Numbers Hypergeometric Functions Associated with...

Funkcialaj Ekvacioj, 41 (1998) 291-308

Intersection Numbers for 1-Forms Associated with ConfluentHypergeometric Functions

By

Keiji MATSUMOTO(Hiroshima University, Japan)

1. Introduction

There are four coffiuent hypergeometric functions of the Gauss hyper-geometric function, which are called the Kummer function, the Bessel function,the Hermite function and the Airy function. They admit the following type ofintegral representation

(1) $¥int_{¥gamma}u(t)¥varphi(t)$ ,

where $u(t)$ is a multi-valued function such that the logarithmic derivative$¥omega:=$ dlog(u(t)) is a rational 1-form on the projective line $P$ with the polardivisor $D¥subset¥{0,1, ¥infty¥}$ , $¥varphi(t)$ is a rational 1-form on $P$ with poles only on $D$ and $¥gamma$

is a curve such that the function $u(t)$ vanishes at the terminals of $¥gamma$ . By theStokes theorem

$¥int_{¥partial_{p}}u(t)¥eta(t)=¥int_{¥rho}d(u(t)¥eta(t))=¥int_{¥rho}u(t)(d¥eta(t)+¥omega¥wedge¥eta(t))$ ,

we have

(2) $¥int_{¥gamma}u(t)V_{¥omega}f(t)=¥int_{¥partial¥gamma}u(t)f(t)=0$ ,

where $V_{¥omega}$ is the derivation $ d+¥omega¥wedge$ and $f(t)$ is a rational function on $P$ withpoles only on $D$ . The equality (2) implies that the integral (1) depends only onthe class of the form $¥varphi(t)¥in¥Gamma(P,¥Omega^{1} (*D))$ modulo $V_{¥omega}¥Gamma(P, ¥mathcal{O}(*D))$ . In this way,we are naturally led to the ffist rational twisted de-Rham cohomology group

$H^{1}$ $(¥Gamma(¥Omega.(*D), V_{¥omega}))=¥Gamma(P,¥Omega^{1} (*D))/V_{¥omega}¥Gamma(P, ¥mathcal{O}(*D))$

with respect to $¥omega$ . This group is important for the study of confluenthypergeometric functions and the differential equations annihilating these.

In this paper, we introduce a pairing between $H^{1}$ $(¥Gamma(¥Omega.(*D), V_{¥omega}))$ and$H^{1}$ $(¥Gamma(¥Omega.(*D),V_{¥_}¥omega))$ for general effective divisors $D$ on $¥mathrm{P}$ , following the idea in

292 Keiji MATSUMOTO

[2], and present intersection matrices with respect to some bases of thesegroups. It is shown in [2], [5], [8] and [10] that intersection matrices of the firsttwisted cohomology groups with respect to $¥pm¥omega$ with at most simple poles playan important role in studying hypergeometric functions with regular singu-larities.

Our intersection matrices appear in [7] when considering the limit (con-fluence) of the intersection matrices associated with Lauricella’s hypergeometricfunctions $F_{D}’ ¥mathrm{s}$ in [2].

Acknowledgment. The auther is grateful to Professor K. Cho for valuablediscussion on hypercohomology groups, to Professor Y. Haraoka for informinghis results in [7] and to Professors H. Kimura, H. Majima and N. Takayama fortheir encouragement.

2. Isomorphisms

Let $n_{1}$ , $¥ldots,n_{m}$ be natural numbers such that

$n_{1}¥geq n_{2}¥geq¥cdots¥geq n_{m}$ , $n=¥sum_{i=1}^{m}n_{i}¥geq 3$

and let $x_{1}$ , $¥ldots,¥chi_{m}$ be distinct $m$ points on the complex projective line $P$ . Put

$X=P¥backslash ¥{x_{1}, ¥ldots,x_{m}¥}$ ,

$¥omega=¥sum_{i=1}^{m}(¥frac{a_{i,1}}{t-x_{i}}.+¥frac{a_{i;2}}{(t-¥chi_{i})^{2}}+¥cdots+¥frac{a_{i;n_{¥mathrm{i}}-1}}{(t-x_{i})^{n_{i}-1}}+¥frac{¥alpha_{i;n_{i}}}{(t-X_{i})^{n_{i}}})dt$,

where $a_{i;k}¥in C$ , $a_{i;n_{i}}¥neq 0$ and $a_{i;1}$ ’s satisfy that $a_{i;1}¥not¥in Z$ if $n_{i}=1$ and that

$¥sum_{i=1}^{m}¥alpha_{i;1}=0$ .

For the connection $ V_{¥omega}=d+¥omega¥wedge$ with respect to $¥omega$ and the polar divisor

$D=¥sum_{i=1}^{m}n_{i}x_{i}$

of the rational 1-form $¥omega$ on $P$, we have a complex:

$(¥Omega.(*D),V_{¥omega}):¥cdots 0¥rightarrow ¥mathcal{O}(*D)¥rightarrow V_{¥omega}¥Omega^{1}$$(*D)¥rightarrow 0¥cdots$ ,

where $¥mathcal{O}(*D)$ and $¥Omega^{1}$

$(*D)$ are the sheave of meromorphic functions and 1-formswith poles only on $¥{x_{1}, ¥ldots,x_{m}¥}$ , respectively. The group

$H^{1}$ $(¥Gamma(¥Omega.(*D), V_{¥omega}))=¥Gamma(P,¥Omega^{1} (*D))/V_{¥omega}¥Gamma(P, ¥mathcal{O}(*D))$

Intersection Numbers for 1-Forms 293

is called the first rational twisted de-Rham cohomology group with respect to$V_{¥omega}$ .

Proposition 2.1. We have the following isomorphism:

$ J¥omega$ : $H^{1}$ $(¥Gamma(¥Omega.(*D), V_{¥omega}))¥rightarrow¥Gamma(P,¥Omega^{1}(D))/C¥cdot¥omega$,

where $¥Omega^{1}(D)$ is the sheaf of meromorphic lforms $¥psi$ such that

$(¥psi)+D¥geq 0$ .

The rank of $H^{1}$ $(¥Gamma(¥Omega.(*D),V_{¥omega}))$ is $n-2$.

