Integral representations and volume forms on Hirzebruch surfaces Текст научной статьи по специальности «Математика»

УДК 517.55

Integral Representations and Volume Forms on Hirzebruch Surfaces

Alexey A.Kytmanov*

Institute of Space and Information Technologies, Siberian Federal University, Kirenskogo 26, Krasnoyarsk, 660074,

Russia

Received 10.01.2008, received in revised form 20.02.2008, accepted 05.03.2008

We construct a class of integral representations for holomorphic functions in a polyhedron in C4, associated with Hirzebruch surfaces. The kernels of the integral representations are closed differential forms in C4 associated with volume forms on Hirzebruch surfaces.

Keywords: integral representation, Hirzebruch surface, toric variety.

Introduction

The kernel of the Bochner-Martinelli integral representation in Cn+1 is well known to be closely connected with the Fubini-Studi form for the projective space Pn = CPn as follows:

= ¿T A (1)

(see, for instance, [1, Ch. 3]; [2, Ch. 4]). Here w is the Bochner-Martinelli form,

-+1

»(*) = (¿Луп+I ^(-1)fc-1 FF+2 ^ л ^

dz = dzi A ... A dzn+i, and dz[k] results from deleting the differential dz in dzk. The form w0([£]) is the volume form for the Fubini-Studi metric in Pn (see [3, p. 21])

n! E(£) A E(£) (2)

where

n+1

E(e) = E(-!)k-1^k d£[k]

k= 1

is the Euler form and £ = (£1,..., £„+1) are the homogeneous coordinates of a point [£] G Pn. Moreover, £, z G Cn+1 and A G C are connected by the relation z = A£.

The Bochner-Martinelli form is a "canonical" form of degree 2n +1 in Cn+1 \ {0}. The latter set is a bundle over Pn whose fiber is the one-dimensional torus C*. In other words,

*e-mail: [email protected] © Siberian Federal University. All rights reserved

Pn = [Cn+1 \ {0}]/G, where G = {(A,..., A) e Cn+" : A e C*} is the transformation group of diagonal matrices. The projective space is a particular instance of a toric variety. In the general case, each n-dimensional toric variety is some quotient space (see [4, 5, 6])

X = [Cd \ Z(E)] /G.

Here Z(E) is the union of some coordinate subspaces in Cd constructed from a fan E C Rn with d generators and G is a group isomorphic to the torus (C*)r, r = d — n, which is also constructed from E.

In his report at the "Nordan" conference on complex analysis (Stockholm, April 1999) A. K. Tsikh posed the problem of calculating the volume forms wo([£]) on toric varieties Xk (the Fubini-Studi forms) and the canonical forms w(z) on Cd \ Z(E) with the property

/ \ 1 dA i dAr / r ^n \

w(z) ~ T" A ... A T" A wo([£]),

(2n«)r A" Ar

generalizing (1), where the sign ~ means that the forms have the same residues with respect to A" = ... = Ar =0. Moreover, he noted that the forms w may serve as kernels of integral representations in Cd.

In the present work we consider a class of toric varieties of complex dimension 2 called Hirzebruch surfaces. We construct volume forms for this class and canonical forms in C4 \ Z' where the set Z' is, in general, not the same as the singular set Z(E). It is shown that the constructed canonical forms define an integral representation in 4-circular polyhedra G C C4. In [7] author considered toric varieties, defined by convex fans. Convexity of a fan provides that the singular set of a canonical form w coincides with Z(E). As we will see below in the case of Hirzebruch surfaces fan fails to be convex if k > 2.

1. Hirzebruch Surfaces, Moment Maps and Integration Cycles

Hirzebruch surface Xk is the toric variety defined by the 2-dimensional fan, spanned by the vectors v"=(1,0), v2=(0,1), v3=( — 1, 0), v4=(—k, —1), where k e Z+.

V2

Fig. 1. The fan of X2.

