CC BY
4
Journal of Siberian Federal University. Mathematics & Physics 2018, 11(6), 686—691
УДК 517.9
Prym Differentials and Teichmiiller Spaces
Alexander V. Chueshev Victor V. Chueshev*
Institute of Fundamental Sciences Kemerovo State University Red str., 6, Kemerovo, 650043 Russia
Received 15.08.2018, received in revised form 25.10.2018, accepted 02.11.2018 In this article we study multiplicative meromorphic functions and differentials on Riemann surfaces of finite type. We proved an analogue 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 factor spaces and prove a theorem on a fiber isomorphism of vector bundles and n!-sheeted mappings over Teichuller spaces. This theorem gives an important relation between spaces of Prym differentials on a compact Riemann surfaces and on a Riemann surfaces of finite type.
Keywords: Teichmuller spaces for Riemann surfaces of finite type, Prym differentials, vector bundles, group of characters, Jacobi manifolds. DOI: 10.17516/1997-1397-2018-11-6-686-691.
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].
In this paper we continue constructing the general theory of functions on Riemann surfaces of the 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.
1. Preliminaries
Let F be a smooth compact oriented surface of genus g > 2, with the marking {ak, bk}gk=l, 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 Pl,...,Pn € F. We assign type (g,n) to a surface F' = F\{Pi,... ,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 F0. Thus, F0 = U/r', where r' has the representation [2]
* vvchueshev@ngs.ru © Siberian Federal University. All rights reserved
r' = {Au...,Ag ,Bu...,Bg ,Cu...,Cn : J] Aj Bj Aj1B-1C\ ...Cn = I).
j=i
Any other complex analytic structure on F' is given by a Beltrami differential p on F0, i.e. by an expression of the form p(z)dz/dz, invariant with respect to the choice of the local parameter on F0, where p(z)is a complex-valued function on F0 and HmIIl^(F') < 1. We denote this structure on F' by F ' .
Since the mapping U ^ F0 = U/r' is a local diffeomorphism, any Beltrami differential p on F0 lifts to a Beltrami r'-differential p on U, i. e. p G L^(U), ||p||TO = esssupzey \p(z)\ < 1, and p(T(z))T'(z)/T'(z) = p(z),z G U,T G r', see [3].
In the work [3, p. 99] there were constructed abelian differentials Zi[p],..., Zg[p] on F[£], that form a canonical base dual to a canonical homotopy base {a£,b£}gk=1 on F£, which depends holomorphically on moduli [p] 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 F£. Here we assume that the class [p] has Bers coordinates hl,h2,... h3g-3 when embedding the Teichmiiller space Tg (F0) of compact Riemann surfaces into C3g-3. Moreover, the matrix of b-periods
i(p) = (njk [p])gjk=l on F£ consists of complex numbers j [p] = f Zj ([p],w)dw, £ G w£ (U),
«
and depends holomorphically on [p].
For any fixed [p] G Tg and £o G w£(U) define a classical Jacobi mapping ^ : w£(U) ^ Cg by «
the rule: j (£) = f Zj ([p], w)dw, j = 1,... ,g. The quotient space J (F) = Cg/L(F) is called the
«0
marked Jacobi manifold for F = F0, where L(F) is a lattice over Z, generated by the columns
of the matrix (Ig, l), where Ig is an identity matrix of order g. The universal Jacobi manifold of order g is a fibered space over Tg, with fiber over [p] G Tg being a marked Jacobi manifold J(F£) for a marked Riemann surface F£ [4].
A character p for F£ is any homomorphism p : (n\(F £), ■) ^ (C*, ■), C* = C\{0}. Further on we shall assume that p(y£) = 1, where y£ is a simple loop around only one puncture Pj on F'£, j = 1,...,n.
Definition 1. A multiplicative function f on F£ for the character p is a mermorphic function f on w£(U) such that f (Tz) = p(T)f (z), z G w£(U), T G r£.
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 G U,T G r', p : r' ^ C*.