Proof. We can eliminate poles of any element of $¥Gamma(P,¥Omega^{1}(*D))$ higherthan $D$ by %,-images; refer to [4] for details. The rank of $¥Gamma(P,¥Omega^{1}(D))/C¥cdot¥omega$ is

$(¥deg(D)-1)$ $-1=(¥sum_{i=1}^{m}n_{i})-2=n-2$ . $¥square $

We have two complexes:

$ F_{1}(¥Omega.(*D),V_{¥omega}):¥cdots 0¥rightarrow ¥mathcal{O}¥rightarrow V_{¥omega}¥Omega^{1}(D)¥rightarrow 0¥cdots$ ,

$ F_{0}(¥Omega.(*D),V_{¥omega}):¥cdots 0¥rightarrow ¥mathcal{O}(-D)¥rightarrow V_{¥omega}¥Omega^{1}¥rightarrow 0¥cdots$ ,

where $¥mathcal{O}$ is the sheaf of holomorphic functions on $P$, $¥Omega^{1}$ the sheaf of hol-omorphic 1-forms on $P$, and $¥mathcal{O}(-D)$ the sheaf of holomorphic functions $f$ suchthat

$(f)$ ? $D¥geq 0$ .

Proposition 2.2. We have two isomorphisms

$¥mathrm{H}^{1}(P,F_{1}(¥Omega.(*D),V_{¥omega}))¥simeq¥Gamma(P,¥Omega^{1}(D))/C¥cdot¥omega$ ,

$¥mathrm{H}^{1}$ $(P,F_{0}(¥Omega.(*D),V_{¥omega}))¥simeq ¥mathrm{k}¥mathrm{e}¥mathrm{r}(V_{¥omega} : H^{1}(P, ¥mathcal{O}(-D))¥rightarrow H^{1}(P,¥Omega^{1}))$ .

Proof. Since the spectral sequence

$E_{1}^{0q}¥simeq H^{q}(P, ¥mathcal{O})$ , $E_{1}^{1q}¥simeq H^{q}(P,¥Omega^{1}(D))$

abuts to $¥mathrm{H}^{p+q}(P,F_{1} (¥Omega.(*D), V_{¥omega}))$ and $E_{1}^{pq}=0$ for $q>0$, we have an iso-morphism

$¥mathrm{H}^{1}$ $(P,F_{1} (¥Omega.(*D),V_{¥omega}))¥simeq¥Gamma(P,¥Omega(D))/C¥cdot¥omega$ .

Since the spectral sequence

$E_{1}^{0q}¥simeq H^{q}(P, ¥mathcal{O}(-D))$ , $E_{1}^{1q}¥simeq H^{q}(P,¥Omega^{1})$$E_{1}^{1q}¥simeq H^{q}(P,¥Omega^{1})$

294 Keiji MATSUMOTO

abuts to $¥mathrm{H}^{p+q}(P,F_{0}(¥Omega.(*D),V_{¥omega}))$ and $E_{1}^{pq}=0$ for $q=0$, we have an iso-morphism

$¥mathrm{H}^{1}(P,F_{0}(¥Omega.(*D), V_{¥omega}))¥simeq ¥mathrm{k}¥mathrm{e}¥mathrm{r}(¥mathrm{V}_{¥omega} : H^{1}(P, ¥mathcal{O}_{P}(-D))¥rightarrow H^{1}(P,¥Omega_{P}^{1}))$ . $¥square $

Consider the following exact sequence of complexes:

0 0 0$¥downarrow$ $¥downarrow$ $¥downarrow$

0?$¥rightarrow ¥mathcal{O}(-D)¥rightarrow l$ $¥mathcal{O}$

$¥rightarrow¥bigoplus_{i=1}^{n}(¥sum_{k=1}^{n_{i}}b_{i;k}(t-x_{i})^{k-1})_{x_{i}}¥rightarrow 0$

$ V_{¥omega}¥downarrow$ $ V_{¥omega}¥downarrow$ $ V_{¥omega}¥downarrow$

$ 0¥rightarrow$ $¥Omega^{1}$ $¥rightarrow l¥Omega^{1}(D)¥rightarrow¥varpi$$¥bigoplus_{i=1}^{n}(¥sum_{k=1}^{n_{i}}¥frac{c_{i,k}dt}{(t-x_{i})^{k}})_{¥chi_{i}}$ $¥rightarrow 0$

$¥downarrow$ $¥downarrow$$¥downarrow$

0 0 0

where $(¥sum_{k=1}^{n_{i}}b_{i;k}(t-x_{i})^{k-1})_{x_{i}}$ is the germ of polynomials of degree $(n_{i}-1)$ at $x_{i}$ ,

$(¥sum_{k=1}^{n_{i}}¥frac{c_{i;k}dt}{(t-x_{i})^{k}})_{x_{i}}$ is the germ of the $n_{i}$ -th principal parts of the Laurent

expansion of meromorphic forms at $x_{i}$ , and $¥varpi$ is defined by taking the principalpart of a meromorphic form at $x_{i}$ .

Proposition 2.3. The inclusion $i$ induces the isomorphism between twohypercohomology groups

$¥iota$ : H. $(P,F_{0}(¥Omega.(*D), V_{¥omega}))¥rightarrow ¥mathrm{H}.(P,F_{1}(¥Omega.(*D),V_{¥omega}))$ .

Proof. Note that

$V_{¥omega}$ : $¥sum_{k=1}^{n_{i}}b_{i;k}(t-x_{i})^{k-1}¥rightarrow¥sum_{k=1}^{n_{i}}¥frac{c_{i,k}dt}{(t-x_{i})^{k}}$

.

is expressed as$(c_{i;n_{i^{ }}},¥ldots, c_{i;1})=(b_{i;1}, ¥ldots,b_{i;n_{i}})A_{i}$

$A_{i}$ $=¥left(¥begin{array}{lllll}a_{i¥cdot n_{i}}. & a_{i¥cdot n_{¥mathrm{i}}-¥mathrm{l}}. & ¥cdots & a_{i¥cdot 2}. & a_{i¥cdot 1}.¥¥ & a_{i¥cdot n_{i}}. & ¥cdots & a_{i¥cdot 3}. & a_{i¥cdot 2},¥¥ & & .. & ¥vdots & ¥vdots¥¥ & 0 & & a_{i¥cdot n_{i}}, & a_{i¥cdot n_{i}-¥mathrm{l}}.¥¥ & & & & a_{i¥cdot n_{i}},¥end{array}¥right)$ ,

$......¥cdot$

.