To each vector Vj we assign a complex variable Zj so that Z = (Zi,(2,(3,(4) plays role of homogeneous coordinates of Hirzebruch surfaces Xk. Each pair of nonneighboring vectors

vi,vj (i.e., those not defining a two-dimensional cone) defines a coordinate plane in Z(E) (see [7]) so that

z (E) = {Z1 = Z3 = 0} u {Z2 = Z4 = 0}.

The group G is determined by the relations ^ pj Vj =0 on the vectors Vj. The following

j

equations

V1 + V3 = 0, kv1 + V2 + V4 = 0,

are all linearly independent relations between the vectors Vk. Consequently, the vectors = (1,0,1, 0), p2 = (k, 1, 0,1) constitute a basis for the lattice of relations. The group G is the 2-parameter surface {(A1Akk, A2,A1, A2) : Aj G C*} C (C*)4, so that

Z ~ n ^ 3A1, A2 : Z = (C1, C2, Z3, C4) = (A1Ak^1, A2^2, Am3, A2^4). The moment map (see, for instance, [5, 8]) p : C4 ^ R4/R2 ^ R2 looks like

M(Z1,Z2,Z3,Z4) = (PbP2^

where

P1 = IZ112 + | Z312,

P2 = k|Z1|2 + |Z2 |2 + |Z4 |2.

(3)

For a fixed p = (p1,p2) G R2, the relations (3) define the set rk(p) = p 1(p). The Kahler cone (see, for instance, [5]) for Xk is defined by the following inequalities:

P1 > k (4)

P2 > kp1.

The fact that the inequalities (4) hold provides that the integration cycle rk does not intersect the singular set Z(E).

2. A Canonical Form and a Volume Form

We write down a form w in Cd \ Z(E) that is an analog of the Bochner-Martinelli form and establish its basic properties.

The sought form has bidegree (4, 2) and looks like

w(Z) = ^. (5)

ff(Z,0

The numerator is a form of type (4, 2), where dZ = dZ1 A dZ2 A dZ3 A dZ4, and

h(Z) = Z3Z4dZ1 A dZ2 - Z2Z3dZ1 A dZ4 + Z1Z4dZ2 A dZ3 + kZ1Z3dZ2 A dZ4 + Z1Z2dZ3 A dZ4 (6)

is an analog of the Euler form. The denominator g is the function

g(Z,Zz) = IZ1141Z2 |4-2k + |Z1 l4IZ4|4-2k + IZ2 |2k+4 IZ314 + IZ314 IZ4 |2k+4.

Here we have to make one important remark.

Note that g may contain negative powers of Z. In this case we define the form w as in (5), whose numerator and denominator are multiplied by the least power of Z such that the denominator of the resulting form contains no negative powers of Z. This procedure does not affect the transformation laws of the form w that we will derive below.

However, the singular set Zw of the form w depends on k. More precisely, we have the following three cases:

1. If k = 0 or k =1 then Zw coincides with Z(E) = {Z" = Z3 = 0} U {Z2 = Z4 = 0};

2. If k = 2 then Zw = Z' := {Zi = Z3 = 0} U {Zi = Z2 = Z4 = 0};

3. If k > 2 then Zw = Z" := {Zi = Z3 = 0}U{Z2 = Z4 = 0}U{Zi = Z2 = 0}U{Zi = Z4 = 0}.

Each fixed element 5 = (AiA|,A2,Ai,A2) e G defines the mapping 5 : C4 \ Z(E) — C4 \ Z(E) by the formula Z — 5 • Z, i.e.,

Zi — AiA2Z1,

Z2 — A2Z2, (7)

Z3 — A1Z3, Z4 — A2Z4.

Proposition 1. The differential form w is invariant under the action of 5.

Proof. By direct substitution, we obtain the following transformation laws for h(Z), dZ, and g(Z, C):

h(C) - A?Ak+2h(C), dZ - A2A2k+2dZ, g(Z,C) - (AiAi)2(A2A2)k+2g(Z,C).

Inserting them in w, we arrive at the assertion of the proposition. □

We now describe the behavior of w under the action of the group G : (C4 \ Z(E)) — C4 \ Z(E), defined by (7).

