,6]. It is known that tq1q2 = ¿rq1q2 and the abelian integral nQ1Q2 can be expressed implicitly via the Riemann theta-function for the surface Fu. It equals to a sum of two functions, one of which depends only on Q1, and another only on Q2

[5, p. 117]. Therefore the derivative —QlQ2 does not depend on Q2, where z1 = z(Q1).

dz1

(2) ( 1 \ The Prym differential tpq admits the expansion ---r +---+ 0(1) \dz in a neigh-

p;Q1 \(z - z1)2 z - z1 J

m 9 '

borhood of Q1, z(Q1) = z1, where c—1 = ^ logp(aj)pj(Q1) [5,6]. To prove this we consider the

j=1

T(2)

abelian differential -pQ, where f0 is a multiplicative unit for p. Its complete sum of residues is f0

zero. In a neighborhood of Q, z(Q) = z0 we have the Laurent expansions

t(2Q =(--^ + + co + ...)dz

P'Q \ (z - zo)2 z - zo

...

f 9 \ 1 (z z )

= exp[ - jC Aj Pj (z)) = - f0(zo) AV1(z0) + ... + X9 f9 (z0^ +

fo(z)

where Aj = logp(aj), j = 1,... ,g. From that we get

(2)

0 = res TPQ = c—1 _ A1f1(z0) + ••• + A9f9(z0)

Zo f0 f0(z0) f0(z0) :

1

g

since Q is the only pole of the abelian differential. Therefore c-) = J2 Xjvj(zo) and c-) = 0

j=i

( g ) g

in a finite number of points Q on A^. Indeed, dfo = exp ( ^ Xjvj(P) ) XjVjdz(P), and the g

equivalency J2 Xjvj(Q) =0 dfo(Q) = 0 holds. Thus, for a non-essential character p there is

j=i

no Prym differential of the second kind with only pole of the second order at an arbitrary point

Q and principal part --^, since the condition X)V) (zo) + ■ ■ ■ + Xgv'„(zo) = 0 holds only for

(z - zo)2 y

a finite number of points Q on A^, i.e. at points Q that are zeroes of the differential df0.

2 ! 1 c{2) \

(2) olao has on evponsinJ--+--—+ 0(1m dz in a neigh-

The Prym differential TpQ2 also has an expansion ^

p;Q2 " - Z2)2 z - Z2

(2) g

borhood of Q2 on Fa, where c_) = logp(aj(Q2)-

j=i

A Prym differentials with two poles of the second order and zero residues at these points may be given in the form

(2) * m \ (2) (1) r fn \ (2) (!) (2)

Note that the principal part for tp q,q2 at Q1 has the form fo(Q1, and at Q2 it is - fo(Q2S).

Z - z1 z - z2

It follows that the differential constructed above tp;qiq2 has poles of the second order at Q1

and Q2, and zero residues at these points. Indeed, in a neighborhood of Q1 its principal part

. ,.:' .. ... f... . c(2)

has the form c(2\f0(Q1)

1 + C-)

-C-lfo(Q2)

(z - z))2 z - z)

„(2) )

MQ1 = ft), analogously at Q2:

r +

|_(z - z2)2 z - z2_j q1q2 depend holomorphically on [p] and p.

- z1 (z - z1)2

))(2) fo(Q2) = C-\ fo(Q2) z - z2 (z - z2)2

+c-[c-{-= —--—. The constructed differential

Theorem 2. Let f be a branch of a function of class M1 for a non-essential character p on a variable Riemann surface Fa of type (g,n), g ^ 2, n > 0, with pairwise distinct poles at Pn+i, ■ ■ ■, Pn+s of multiplicities qn+i, ■ ■ ■, qn+s with given principal parts (1). Assume that for

an analytic continuation of f to Fa the conditions (f) ^ —q-uqn+s, qj ^ 0, j = 1, ■ ■ ■ ,n, and

P1 ■■■Pn+s

g '

Y^, logp(aj(Pn+s) = 0 are fulfilled. Then

j=i

m= jSUo foCj + ^ -dfV + m=2 (m - 1)! dPr-1 +

n+s

M - o d A A n . d ij I d. d.