$a_{i¥cdot n_{i}}a_{i;3}a_{i;2}...$

,

$a_{i;n_{i}-1}a_{i;n_{i}}a_{i;1}a_{i,2}....)$ ,


Since $¥det(A_{i})=a_{i¥cdot n_{¥mathrm{i}}}^{n_{i}},¥neq 0$ for any $i$, $l$ induces the isomorphism between thehypercohomology groups. $¥square $

By Propositions 2.2 and 2.3, we have the isomorphism

$¥mathrm{k}¥mathrm{e}¥mathrm{r}(¥mathrm{V}_{¥omega} : H^{1}(P, ¥mathcal{O}(-D))¥rightarrow H^{1}(P,¥Omega^{1}))¥rightarrow¥Gamma(P,¥Omega(D))/C¥cdot¥omega$ .

In order to give the inverse of the above explicitly, we prepare two propositions.

Proposition 2.4. We have an isomophism

$¥mathrm{H}^{1}(P,F_{0}(¥Omega.(*D),V_{¥omega}))¥simeq.¥frac{¥mathrm{k}¥mathrm{e}¥mathrm{r}(V_{¥omega}.¥Gamma(d^{01}(-D)¥oplus¥ovalbox{¥tt¥small REJECT}^{10})¥rightarrow¥Gamma(d^{11}))}{V_{¥omega}¥Gamma(d^{00}(-D))}$ ,

where $d^{pq}$ stands for the sheaf of smooth $(p, q)$ forms on $P$ and $¥theta^{pq}(-D)$ thesheaf of smooth $(p, q)$ forms $¥zeta$ on $P$ such that $¥zeta/(t-x_{i})^{n_{i}}$ is smooth around $x_{i}$ .

Proof. A fine resolution of $F_{0}(¥Omega.(*D),V_{¥omega})$

$0¥rightarrow ¥mathcal{O}(-¥downarrow D)¥rightarrow V_{¥omega_{¥mathrm{I}}}¥Omega^{1}¥rightarrow 0$

$0¥rightarrow d00_{¥overline{¥partial}}¥downarrow¥downarrow+¥omega_{¥overline{¥partial}}(-D)^{¥partial}¥rightarrow d^{10}¥rightarrow 0$

$0¥rightarrow¥ovalbox{¥tt¥small REJECT}^{01}(-D)¥rightarrow¥partial+¥omega d^{11}¥rightarrow 0$

$¥downarrow$ $¥downarrow$

0 0

yields the associated single complex

$0¥rightarrow ¥mathcal{O}(-¥downarrow D)$

$¥rightarrow V_{¥omega}$

$¥Omega^{1}¥downarrow$ $¥rightarrow 0¥downarrow¥rightarrow 0$

$0¥rightarrow¥ovalbox{¥tt¥small REJECT}^{00}(-D)¥rightarrow V_{¥omega}g^{01}(-D)¥oplus¥ovalbox{¥tt¥small REJECT}^{10}¥rightarrow V_{¥omega}d^{11}¥rightarrow 0$

which implies this proposition. $¥square $

Note that the Dolbeault theorem implies

$H^{1}(P, ¥mathcal{O}(-D))¥simeq¥frac{¥Gamma(¥theta^{01}(-D))}{¥overline{¥partial}¥Gamma(d^{00}(-D))}$ , $H^{1}(P,¥Omega^{1})¥simeq¥frac{¥Gamma(d^{11})}{¥overline{¥partial}¥Gamma(¥theta^{10})}$ .

296 Keiji MATSUMOTO

The connection $V_{¥omega}$ annihilates an element

$¥eta¥in¥frac{¥Gamma(ff^{01}(-D))}{¥overline{¥partial}¥Gamma(d^{00}(-D))}¥simeq H^{1}(P, ¥mathcal{O}(-D))$

means that $ V_{¥omega}¥eta$ is zero in

$H^{1}(P,¥Omega^{1})¥simeq¥frac{¥Gamma(g^{11})}{¥overline{¥partial}¥Gamma(d^{10})}$ ,

i.e. there exists $¥xi(¥eta)¥in¥Gamma(d^{10})$ such that

$¥nabla_{¥omega}¥eta-¥overline{¥partial}¥xi(¥eta)=0$ .

This gives an explicit expression of the isomorphism

$¥mathrm{k}¥mathrm{e}¥mathrm{r}(¥mathrm{V}_{¥omega} : H^{1}(P, ¥mathcal{O}(-D))¥rightarrow H^{1}(P,¥Omega^{1}))¥ni¥eta$

$¥mapsto¥eta-¥xi(¥eta)¥in.¥frac{¥mathrm{k}¥mathrm{e}¥mathrm{r}(V_{¥omega}.¥Gamma(d^{01}(-D)¥oplus¥theta^{10})¥rightarrow¥Gamma(d^{11}))}{V_{¥omega}¥Gamma(d^{00}(-D))}$ .

Proposition 2.5. We have an isomorphism

$¥mathrm{H}^{1}(P,F_{1}(¥Omega.(*D),V_{¥omega}))¥simeq.¥frac{¥mathrm{k}¥mathrm{e}¥mathrm{r}(V_{¥omega}.¥Gamma(d^{01}¥oplus d^{10}(D))¥rightarrow¥Gamma(¥#^{11}(D)))}{V_{¥omega}¥Gamma(d^{00})}$ ,

there $d^{pq}(D)$ stands for the sheaf of smooth $(p,q)$ forms $¥zeta$ on $X$ such that$(t -x_{i})^{n_{i}}¥zeta$ is smooth around $x_{i}$ .