Lemma 1. The form dZ transforms as follows under the action of (7):

dZ — A1Ak+1dA1 A dA2 A h(Z) + -(A, Z), where h is determined by (6), and the form — has higher degree in Z than h(Z). Lemma 2. The form h(C) transforms by the following rule under the action of (7):

h(C) — A2A2+2h(C).

It is not hard to prove lemmas 1 and 2 by direct substitution of the action of G into the forms dZ and h(C).

Let us note that since the denominator g is a function (not differential form), it transforms by the same rule as in Proposition 1 under the action of (7). We thus come to the following

Theorem 1. Under the action of (7) the form w transforms as follows:

dA1 dA2

w ^ —1 A —- A wo + W1 (8)

A1 A2

with the positive form

_ h(Z) A h(Z)

wo = ^ccT

of homogeneity degree zero under the action of the group G and with some form w1, involving no conjugate differentials dAj and having at most one differential dAj in each summand.

The form wo is an analog of the Fubini-Studi form (2) for the projective space. Recall that rk = rk (p) is the set (3). We now treat it as an integration cycle. The cycle rk foliates over Xk with fibers isomorphic to the real tori T2 (T = {z G C : |z| = 1}), i.e.,

rk (P)/Gr = Xk, (9)

where GR := |(AiAk,A2,Ai,A2) : |A¿| = 1, j = 1, 2} (see [5, Theorem 4.1]). From this and Theorem 1 we see that the form wo depends only on the orbits of the group G and consequently is well defined on Xk. Moreover, rk is not homologous to zero in C4 \ Z(E).

At this point let us note that if k ^ 2 then the singular set Zw does not coincide with Z(E). (This happens because the fan E is not strictly convex.) If k = 2 then the singular set Zw is a subset of Z (E), and therefore the cycle rk does not intersect Zw .If k > 2 then the cycle rk can intersect the planes {Zi = Z2 =0} H {Zi = Z4 = 0}. In this case we need to prove the following

Proposition 2. The form w is bounded in the neighborhood of the planes {Zi = Z2 = 0} and {Zi = Z4 = 0}.

Proof. Let us show that the form w is bounded in the neighborhood of the plane {Zi = Z2 = 0}. Let |Zi| = £i, and |Z21 = £2. Equalities (3) imply |Z3|2 = Pi - e2 > pi and

|Z4|2 = p2 — kei — £2 > pi when £i and e2 are sufficiently small. Note that for such |Zk| we

p2pk+2

have that g ^ k+4 , and the numerator h(£) A dZ is bounded. Therefore, the form w is bounded in the neighborhood of {Zi = Z2 = 0}. Similarly one can show that w is bounded in the neighborhood of {Zi = Z4 = 0}. □

Proposition 2 implies that the form w is integrable over the cycle rk.

Corollary 1. The equality Jrk w = C holds, where C is some nonzero constant.

Proof. (8) and (9) imply

f f dAi f dA2 f ■)2 f

w = ^ -T— wo = (2n«) wo.

./rk J |Ai| = 1 Ai J | A2| = i A2 JXk JXk

The last integral is a positive number by positivity of the form wo, as required. □

Now, we prove the following

Proposition 3. The form w is closed.

Proof. In fact we have to demonstrate that (g/<?)dh — d (g/g) A h = 0. This would imply that

(g/g)d(hAdZ)—d(g/^)A(hAdZ) = (g/g)dhAdZ—d(g/g)AhAdZ = ((g/ff)C h—d (g/g)Ah)AdZ = 0,

i.e., the form w is closed. By direct calculation of d h and d(g/g) we get the statement of the proposition. □

Proposition 4. Let f (Z) be a holomorphic function in a neighborhood U about the origin and let pi, p2 be small enough to guarantee r[; C U. Then the following integral representation is valid:

f(0) = 1 f f(Z)w(Z), (10)

C J rk

where C is the normalization constant: Lk w = C = 0.

Ji 0

Proof. Since the form fw is d-closed, the integral in (10) is independent of pi,... ,pr. We rewrite it as

f f(Z)w(Z)=/ f(0)w(Z)+/ (f(Z) — f(0))w(Z) = Jrk ./rj ./rj

= Cf(0)+ / (f(Z) — f(0))w(Z).