If a multipicative function f0 on F£ for p does not have zeroes or poles, then the character p is called non-essential and fo is called a unit. The characters which are not non-essential are called essential on n\(F£). The set Lg of non-essential characters form a subgroup in the group Hom(r£, C*) of all characters on r£. A divisor on F£ is a formal product D = Pn .. .Pkik, Pj G F£,nj G Z, j = 1,...,k.
Theorem (Abel's theorem for characters [2,5]). Let D be a divisor on a marked variable compact Riemann surface [F £, {a£,..., a£, b£, .. ., b£}] of genus g ^ 1, and p be a character on n\(F£). Then D is a divisor of a multiplicative function f on F £ for p if and only if degD = 0 and
1 g 1 g *(D) = 2^12l0g p(b£ )e(j) - p(a£ )nU)[p]( = Hp, M)),
j=i j=i
where ^[p] : F^ ^ J(F^) 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 F^.
2. An analog of Appel's decomposition formula
for a multiplicative function on a variable Riemann surface of finite type
p
Denote by T(1Q = — J r(2Q an elementary Prym integral of second kind on F^ for an essential
Qo
character p with only simple pole at Q with residue that depends holomorphically on [y] and p, where r^Q has zero residue at Q [5,6].
Theorem 1. Let f be a branch of a function of class M\ for an essential character p on a variable Riemann surface F^ of type (g, n), g ^ 2, n > 0, with pairwise distinct poles at Pn+1,..., Pn+s of multiplicities qn+i,..., qn+s with given principal parts:
( A% + ... + ( AjP )), j = n +l,...,n + s. (1)
(z — z(Pj))q (z — z(Pj))
Then for an analytic continuation of f we have (f) ^ -pq-nqn+s, qj ^ 0, j = l,... ,n, on F^
Pf ■ ■ ■ Pn+s
and
n+s [ . dm—It(1) ] 9—1 rP
f = j ^
j=1 m=l
j,m__p;pj
(m - 1)! dPm—1
+ j / C:
j=i -'Qo
where f = --..--+ • • + --,,o +--^^r+0(1) for some branch in a neighborhood
J (z - z(Pj ))f (z - z(Pj ))2 z - z(Pj ) KJJ y
of Pj, j = 1, ■ ■ ■ ,n + s, m Fa, 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 second kind with only simple pole on F ^ 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 Aa
with zero residues at Q1 and Q2 [5,6]. In this case one should use as basic elements of expansion
p
the Prym integrals tp;q1q2 = - j tp;q2q2 of second kind with two simple poles Q1 and Q2.
Qo '12
(2) ( 1 cL1) \
The Prym differential t\/q admits the expansion --^ +---+ 0(1) \dz in a neigh-
P'Q1 \(z - z1)2 z - z1 J
(1) 9 ' borhood ofQ1, z(Q1) = z1, where c—1 = logp(aj(Q1) [5,6].
j=1
(2) ( 1 c(2)1 \ The Prym differential t\q als has an expansion ---r +---+ 0(1) \dz in a neigh-
P;Q2 \(z - z2)2 z - z2 J
(2) 9 '
borhood of Q2 Ha Fa, where c_ 1 = logp(aj(Q2)■
j=1
Prym differentials with two poles of the second order and zero residues at these points my be given in the form
TpQQ = c—f o(Q1)tP2qQi - c—lfo(Q2)Tp2Q2 - c— c—%;QiQ2 ■
Note that the principal part for tp q,q2 at Q1 has the form fo(Q1), and at Q2 it is — fo(Q2),
z — z1 z — z2
It follows that the differential constructed above tp;q2q2 has poles of the second order at Q1 and
Q2, and zero residues at these points.