Proof. A fine resolution of $F_{1}$ $(¥Omega.(*D),V_{¥omega})$

$0¥rightarrow ¥mathcal{O}¥downarrow¥rightarrow V_{¥omega_{¥mathrm{I}}}¥Omega^{1}(D)¥rightarrow 0$

$0¥rightarrow d^{00}¥overline{¥partial}¥downarrow¥rightarrow¥partial+¥omega_{¥mathrm{I}}d^{10}¥overline{¥partial}(D)¥rightarrow 0$

$0¥rightarrow¥theta^{01}¥rightarrow¥partial+¥omega d^{11}(D)¥rightarrow 0$

$¥downarrow$ $¥downarrow$

0 0


yields the associated single complex

$0¥rightarrow ¥mathcal{O}¥rightarrow V_{¥omega}$$¥Omega^{1}(D)$ $¥rightarrow$ 0 $¥rightarrow 0$

$¥downarrow$ $¥downarrow$ $¥downarrow$

$0¥rightarrow g^{00}¥rightarrow V_{¥omega}¥ovalbox{¥tt¥small REJECT}^{01}¥oplus ff^{10}(D)¥rightarrow V_{¥omega}d^{11}(D)¥rightarrow 0$

which implies this proposition. $¥square $

Note that an explicit expression of the map

$¥Gamma(P,¥Omega(D))/C$ . $¥omega¥rightarrow.¥frac{¥mathrm{k}¥mathrm{e}¥mathrm{r}(V_{¥omega}.¥Gamma(¥theta^{01}¥oplus d^{10}(D))¥rightarrow¥Gamma(¥theta^{11}(D)))}{V_{¥omega}¥Gamma(¥theta^{00})}$

is given by

$¥psi¥mapsto 0¥oplus¥psi$ .

The isomorphism $¥iota$ in Proposition 2.3 is expressed as the natural map

$.¥frac{¥mathrm{k}¥mathrm{e}¥mathrm{r}(V_{¥omega}.¥Gamma(¥ovalbox{¥tt¥small REJECT}^{01}(-D)¥oplus^{g^{10}})¥rightarrow¥Gamma(g^{11}))}{V_{¥omega}¥Gamma(¥theta^{00}(-D))}$

$¥rightarrow.¥frac{¥mathrm{k}¥mathrm{e}¥mathrm{r}(V_{¥omega}.¥Gamma(d^{01}¥oplus d^{10}(D))¥rightarrow¥Gamma(d^{11}(D)))}{¥Gamma(V_{¥omega}d^{00})}$ .

Let us express the inverse of $ 0¥oplus¥psi$ $(¥psi¥in¥Gamma(P,¥Omega(D))/C. ¥omega)$ of the aboveisomorphism. Let

$¥sum_{k=1}^{n_{i}}¥frac{c_{i;k}dt}{(t-X_{i})^{k}}$

be the principal part of the Laurent expression of $¥psi$ at $x_{i}$ . Put

$¥eta_{i}=¥sum_{k=1}^{n_{i}}b_{i;k}(t-x_{i})^{k-1}$ ,

where

$(b_{i,1}, ¥ldots,b_{i;n_{i}})=(c_{i;n_{i^{ }}},¥ldots, c_{i;1})A_{i}^{-1}$ .

A smooth function on $P$

$¥eta=¥sum_{i=1}^{m}h_{i}¥eta_{i}$

298 Keiji MATSUMOTO

satisfies

$V_{¥omega}¥eta=¥sum_{i=1}^{m}(dh_{i}¥eta_{i}+h_{i}V_{¥omega}¥eta_{i})$ ,

where $h_{i}$ is a smooth function such that

$h_{i}(t)=1$ $t¥in V_{i}$ ,

$0¥leq h_{i}(t)¥leq 1$ $t¥in U_{i}¥backslash V_{i}$ ,

$h_{i}(t)=0$ $t¥in U_{i}^{c}$ .

Since $V_{¥omega}¥eta_{i}$ is the principal part of the Laurent expression of $¥psi$ at $x_{i}$ and

$dh_{i}(t)=0$ , $h_{i}(t)=1$

for $t¥in V_{i}$ , we have$¥psi-V_{¥omega}¥eta¥in¥Gamma(d^{01}(-D)¥oplus d^{10})$ ,

which represents $¥iota^{-1}$$(¥psi)$ in

$.¥frac{¥mathrm{k}¥mathrm{e}¥mathrm{r}(V_{¥omega}.¥Gamma(_{(}g^{01}(-D)¥oplus ff^{10})¥rightarrow¥Gamma(d^{11}))}{V_{¥omega}¥Gamma(d^{00}(-D))}$ .

We conclude this result in the following Proposition.

Proposition 2.6. For any $¥psi¥in¥Gamma(P,¥Omega(D))$ , construct a smooth function

$¥eta=¥sum_{i=1}^{m}h_{i}¥eta_{i}$ ,

where $h_{i}$ is a smooth function with small support to be identically 1 on the smallneighborhood $V_{i}$ of $¥chi_{i}$ , and $¥eta_{i}$ is the polynomial such that $V_{¥omega}¥eta_{i}$ is equal to theprincipal part of the Laurent expression of $¥psi$ at $x_{i}$ . Then the isomorphism

$¥iota_{¥omega}$ : $¥Gamma(P,¥Omega(D))/C$ . $¥omega¥rightarrow ¥mathrm{k}¥mathrm{e}¥mathrm{r}(V_{¥omega} : H^{1}(P, ¥mathcal{O}(-D))¥rightarrow H^{1}(P,¥Omega^{1}))$

is given by$¥psi¥mapsto¥iota_{¥omega}(¥psi)=¥psi-V_{¥omega}¥eta$ ,

of which (01)-part is $-¥sum_{i=1}^{m}(¥eta_{i}dh_{i})^{(01)}$ , where $¥zeta^{(01)}$ stands for the (01)-part ofsmooth lform $¥zeta$ .


3. Duah.ty

By Propositions in the previous section, we have the isomorphisms

$H^{1}$ $(¥Gamma(¥Omega.(*D),V_{¥omega}))¥rightarrow J¥omega¥Gamma(P,¥Omega^{1}(D))/C¥cdot¥omega$

$¥rightarrow¥iota_{¥omega}¥mathrm{k}¥mathrm{e}¥mathrm{r}(V_{¥omega} : H^{1}(P, ¥mathcal{O}(-D))¥rightarrow H^{1}(P,¥Omega^{1}))$ ,

$H^{1}$ $(¥Gamma(¥Omega.(*D),V_{¥_}¥omega))¥rightarrow J-¥omega¥Gamma(P,¥Omega^{1}(D))/C$ . $(-¥omega)$ .

We define the intersection number $¥langle¥psi^{+}, ¥psi^{-}¥rangle_{¥omega}$ of $¥psi^{+}¥in H^{1}$ $(¥Gamma(¥Omega.(*D), V_{¥omega}))$ and$¥psi^{-}¥in H^{1}$ $(¥Gamma(¥Omega.(*D), V_{-¥omega}))$ as

$¥langle¥psi^{+}, ¥psi^{-}¥rangle_{¥omega}=¥int_{P}¥iota_{¥omega}oJ¥omega(¥psi^{+})¥wedge¥gamma_{-¥omega}(¥psi^{-})$ .