+ E

j=2

A dtp;p3Pi + Aj,3 d2tp;p3Pi + + Aj,g3 dqj-)TPPPi

j'2 dPj 2! dP2 +... + (qj -1)! dPqj-)

j

where

f =_j_+ +_j_+ Aj'1 + O(l)

1 (z - z(Pj))q^"- +(z - z(Pj))2+ z - z(Pj)+

for some branch in a neighborhood of Pj, j = 1, ■ ■ ■ ,n + s, on Fa; C = 0 for p = 1; dk =

g '

= Y1 logp(am)pm(Pk), k = 1, ■ ■ ■ ,n + s, on Fa, and all summands depend holomorphically on

m=1

[p] and p■

1

Proof. It is enough to check that principal parts of both parts of the formula coincide. For a neighborhood of Pr, r = 1,... ,n + s — 1, on we have the Laurent expansion

( dn+sf0(Pr ) dr f0(Pn+s) ) Arl = Arl +

r Z — Pr Z — pn+s dn+sf0(Pr) z — Pr

For a neighborhood of Pn+s on A^ we have

y- — dr f0(Pn+s) Arl = 1 f0(Pn+s) y- — dr Arl + = An+s,l +

z — Pn+s dn+sf0(Pr) z — Pn+s dn+s f0 (Pr) z — Pn+s

n+s —Arldr = 0 fo(Pn+s) n+s-1 —drArl

0(1 f

since £ = 0, —-- J2 r fr> 1 = An+st1, according to the complete sum of

r=1 J0 (Pr) dn+s r=1 f0 (Pr*

residues formula for an abelian differential d( J2 logiJ(ajof the third kind on Fp, which

J0 \j=1 J

at Pj has the residues „ , j = 1,... ,n + s. Thus, the coefficients at the power -1 in principal

fo (Pj)

parts at P1,P2,..., Pn+s, are the same.

The third sum shows that the coefficients coincide in principal parts at P1 for powers starting from -2. The fourth sum shows that the coefficients coincide in principal parts at P2,P3,..., Pn+s for powers starting from -2.

If p = 1 then all the summands in the formula become abelian integrals, which differ by a constant C. If p = 1 and p is a non-essential character, then C = 0, since a constant is neither a multiplicative function, nor a Prym integral for this character on Fp of genus g ^ 2. □

Remark. P. Appel [6, see p. 118] proved Theorem 2 for a fixed compact Riemann surface and simple poles with every simple element (summand) depending on additional g — 1 poles. Our theorem is proved for a variable Riemann surface F' of finite type (g,n), g > 2, n > 0, and poles of any order with any summand having either one or two poles. Moreover, if p = 1, n = 0 we recover the classical fact on decomposition of a single-valued meromorphic function into a sum of abelian integrals on a compact Riemann surface.

Corollary. For any non-essential character p on a variable compact Riemann surface Fp of genus g ^ 2 at Q1, which are 'zeroes of the differential df0, there exists a differential t^qi of the second kind with only pole of the second order at Q1 that depends holomorphically on [p] and p, and having zero residue at Q1.

3. Vector bundles of Prym differentials over a Techmiiller space of Riemann surfaces of finite type

Denote by QqJ t 1 a; F^j the vector space of (p, q)—differentials that are multiples

\Qi ■ ■ ■ Qs " J

of the divisor -—{----, where a.j > 1, a.j G N, j = l,...,s, s > 1, q > 1, q G N, and

Q1 ■ ■ ■ Qss

by Qq(1; Fp) the vector subspace of holomorphic (p, q)-differentials on Fp [3]. Here the divisor Q1.. .Qs on Fp is understood as a constant set of points on a surface F of genus g > 2.

Let E be the principal Hom(r, C*)-bundle over Ts(F0) with the fiber Hom(rp, C*) over F[p] from Ts(F0). Here F0 = U/r, r is a Fuchsian group uniformizing F0 over the circle U, and F[p] = wp(U)/rp = Ap/rp, rM be a quasi-Fuchsian group uniformizing the compact Riemann surface over wp(U).

Lemma 1 ([5], pp. 105-106). A holomorphic principal Hom(T, C*)-bundle E is biholomorphic to the trivial bundle Tg(F0) x Hom(r, C*) over Tg(F0).

Proposition 1. The vector bundle E = U QqJ 1 as; F^j / Qqp(1; Fover Tg x

^ Q1 • • • Qs s '

(Hom(r, C*)\1) for q > 1 (over Tg x (Hom(F, C*)\Lg) when q = l) and g ^ 2 is a holomorphic vector bundle of rank a1 + • • • + as = d, while the co-sets of (p, q)-differentials

r(l) (1) (a.)

P,q;Qi7 ''' 7 p,q;Qi ' ' ' ' ' p,q;Q. ' ' ' ' ' p,q;Q.