Theorem 2. Let f be a branch of a function of class M1 for a non-essential character p on a variable Riemann surface F £ of type (g,n),g ^ 2, n > 0, with pairwise distinct poles at Pn+1, ■ ■ ■, Pn+s of multiplicities qn+i, ■ ■ ■, qn+s with given principal parts 1. Assume that for an
analytic continuation of f to F£ the conditions (f ) ^ —q-uqn+s, qj ^ 0,j = l^^/n, and
pqi pq
P1 ■ ■ ■ Pn+s
J2 logp(aj(Pn+s) = 0 are fulfilled. Then j=1
B f P n+s-1 A rn qi A am-lm
7 (P ^ CjL f°Cj + g -dn+fM + £ (m - 1)! OPr1 +
n+ s
+E
j=2
A dTp;Pj Pi + j d2Tp;Pj Pi + + 3 dqj-1 TP;Pj Pi
j'2 dPj + 2! dP2 + + (qj - 1)! dPqi-1
j
+C,
where
Ajq + ••• + , + + o(i)
(z - z(Pj))qj (z - z(Pj))2 z - z(Pj)
for some branch in a neighborhood of Pj, j = 1,... ,n + s, on F£; C = 0 for p = 1; dk =
g ,
= Y1 log p(am)^m(Pk), k = 1,... ,n + s, on F£, and all summands depend holomorphically on
m=l
[p] and p.
3. Vector bundles of Prym differentials over a Techmiiller space of Riemann surfaces of finite type
Denote by ilqp(yar~?as> f£] the vector space of (p, q)-differentials that are multiples of
1 ■ ■ ■ Qs s '
the divisor -a-jas, where a.j > 1, a.j G N, j = 1,... ,s, s > 1,q > 1,q G N, and by lqp(1; F£)
Q1 ■ ■ ■ Qs s
the vector subspace of holomorphic (p, q)-differentials on F £ [2]. Here the divisor Q\ ...Qs on F£ is understood as a constant set of points on a surface F of genus g > 2.
Lemma 1 ( [5], 105-106). A holomorphic prinsipal Hom(r, C*)-bundle E is biholomorphic to the trivial bundle Tg(F0) x Hom(r, C*) over Tg(F0).
Proposition 1. The vector bundle E = ; fA/Qqp(l; F^) over Tg x(Hom(r, C*)\1)
^ Q1 ' ' ' QS '
for q > 1 (over Tg x (Hom(r, C*)\Lg) when q = 1) and g ^ 2 is a holomorphic vector bundle of rank a.\ + ■ ■ ■ + as = d, while the co-sets of (p, q)-differentials
T w T(ai) a) T(as)
p,q;Qi' ' ' ' ' p,q;Qi' ' ' ' ' p,q;Qs'''' ' p,q;Qs'
(2)
form a basis of locally holomorphic sections of this bundle.
Lemma 2. For any divisor Pqi • ■ ■ ■ • Pqn, 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 üj G Q.qp ^-p^-P^qn, F£^j
with the divisor (ü)
R1
, R
N
pqi ^ ^ pqn
, where Rj = P¡ ,l = 1,
N = (2g - 2)q+
+qi +... + qn, and any given principal parts of Laurent series at Pj,j = l,..., n, for its branches. This differential depends locally holomorphically on moduli [y] of the surface F^ and the character p.
Consider the diagram
E' = U
(q?1 -Q?"
Qqp\Qa
n Mi
üqP(\,F' ) n Mi
i
T" x Hom(r, C*)\X
u
(q?1 ^-Q?8 ,Fß)
Qqp\Qa
I
=E
Ts x Hom(r, C*)\X,
(3)
where Tn is a part of the Teichmiiller space Ts,n [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 [2].
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 = l when q > l, and X = Lg when q = l.
It should be noted that analogous results hold true for the spaces of single-valued (Abelian) differentials.
The work was finacially 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 the 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] H.M. Farkas, I. Kra, Riemann surfaces, Springer, New-York, Grad. Text's Math., vol. 71, 1992.
[3] L.Ahlfors, L.Bers, The spaces of Riemann surfaces and quasi-conformal mappings, IL, 1961 (in Russian).
[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, Math.Notes, 95(2014), no/ 3-4, 418-433.
Дифференциалы Прима и пространства Тейхмюллера
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.