Since $¥iota_{¥omega^{¥circ}J¥omega}(¥psi^{+})$ and $¥gamma_{¥_}¥omega(¥psi^{-})$ are represented by elements of $¥Gamma(¥theta^{01}(-D))$ and$¥Gamma(d^{10}(D))$ , the above integral converges. Since we have

$¥int_{P}¥iota¥omega J¥omega ¥mathrm{o}(¥psi^{+})¥wedge¥omega=¥int_{P}¥{V_{¥omega}(¥iota_{¥omega^{¥circ}J¥omega}(¥psi^{+}))-d(l¥omega J¥omega ¥mathrm{o}(¥psi^{+}))¥}$

$=¥int_{P}¥overline{¥partial}¥xi(¥iota_{¥omega J¥omega}¥mathrm{o}(¥psi^{+}))=¥int_{P}d¥xi(¥iota_{¥omega^{¥mathrm{O}}J¥omega}(¥psi^{+}))=0$,

where $¥xi(¥iota_{¥omega^{¥mathrm{O}}J¥omega}(¥psi^{+}))$ is an element $¥Gamma(¥theta^{10})$ such that

$V_{¥omega}(¥iota_{¥omega^{¥circ}J¥omega}(¥psi^{+}))-¥overline{¥partial}¥xi(¥iota_{¥omega}¥mathrm{o}J¥omega(¥psi^{+}))=0$ ,

and

$¥int_{P}(V_{¥omega}(¥zeta^{01}))^{(01)}¥wedge g_{-¥omega}(¥psi^{-})=¥int_{P}(¥overline{¥partial}¥zeta)¥wedge g_{-¥omega}(¥psi^{-})=¥int_{P}d(¥zeta¥wedge g_{-¥omega}(¥psi^{-}))=0$ ,

for $¥zeta¥in¥Gamma(d^{00}(-D))$ , the intersection number $¥langle¥psi^{+}, ¥psi^{-}¥rangle_{¥omega}$ is well-defined. Notethat this pairing is naturally induced by the Serre duality for $H^{1}(P, ¥mathcal{O}(-D))$ and$¥Gamma(P,¥Omega^{1}(D))$ .

4. Intersection matrices

To state the main theorem, we recall notations: $n_{1}$ , $¥ldots,n_{m}$ are naturalnumbers such that

$n_{1}¥geq n_{2}¥geq¥cdots¥geq n_{m}$ , $n=¥sum_{i=1}^{m}n_{i}¥geq 3$

300 Keiji MATSUMOTO

and $x_{i}$ , $¥ldots,x_{m}$ are distinct $m$ points on $P$ . For

$¥omega=¥sum_{i=1}^{m}(¥frac{a_{i;1}}{t-¥chi_{i}}+¥frac{a_{i_{¥mathrm{i}}2}}{(t-x_{i})^{2}}+¥cdots+¥frac{a_{i;n_{i}-1}}{(t-x_{i})^{n_{i}-1}}+¥frac{a_{i;n_{¥mathrm{i}}}}{(t-x_{i})^{n_{i}}})dt$,

where $a_{i;k}¥in C$ , $a_{i;n_{¥mathrm{i}}}¥neq 0$ and $a_{i;1}$ ’s satisfy that $a_{i;1}¥not¥in Z$ if $n_{i}=1$ and that

$¥sum_{i=1}^{m}a_{i;1}=0$ ,

we put $ V_{¥omega}=d+¥omega¥wedge$ , the polar divisor

$D=¥sum_{i=1}^{n}n_{i}x_{i}$

of the meromorphic 1-form $¥omega$ on $P$ and $(¥mathrm{n}_{¥mathrm{i}},¥mathrm{n}_{¥mathrm{i}})$-matrix

$A_{i}$ $=¥left(¥begin{array}{lllll}a_{i¥cdot n_{i}}. & a_{i¥cdot n_{i}-¥mathrm{l}}. & ¥cdots & a_{i¥cdot 2}. & a_{i¥cdot ¥mathrm{l}}.¥¥ & a_{i¥cdot n_{i}}, & ¥cdots & a_{i¥cdot 3}. & a_{i¥cdot 2}.¥¥ & & .. & ¥vdots & ¥vdots¥¥ & 0 & & a_{i¥cdot n_{i}}. & a_{i¥cdot n_{i}-¥mathrm{l}}.¥¥ & & & & a_{i¥cdot n_{i}}.¥end{array}¥right)$ .

Note that

$A_{i}^{-1}=¥left(¥begin{array}{lllll}¥lambda_{n_{i}}(a_{i}) & ¥lambda_{n_{i}-1}(a_{i}) & ¥cdots & ¥lambda_{2}(a_{i}) & ¥lambda_{1}(a_{i})¥¥ & ¥lambda_{n_{i}}(a_{i}) & ¥cdots & ¥lambda_{3}(a_{i}) & ¥lambda_{2}(a_{i})¥¥ & & ¥ddots & ¥vdots & ¥vdots¥¥ & 0 & & ¥lambda_{n_{i}}(a_{i}) & ¥lambda_{n_{i}-¥mathrm{l}}(a_{i})¥¥ & & & & ¥lambda_{n_{i}}(a_{i})¥end{array}¥right)$ ,

where $a_{i}=(a_{i;n_{i^{ }}},¥ldots, a_{i;1})$ and

$¥lambda_{n_{i}}(a_{i})=¥frac{1}{a_{i;n_{i}}}$ , $¥lambda_{n_{i}-1}(a_{i})=-¥frac{a_{i;n_{i}-1}}{a_{i;n_{¥mathrm{i}}}^{2}}$ , $¥lambda_{n_{i}-2}(a_{i})=-¥frac{a_{i;n_{i}}a_{i;n_{i}-2}-a_{i,n_{i}-1}^{2}}{a_{i;n_{i}}^{3}}.$ ,

$¥lambda_{n_{i}-3}(a_{i})=-¥frac{a_{i;n_{i}}^{2}a_{i;n_{i}-3}-2a_{i;n_{i}}a_{i;n_{i}-1}a_{i;n_{i}-2}+a_{i;n_{i}-1}^{3}}{a_{i;n_{i}}^{4}}$ , $¥ldots$ .