---Q. , , ---Q. , , ---Q , , p,q;Q. I V !

form a basis of locally holomorphic sections of this bundle.

Proof. With given conditions on q for the character p we have the equality dim^p(1; Fp) = = (g — 1)(2q — 1). By the Riemann-Roch theorem for (p, q)-differentials we find the dimension

P,q (Q-ai ... Q-as ; Fp) = (g — 1)(2q — 1) + «1 + ... + as + r( (f [p])Zq-1Qa11 ...Q,

where f [p] is a function for p, Z is the canonical class for abelian 1-differentials on F^. Here r((f[p])Zq-1 Qa ••• Q^s) =0, since deg((f[p])Zq-1^a ••• Q> a! > 0. Thus,

dim Qq( Qai ^ Qfs; F^j /Qqp (1; F„) = ai + ... + as = d.

It follows from Theorems 2.1 and 2.2 of [6] that there exist differentials from the set (2) that depend locally holomorphically on [p] and p.

Let us show that the set (2) of equivalency classes of (p, q)-differentials that depend locally holomorphically on [p] and p is linearly independent over C for given characters p. Consider a linear combination of the form

Cf^k + ... + Cf ^1 + ... + + ... + s =

where w is a holomorphic (p, q)-differential on F^. Since the right hand side does not have singularities, all the coefficients are zeroes. All these differentials depend holomorphically on [p], p and divisors Q1 . . . Qs, which are locally holomorphic (constant) sections of the bundle of integer divisors of degree s over the Teichmiiller space Tg of genus g [4]. Therefore, this set gives the base of locally holomorphic sections of this bundle. □

Lemma 2. For any divisor Pqi •... • Pq—, qj ^ 0,j = 1,... ,n, q > 1 and any p (or q = 1 and an

essential character p) on F^ of genus g ^ 2, there exists a differential w € Qq ^ pqi-pq—, F^j

Ri Rn 1

with the divisor (w) = pq^-' pq— , where Rj = Pi, l = 1,... ,n, j = 1,..., N, N = (2g — 2)q+

+qi + ... + qn, and any given principal parts of Laurent series at Pj, j = 1,... ,n, for its branches. This differential depends locally holomorphically on moduli [p] of the surface F^ and the character p.

Proof. If qj = 0 for all j, there exists a holomorphic (p, q)-differential w = 0 on F^ for every q > 1 and p, since ipq(1) = (2q — 1)(g — 1) > 3 if q > 1 and ip(1) > g — 1 > 1 if q = 1.

Fix q1,... ,qn as possible order of poles at punctures P1,... ,Pn on F^ respectively and assume that for at least one j, qj > 1.

If q = 1 and q1 = 1, q2 =0,... ,qn = 0 for an essential character p there exists a differential

w = 0 such that (w) > p~ [6]. Further on, if q = 1 we shall assume that q1 + q2 + • • • + qn > 2.

For any (p, q)-differential w the degree of its divisor deg(w) = (2g — 2)q on F^. It follows that N = (2g — 2)q + q1 + ... + qn. By Proposition 1.4.4 [5] and Abel's theorem

R R

there exists a differential oj = 0 with the divisor (oj) = qi'''' ^^ if and only if the

equality pR. ..Rn ) — vP?1 . ..Pt) = —2Kq + '(p) holds in the Jacobi manifold J (F^), where K is the vector of Riemann constants. From this it follows that R1... Rg) = = —2Kq+ ^(Pq1... Pqqn)+ '(p)— Rg+1... Rn). Thus, to determine the zeroes of the differential we have N — g = (2g — 2)q — g + q1 +... + qn > g — 1 > 1 free parameters that can be chosen so that they depend locally holomorphically on moduli [p]. Solving the Jacobi inversion problem we find the divisor R1.. .Rg, which is the only holomorphic solution to the previous equation if the right hand side does not belong to W^ [3,5]. This can be done since dim W^ < g — 2, but N — g > g — 2