Jrj

Let us show that the last integral vanishes. By substituting Z — "Z, we obtain:

Zi — T k+1Zi, Z2 — тZ2, Z3 — Z4 — "Z4.

Then the cycle r[; goes into the cycle :

f |Tk+1Zi|2 + |"Z3|2 = pi, I k|Tk+1Zi|2 + |"Z212 + |"Z4|2 = P2.

The integral goes to

/ (f (Z) — f (0))w(Z ) = lim/ (f (Z) — f (0))w(Z ) = limo/ (f (Z") — f (0))w(Z").

By Proposition 1 the form w is invariant under the substitution w(Z") = w(Z). Since all sk are positive, we have lim f (Z") = f (0). Thus

lim / (f(Zt) — f(0))w(Z") = lim / (f(Z") — f(0))w(Z) = 0. The proof of the proposition is now completed. □

3. Integral Representation

We now consider the question of finding a domain D, such that the following integral representation is valid for every point z G D

f (z) = ^f f (CMC - z). (11)

Consider the domain D = Dp:

Ki|2 + IZ3|2 <P1, (12)

IC2I2 + |Z4|2 <P2 - kpi.

We will show that it is the required domain. Note that D is nonempty if the Kàhler conditions (4) are satisfied.

Denote by Zz(E) the translate z + Z(E):

Zz(£) = (Zi - zi = Z3 - Z3 = 0} U {C2 - Z2 = Z4 - Z4 = 0}, and let rk be the translate z + T^:

rk

|Ci - zi|2 + |C3 - Z3|2 = Pi,

k|Ci - zi|2 + |C2 - Z2I2 + |C4 - Z412 = p2.

Denote by W = Wp 2-circular polyhedron defined by the system

ICi|2 + IC3|2 <P1, k|Ci|2 + IC212 + IC412 <P2.

(13)

By W2p we denote the domain like (13), where the right-hand sides of the inequalities are 2pi, 2p2.

Lemma 3. For each z G D the cycle lies in W2p. Moreover, if the Kahler conditions (4) are satisfied then the homology rz ~ holds in the domain W2p \ Zz (E).

Proof. Consider the following homotopy of the cycles rk and

|Zi - tzi|2 + |Cs - tz3|2 = Pi, (14)

k|Zl - tzi|2 + |Z2 - tZ2|2 + |Z4 - tZ4|2 = P2, ( )

where 0 ^ t ^ 1. We will prove that the cycle (14) is disjoint from Zz(E) for any t in the interval [0,1].

Let us show that the cycle (14) is disjoint from the plane {Zi - zi = Z3 - z3 = 0} in Zz(E). Substituting it to (14) we get (1 - t)2(|Zi|2 + |Z3|2) = Pi. The last equality is false since (1 -1)2 < 1 and |Zi|2 + |Z312 < Pi.

Similarly we show that the cycle (14) is disjoint from the plane {Z2 - z2 = Z4 - z4 = 0} in Zz(E). Substituting it to (14) we get k|Z3 - tz3|2 = -(p2 - kpi) + (1 - t)2(|Z212 + |Z412) that never holds since (1 -1)2 ^ 1 and |Z2|2 + |Z4|2 < P2 - kpi. This completes the proof of the lemma. □

We have thus proven the integral representation (11) for functions holomorphic in W2P. Note that it suffices to take the holomorphy domain of the function f (z) in (11) to be W = Wp, since the latter is a convex domain whose boundary contains the cycle Гк. It follows from convexity of W that a function holomorphic in W and continuous in the closure of W can be approximated by polynomials in the closure of W for which the integral representation (11) is proven. Thus, we arrive at the following

Theorem 2. Suppose that f (Z) is a holomorphic function in the domain W defined by (13) and f is continuous in the closure of W. Then the integral representation (11), with the cycle Гк defined by (3) is valid in the domain D defined by (12).

The author was supported by Grants MK-914.2007.1 and NSh-2427.2008.1 from the President of Russian Federation.

References

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