Take a base

Intersection Numbers for 1-Forms

of $(n -1)$ -dimensional vector space of $¥Gamma(P,¥Omega^{1}(D))$ as

$¥varphi_{i;k}=¥left¥{¥begin{array}{l}¥frac{(t-x_{i})^{k-1}dt}{(t-x_{i})^{n_{¥mathrm{i}}}},¥¥¥frac{dt}{(t-x_{i})}-¥frac{dt}{(t-x_{i+¥mathrm{l}})},¥end{array}¥right.$ $ 11¥leq i¥leq i¥leq¥leq m-1m,1¥leq$

,

$kk¥leq n_{i}-=n_{i}.1$

,

Let $¥varphi_{i;k}^{¥pm}¥in¥Gamma(P,¥Omega^{1}(D))/C$ . $(¥pm¥omega)$ be the natural projections of $¥varphi_{i;k}$ .

Theorem 4.1. The intersection matrix $¥langle¥varphi_{i;k}^{+}, ¥varphi_{j;l}^{-}¥rangle_{¥omega}$ is

$2¥pi¥sqrt{-1}$$¥left(¥begin{array}{lllll}¥Lambda_{1} & & & & E_{1}¥¥ & ¥Lambda_{2} & & & E_{2}¥¥ & & ¥ddots & & ¥vdots¥¥ & & & ¥Lambda_{p} & E_{p}¥¥{}^{t}E_{1} & {}^{t}E_{2} & ¥cdots & {}^{t}E_{p} & H¥end{array}¥right)$ ,

of which components are given as follows: the $(n_{i}-¥mathit{1},n_{i}-¥mathit{1})$ -matrix $¥Lambda_{i}$

$¥left(¥begin{array}{llllll}¥lambda_{¥mathrm{l}}(a_{i}) & ¥lambda_{2}(a_{i}) & ¥lambda_{3}(a_{i}) & ¥cdots & ¥lambda_{n_{i}-2}(a_{i}) & ¥lambda_{n_{i}-¥mathrm{l}}(a_{i})¥¥¥lambda_{2}(a_{i}) & ¥lambda_{3}(a_{i}) & ¥lambda_{4}(a_{i}) & ¥cdots & ¥lambda_{n_{i}-¥mathrm{l}}(a_{i}) & ¥lambda_{n_{i}}(a_{i})¥¥¥lambda_{3}(a_{i}) & ¥lambda_{4}(a_{i}) & ¥lambda_{5}(a_{i}) & ¥cdots & ¥lambda_{n_{i}}(a_{i}) & 0¥¥¥vdots & ¥vdots & ¥vdots & & & ¥vdots¥¥¥lambda_{n_{i}-2}(a_{i}) & ¥lambda_{n_{i}-1}(a_{i}) & ¥lambda_{n_{i}}(a_{i}) & ¥cdots & 0 & 0¥¥¥lambda_{n_{i}-1}(a_{i}) & ¥lambda_{n_{i}}(a_{i}) & 0 & ¥cdots & 0 & 0¥end{array}¥right)$ ,

the $(n_{i}-¥mathit{1},m-¥mathit{1})$ -matrix $E_{i}$ is

$¥mathrm{i}$ -th$(i-1)$-th

$¥left(¥begin{array}{llllll}0 & ¥cdots & -¥lambda_{n_{¥mathrm{i}}}(a_{i}) & ¥lambda_{n_{i}}(a_{i}) & ¥cdots & 0¥¥0 & ¥cdots & 0 & 0 & ¥cdots & 0¥¥¥vdots & ¥cdots & ¥vdots & ¥vdots & ¥cdots & ¥vdots¥¥ 0 & ¥cdots & 0 & 0 & ¥cdots & 0¥end{array}¥right)$,

and the $(m -¥mathit{1},m -¥mathit{1})$ -matrix $H$ is

$p-1m-p$

$H=mp-1-p¥left(¥begin{array}{ll}0 & 0¥¥0 & H,¥end{array}¥right)$’

302 Keiji MATSUMOTO

$H^{¥prime}=(_{0}^{¥frac{}{a_{p+1}}}¥frac{1}{a_{p+1}1}00$

$¥frac{1}{a_{p+1}}...¥cdot¥frac{1}{a_{p+2}}-..¥frac{1}{a_{p+1}}+...¥frac{1}{a_{p+2}}$

$¥frac{1}{a_{p+2}}.¥frac{1}{a_{p+2}}+0..¥frac{1}{a_{p+3}}$$..........$

.

$¥frac{1}{a_{m-2}}¥frac{+1}{a_{m-1}}000¥frac{1}{a_{m-1}}$

$¥frac{1-}{a_{m-1}}¥frac{1}{a_{m-}}000+¥frac{11}{a_{m}}]$ ,

where$p=¥neq¥{j|n_{j}>1¥}$

and we regard $a_{i}$ as $a_{i;1}$ for $i¥geq p+1$ . When $p=1$ , we regard $H$ as $H^{/}$, andwhen $p=0$, we regard $H$ as the submatrix of $H^{¥prime}$ obtained by deleting the firstrow and column.

Proof. By Proposition 2.6, the (01)-part $¥iota_{¥omega}(¥varphi_{i;k})$ is

$¥{-¥eta_{i}(¥varphi_{i;k})dh_{i}+¥delta(n_{i},k)¥eta_{i+1} (¥varphi_{i;k})dh_{i+1}¥}^{(01)}$ ,

where $¥delta(n_{i},k)$ is Kronecker’s symbol and $¥eta_{i}(¥varphi_{i;k})$ and $¥eta_{i+1}$ $(¥varphi_{i;k})$ are polynomialssuch that the principal parts the Laurent expressions of $V_{¥omega}¥eta_{i}(¥varphi_{i;k})$ at $x_{i}$ and$V_{¥omega}¥eta_{i+1}$ $(¥varphi_{i;k})$ at $x_{i+1}$ coincide with those of $¥varphi_{i;k}$ at $x_{i}$ and $x_{i+1}$ , respectively. Notethat the coefficients of polynomial $¥eta_{i}(¥varphi_{i;k})$ are obtained from

$¥mathrm{k}$-th

$(0, ¥ldots,0,1,0, ¥ldots,0)A^{-1}$ ,

which is the $¥mathrm{k}$-th row vector of $A^{-1}$ , and that

$¥eta_{i+1}(¥varphi_{i;n_{i}})=¥lambda_{n_{i+1}}(a_{i+1})t^{n_{i+1}-1}$ .