R R

under our hypothesis. Therefore the divisor of the differential (oj) = ^qi''' Pqn has exactly required singularities, if Rj = Pi for all indices. In order to do this we choose points Rg+1,..., RN = P1,... ,Pn. We shall show that after a specific choice of the divisor Rg+1... RN we can satisfy the condition Rj = Pl for any j and l. Assume the converse, if R1 = P1 on Fp, then from the previous equality we get pR... Rg ) = —2Kq + ^(Pi1-1P22 ...P^r) + '(P) — V>(Rg+i ...Rn ) or ^R ... Rg Rg+1... R2g-1) = —2Kq + ^(Pqi-1Pq2 ... P^) + '(p) — ?(R2g ... Rn ). Consider the integer divisor D = R2... RgRg+1... R2g-1 of degree 2g — 2. It has g — 1 free points Rg+1,... ,R2g-1. By the free points theorem [3] we get the inequality i(D) > 1, and therefore ^(D) = —2K. Then the previous inequality can be rewritten as

—2K (q — 1) + v(Pqq1-1... Pqn) + 'M = *>(R2g ...rn ). (3)

Note that N — (2g — 1) = (2g — 2)q + q1 + ... + qn — 2g + 1 > 1 in these conditions. Thus, we see that the sets defined by both sides of this equality in J(Fp) has different dimensions. Therefore we can choose R2g,..., RN on Fp such that (3) does not hold. This is a contradiction.

It is known that under our conditions on q and character p there exist elementary (p, q)-differentials of the form t^.q and Tp^Qm > 1 on Fp [6]. Therefore we can construct any principal parts for Laurent series of the differential oj at all points Pj, j = 1,... ,n, on Fp. □

Further on, we shall assume that the character p' on r' such that p'(Yj) = 1, j = 1,.. .,n, i.e. p' = p G Hom(r, C*). Consider the diagram

E' = nQp( Qa1 1-Qas ,Fp) n M1 Qp( Qa11..Qfs ,Fp) = E

" nqp(1,Fp) n M1 U m.1,Fp)

I I (4)

T; x Hom(r, C*)\X ^ Tg x Hom(r, C*)\X,

where Tn is a part of the Teichmiiller space Tgn [6, p.81, p.88], the vertical arrows are projections in vector bundles, and the lower horizontal arrow is related to the operation of gluing the punctures, which makes the surface F\{P1,... ,Pn} into a compact surface F [3]. The upper horizontal arrow will be explained later.

Theorem 3. The diagram above is a commutative diagram of vertical holomorphic vector bundles with isomorphic corresponding fibers and horizontal holomorphic n—sheeted mappings, where X = 1 when q > 1, and X = Lg when q = 1.

Proof. By the Riemann-Roch theorem we find the dimension p*i—^ Pqn , F^j = (2q — 1)(g—

— 1) + q1 + ... + qn. Therefore, Qqp(1, F') is an infinite-dimensional vector space.

Now we prove the isomorphism of fibers for fixed F' and F, where F is obtained from F' by glueing up the punctures. For any fixed p =1 we define the map d of a fiber of E' over F into

a fiber of E over F, which puts in correspondence to the class < w >= w + &p(1, F') n Mi the

class <w — w >= w — w + Qqp(1, F) in the following way. If w G &qp \ , Fj n Mi, i.e.

\Qi .. -Qs" J

(w) > —a-yas • To!-rqn , then we put into correspondence w — w, since by Lemma 2 we

Q- ... QS P- . . . Pn

R R

can choose the differential w such that (w) = ''' n , Rj = Pi, j = 1,... ,N, l = 1,... ,n, and

P 1 . . . Pn

having the same principal parts of Laurent series at all points Pj, j = 1,... ,n, as an analytic

continuation of the differential w to F. Then w — w G QqJ 1 a, Fj. Let 0(w + Qqp(1, F') n

VQi .. .Qs" J

M-i) = w — w + Qqp(1,F).

We shall show that this mapping is well-defined. Consider another differential w' from the same equivalency class < w + Qqp(1,F') n Mi >. It has the same singularities as w at all

points Qi,..., Qs, and its own singularities at the punctures, i.e. (w') >-1—-,--.

Qa ...Qa pq1 ...Pt

Then we choose w' such that w' — w' G Qp[ —a-—^,F I. Therefore, on the one hand we

pV QI1 ...Q"s J

have 9(< w >) =< w — w >, on the other hand we have 9(< w' >) =< w' — w' >. Consider the difference of representatives of both classes (w—w') + (w' — w) = (w—w) — (w' —w') = $ G Qqp(1, F), i.e. $ is a holomorphic (p, ^-differential on F. Therefore,

9(< w >) = w — w + Qqp(1, F) = w' — + $ + Qqp(1, F) = w' — w' + np(1, F) = 6(< w' >).

Thus, the map e is well defined on the equivalency classes.

Let us establish that e is surjective. For any equivalency class w0+np(1, F) we define the class

R R ( 1 \

wo + w + Qqp(1, F'), where w = 1''' ^ on F and (w0 + w) G ^^-q^s, F'J D Mi

for some qj > 0, j = 1,...,n. Thus, d(wo + w + ^qp(1,F') p Mi) = wo + Q,p(1,F). This can be proved differently. Take a Prym differential with required singularities holomorphic at punctures on F' from the bundle in the right hand side of (4) and consider it on the surface with punctures. The map e takes it back.

Now let us prove that the mapping of a fixed fiber over F' and a fiber over the corresponding surface F is 1-to-1. Assume that different equivalency classes are mapped by e to one class,

i.e. 6(< wi >) =< wi — wi >=< w2 — w2 >= 6(< w2 >), where w- G Qqp(^pqi 1 p>qn ,F^j, a w2 G Qp(——1-r,F) and (w2 — wi) does not belong to Qqp(1,F') n Mi. Consider the

KPl1 ...Pt J

difference (w2 — wi) + (wi — w2) = (w2 — w2) — (wi — wij = $ G ^p(1, F). Therefore w2 — wi = = $ + w2 — wi G ^p(1, F') n Mi. This is a contradiction.

Now we prove that d is linear. Indeed, for cj G C, j = 1,2, we have equalities

e\c-(w- + Qqp(1, F') n Mi) + C2(w2 + Qqp(1, F') n Mi)] =

= e[ciwi + C2w2 + nqp(1, F') n Mi] = ciwi + C2w2 — (ciwi + C2^2) + nqp(1, F) = = ci(wi — + ciQqp(1,F)+ c2(w2 — w2) + c2^qp(1,F) = c-e(< w- >) + c2e(< w2 >).

Thus, e is linear, and we get an isomorphism

^(go:1 ,F'v) n Mi KioSQF,Fp)

e. \"C1 ■■■ -is • /__

: ttqp(1,FL) n Mi ttqp(1,F')

of fibers under these conditions.

Now we lift the set (2) to the set of Prym differentials on F^, which is obtained from (2) by adding some differential from ilqp(1,FM) n Mi. All these differentials can be chosen so that they depend holomorphically on [p] and p on FM. Thus, we obtain a set of equivalency classes of differentials

T^Q + W1, K )> tP2L + W1, K), ...> T-paqQ! + nqp(i, f, ), t(H;Q2 + Wp(i,F':),...

...'t(7Q2 + W1,F'tPIU+npa,K)...'TPasQs + W1,F:) (V)

which correspond to Prym differentials from (2), on FM. It is a basis of locally holomorphic sections of the vector bundle E'. Consequently, both these bundles E and E' are holomorphic vector bundles of rank d over mentioned bases.

The operation of gluing up the punctures that makes F' into F defines an n!-sheeted holomorphic mapping from T" onto Ts. Here, over each surface F with fixed points Pi,... ,Pn there are n! surfaces F'.

Thus, we have proved commutativity of the diagram (4) with required properties. □

4. Spaces of univalent differentials

Lemma 3. For each divisor Pqi • ... • Pqn, qj ^ 0,j = 1,...,n, and q > 1 on FM of genus g ^ 2

/ 1 \ R R

there exists a differential uj £ Qq I —qi-—qn ,FM ) with the divisor (lj) = ' '

pqi pq™ ' M / wuvjv, v ) qi qn ,

i • ... • P n J • ... • Pn

where Rj = Pi, l = 1,... ,n, j = 1,..., N, N = (2g — 2)q + qi + ... + qn, and any given principal

parts of Laurent series at Pj, j = 1,... ,n. This differential depends locally holomorphically on

moduli [p] of the surface FM.

The proof is analogous to the proof of Lemma 2.

1

Denote by Qq I —«- a -—; FM \ for q > 1 the space of q-differentials on FM that

V^ .. .Ql Qi+i.. .Qs J

are multiple of the divisor —«- -—, where ai,...,ai > 2, s > 1, 0 < l < s

Qi .. .Ql Qi+i .. .Qs

and the points Qi,...,Qs are distinct, and by Qq (1; FM) denote the subspace of holomorphic q-differentials on FM.