By the Stokes theorem and the residue theorem, for $¥varphi¥in¥Gamma(P,¥Omega^{1}(D))$ , we have

$¥langle¥varphi^{+}i¥cdot,k’¥varphi^{-}¥rangle_{¥omega}=¥int_{P}(-¥eta i(¥varphi i;k)dh_{i}+¥delta(n_{i},k)¥eta i+1(¥varphi i;k)dh_{i+1})_{¥Lambda}¥varphi$

$=¥int_{U_{i}+U_{i+1}}$ $(-¥eta_{i}(¥varphi_{i;k})dh_{i}+¥delta(n_{i},k)¥eta_{i+1} (¥varphi_{i;k})dh_{i+1})¥wedge¥varphi$

$=¥int_{¥partial U_{i}-¥partial V_{i}}-h_{i}¥eta_{i}(¥varphi_{i;k})¥varphi+¥delta(n_{i},k)¥int_{¥partial U_{i+1}-¥partial V_{r+1}}h_{i+1}¥eta_{i+1}(¥varphi_{i;k})¥varphi$


$=¥int_{¥partial V_{i}}¥eta i(¥varphi i;k)¥varphi-¥delta(n_{i},k)¥int_{¥partial V_{i+1}}¥eta i+1(¥varphi i;k)¥varphi$

$=2¥pi¥sqrt{-1}¥{({¥rm Res}_{t=x_{i}}¥eta_{i}(¥varphi_{i;k})¥varphi)-¥delta(n_{i},k)({¥rm Res}_{t=x_{i+1}}¥eta_{i+1} (¥varphi_{i;k})¥varphi)¥}$ .

Note that

${¥rm Res}_{t=x_{i}}¥eta_{i}(¥varphi_{i;k})¥varphi_{j;l}=¥left¥{¥begin{array}{l}¥lambda_{k+l-¥mathrm{l}}(a_{i}),¥¥-¥lambda_{n_{i}}(a_{i}),¥¥0,¥end{array}¥right.$ $¥mathrm{o}ij=i-1,k=1,l’==j,k+l¥leq n_{i}+1¥mathrm{t}¥mathrm{h}¥mathrm{e}¥mathrm{r}¥mathrm{w}¥mathrm{i}¥mathrm{s}¥mathrm{e},n_{i-1}$ ,

${¥rm Res}_{t=x_{i+1}}¥eta_{i+1}(¥varphi_{i;n_{i}})¥varphi_{j;l}=¥left¥{¥begin{array}{l}¥frac{¥delta(1,n_{i+1})}{a_{i+¥mathrm{l}}},¥¥-¥lambda_{n_{i+1}}(a_{i+1}),¥¥0,¥end{array}¥right.$

$ji¥mathrm{o}¥mathrm{t}¥mathrm{h}¥mathrm{e}¥mathrm{r}¥mathrm{w}¥mathrm{i}¥mathrm{s}¥mathrm{e}=j,l=,n_{i}=i+1,l’=1$ ,

which imply our results. $¥square $

5. Examples

1. Intersection matrix associated with the gamma function $¥Gamma(a)$ . We put

$¥omega=-dt+a¥frac{dt}{t}(a¥neq 0)$ , $ x_{1}=¥infty$ , $x_{2}=0$ .

Note that the gamma function is defined by

$¥Gamma(a)=¥int_{0}^{¥infty}e^{-1}¥exp(¥int_{1}^{t}¥omega)¥frac{dt}{t}=¥int_{0}^{¥infty}e^{-t}t^{¥alpha}¥frac{dt}{t}$

for $¥mathrm{R}¥mathrm{e}(a)>0$ . Since $¥omega$ is expressed as

$¥omega=¥frac{ds}{s^{2}}-a¥frac{ds}{s}$

around $ t=¥infty$ by the local coordinate $s=1/t$, the polar divisor $D$ of $¥omega$ is$2x_{1}+x_{2}$ and $A_{i}$ ’s are

$A_{1}=¥left(¥begin{array}{ll}1 & -a¥¥0 & 1¥end{array}¥right)$ , $A_{2}=a$ .

The vector space $¥Gamma(P,¥Omega^{1}(D))$ is 2-dimensional; we give its base as

$¥varphi_{1,1}=¥frac{ds}{s^{2}}=-dt$ , $¥varphi_{1,2}=¥frac{ds}{s}=¥frac{-dt}{t}$ .$¥varphi_{1,2}=¥frac{ds}{s}=¥frac{-dt}{t}$ .

304 Keiji MATSUMOTO

Since$A_{1}^{-1}=¥left(¥begin{array}{ll}1 & a¥¥0 & 1¥end{array}¥right)$ , $A_{2}^{-1}=¥frac{1}{a}$ ,

the intersection matrix with respect to this base is

$2¥pi¥sqrt{-1}$$¥left(¥begin{array}{ll}a & 1¥¥1 & 1/a¥end{array}¥right)$

of which determinant is 0.

2. Intersection matrix associated with the integral $¥int_{-¥infty}^{¥dashv¥infty}e^{-t^{2}/2}dt$ . We put

$¥omega=-tdt$, $ x_{1}=¥infty$ .

Note that

$¥int_{-¥infty}^{+¥infty}e^{-t^{2}/2}dt=¥int_{-¥infty}^{¥dashv¥infty}¥exp(¥int_{0}^{t}¥omega)dt$ .

Since $¥omega$ is expressed as

$¥omega=¥frac{ds}{s^{3}}$

around $ t=¥infty$ by the local coordinate $s=1/t$, the polar divisor $D$ of $¥omega$ is $3x_{1}$

and $A_{1}$ is

$A_{1}=¥left(¥begin{array}{lll}1 & 0 & 0¥¥0 & 1 & 0¥¥0 & 0 & 1¥end{array}¥right)$ .

The vector space $¥Gamma(P,¥Omega^{1}(D))$ is 2-dimensional; we give its base as

$¥varphi_{1,1}=¥frac{ds}{s^{3}}=-tdt$ , $¥varphi_{1,2}=¥frac{ds}{s^{2}}=-dt$ .

The intersection matrix with respect to this base is

$2¥pi¥sqrt{-1}$$¥left(¥begin{array}{ll}0 & 0¥¥0 & 1¥end{array}¥right)$ .