By the Riemann-Roch theorem for q-differentials we find the dimensions of these spaces. It is known that dim^q(1; FM) = (2q — 1)(g — 1) for q > 1. Moreover,

iq{ Q? ...QO'Qi+i ...Q

q«1 ...QTQi+i ■■■q))+

+r(Zq-i Q« ■ ■ ■ Q«1 Qi+i ...Qs) = (g — 1)(2q — 1) + ai + ... + ai + s — l & 4). Therefore, dimnq(—--F^j/Qq (1; F„) = ai + ... + ai + s — l & 1).

yQ ■ ■ ■Qi Qi + i ■ ■ .Qs J

Consider the sets of q-differentials:

T1 T(2) T(«1) T(i) T(«l) T QQ T QQ (5) Tq; Qi, q ; Qi, ■ ■ ■ ,Tq ; Qi, ■ ■ ■ ,Tq ; Qi, Tq ; Qi, ■ ■■, Tq ; Qi, Tq; QiQi+i, ■ ■ ■ ,Tq; QiQs (5)

for l > 1, q > 1;

(i)

Tq;Qi, Tq;QiQ2 , ■ ■ ■ , Tq;QiQs (6)

for l = 0, q > 1 on FM.

Proposition 2 ([6]). The bundle

uai Q? ...Q?QH1...Q./№ (1F')

is a holomorphic vector bundle of rank a.\ + ... + ai + s — l over Ts, where g ^ 2, a.\,... ,ai ^ 2, s ^ 1, 0 ^ l ^ s, q > 1 and the points Q\,... ,Qs are distinct. The equivalency classes of q-differentials from (5), (6) form a base of locally holomorphic sections of this bundle over Tfl.

Consider the diagram

№ (1,Fp П Mx № (1,F)

I I (7)

Tn ^ rg.

Теорема 4. The diagram (7) is commutative; vertical arrows stand for holomorphic vector bundles with isomorphic corresponding fibers, horisontal arrows are for holomorphic n!-sheeted mappings over bases from T^ (apart of the Teichmuller spaces TSj„) and a Teichmiiller space Tg.

The proof follows the proof of Theorem 3 together with Lemma 3 and Proposition 2.

The work was financially supported by RFBR (grants no. 15-01-07906, no. 18-01-00420) and the grant of the Russian Federation Government for scientific research under the supervision of leading scientist at Siberian Federal University (contract no. 14.Y26.31.0006).

References

[1] A.B.Bogatyrev, Real meromorphic differentials: a language for describing meron configurations in planar magnetic nanoelements, Theoretical and Mathematical Physics, 193(2017), no. 1, 1547-1559.

[2] L.Ahlfors, L.Bers, The spaces of Riemann surfaces and quasi-conformal mappings, IL, Moscow, 1961 (in Russian).

[3] H.M.Farkas, I.Kra, Riemann surfaces, Springer, New-York, Grad. Text's Math. V. 71, 1992.

[4] C.J.Earle, Families of Riemann surfaces and Jacobi varieties, Annals of Mathematics, 107(1978), 255-286.

[5] V.V.Chueshev, Multiplicative functions and Prym differentials on a variable compact Riemann surface, Part 2, KemGU, Kemerovo, 2003 (in Russian).

[6] V.V.Chueshev, M.I.Tulina, Prym differentials on a variable compact Riemann surface, Matematicheskie Zametki, 95(2014) no. 3, 457-474 (in Russian).

[7] A.S.Mischenko, Vector bundles and their applications, Nauka, Moscow, 1984 (in Russian).

[8] A.V.Chueshev, V. V.Chueshev, The residue theorem and an analog of P. Appell's formula for finite Riemann surface, Science Evolution, 1(2016), no. 1, 40-45.

Векторное расслоение дифференциалов Прима над пространствами Тейхмюллера поверхностей с проколами

Александр В. Чуешев Виктор В. Чуешев

Институт фундаментальных наук Кемеровский государственный университет Красная, 6, Кемерово, 650043 Россия

В работе исследуются мультипликативные мероморфные функции и дифференциалы на рима-новых поверхностях конечного типа. Доказан аналог формулы П. Аппеля о 'разложении мультипликативной функции с полюсами любых кратностей в сумму элементарных интегралов Прима. Построены явные базисы для ряда важных фактор-пространств. Доказана теорема о послойном изоморфизме векторных расслоений и и\-листных отображений над пространствами Тейхмюллера. Эта теорема дает важную связь между пространствами дифференциалов Прима (абелевых дифференциалов) на компактной римановой поверхности и на римановой поверхности конечного типа.

Ключевые слова: пространства Тейхмюллера римановых поверхностей конечного типа, дифференциалы Прима, векторные расслоения, группа характеров, многообразия Якоби.