Note that $¥varphi_{1,1}^{+}=0$ as an element of the quotient the space $¥Gamma(P,¥Omega^{1}(D))/¥mathrm{C}¥cdot¥omega$ .

3. The case $n=4$, intersection matrices associated with confluenthypergeometric functions.

There are four confluent hypergeometric functions of the Gauss hyper-geometric function

$F(a,b, c;z)=¥frac{¥Gamma(c)}{¥Gamma(a)¥Gamma(c-a)}¥int_{0}^{1}t^{a}(1-t)^{c-a}(1-zt)^{-b}(¥frac{dt}{t}-¥frac{dt}{t-1})$ :


the Kummer function

$K(a, c;z)=¥frac{1}{¥Gamma(a)}¥int_{0}^{¥infty}e^{-zt}t^{a}(1-t)^{c-a}(¥frac{dt}{t}-¥frac{dt}{t-1})$ ,

the Bessel function

$J(a;z)=¥frac{1}{2¥pi¥sqrt{-1}}¥int_{¥gamma}¥exp(¥frac{Z}{2}(t-¥frac{1}{t}))t^{-a}¥frac{dt}{t}$ ,

the Hermite function

He(a, $z$) $=¥frac{1}{2¥pi¥sqrt{-1}}¥int_{0}^{¥infty}¥exp(-¥frac{1}{2}t^{2}+zt)t^{-a}¥frac{dt}{t}$ ,

and the Airy function

$Ai(z)=¥frac{1}{2¥pi¥sqrt{-1}}¥int_{t_{0}-¥swarrow 1}¥exp(-¥frac{t^{3}}{3}+zt)dt$,

where $¥gamma$ is a path starting from $-¥infty$ , turning around 0 counterclockwise andgoing to $-¥infty$ , and $¥swarrow j$ is a ray starting from 0 with argument $2j¥pi/3$ . Intersectionmatrices associated with above five functions are obtained as follows:

(i) the Gauss hypergeometric function

$¥omega=a¥frac{dt}{t}+(c-a)¥frac{dt}{t-1}-b¥frac{dt}{t-1/z}$ ,

$D=¥sum_{i=1}^{4}x_{i}$ , $ x_{1}=¥infty$ , $x_{2}=0$ , $x_{3}=1$ , $x_{4}=1/z$,

$A_{1}=b-c$, $A_{2}=a$, $A_{3}=c-a$, $A_{4}=-b$,

$¥varphi_{1;1}=¥frac{-dt}{t}$ , $¥varphi_{2;1}=¥frac{dt}{t}-¥frac{dt}{t-1}$ , $¥varphi_{3;1}=¥frac{dt}{t-1}-¥frac{dt}{t-1/z}$ ,

$¥langle¥varphi_{i;k}, ¥varphi_{j;l}¥rangle_{¥omega}=2¥pi¥sqrt{-1}$( $¥frac{¥frac{-1}{ac}}{(c¥frac{--1a)}{c-a}a}$

$¥frac{-a+c}{(c-a)b}¥frac{-01}{c-b-a}$ ).

Keiji MATSUMOTO

$¥vee ¥mathrm{h}¥mathrm{e}$ Kummer function

$¥omega=-zdt+a¥frac{dt}{t}+(c-a)¥frac{dt}{t-1}$ ,

$D=2x_{1}+x_{2}+x_{3}$ , $ x_{1}=¥infty$ , $x_{2}=0$ , $x_{3}=1$ ,

$A_{1}=¥left(¥begin{array}{ll}Z & -c¥¥0 & Z¥end{array}¥right)$ , $A_{2}=a$ , $A_{3}=c-a$ ,

$¥varphi_{1;1}=-dt=¥frac{ds}{s^{2}}$ , $¥varphi_{1;2}=¥frac{-dt}{t}$ , $¥varphi_{2;1}=¥frac{dt}{t}-¥frac{dt}{t-1}$ ,

$¥langle¥varphi_{i;k}, ¥varphi_{j;l}¥rangle_{¥omega}=2¥pi¥sqrt{-1}$ ( $¥frac{}{z_{0}}¥frac{c}{z1}$

$¥underline{-a1¥underline{1}1}$

$¥frac{}{a(c-a)}¥frac{-01}{ac}$ ).the Bessel function

$¥omega=¥frac{Z}{2}dt-a¥frac{dt}{t}+¥frac{Z}{2}¥frac{dt}{t^{2}}$ ,

$D=2x_{1}+2x_{2}$ , $ x_{1}=¥infty$ , $x_{2}=0$ ,

$A_{1}=¥left(¥begin{array}{ll}¥frac{Z}{2} & a¥¥0 & -¥frac{Z}{2}¥end{array}¥right)$ , $A_{2}=¥left(¥begin{array}{ll}¥frac{Z}{2} & -a¥¥0 & ¥frac{Z}{2}¥end{array}¥right)$ ,

$¥varphi_{1;1}=-dt=¥frac{ds}{s^{2}}$ , $¥varphi_{1;2}=¥frac{-dt}{t}$ , $¥varphi_{2;1}=¥frac{dt}{t^{2}}$ ,

$¥langle¥varphi_{i;k}, ¥varphi_{j;l}¥rangle_{¥omega}=2¥pi¥sqrt{-1}$ $¥left(¥begin{array}{lll}¥frac{-4a}{z^{2}} & 0 & ¥frac{-2}{Z}¥¥0 & ¥frac{4a}{z^{2}} & ¥frac{-2}{Z}¥¥¥frac{-2}{Z} & ¥underline{-2} & 0¥end{array}¥right)$ .

the Hermite function

$¥omega=-tdt+zdt$ $-a¥frac{dt}{t}$ ,

$D=3x_{1}+x_{2}$ , $ x_{1}=¥infty$ , $x_{2}=0$ ,


$A_{1}=¥left(¥begin{array}{lll}1 & -z & a¥¥0 & 1 & -z¥¥0 & 0 & 1¥end{array}¥right)$ , $A_{2}=-a$,

$¥varphi_{1;1}=-tdt=¥frac{ds}{s^{3}}$ , $¥varphi_{1;2}=-dt=¥frac{ds}{s^{2}}$ , $¥varphi_{1;3}=¥frac{-dt}{t}$ ,

$¥langle¥varphi_{i;k}, ¥varphi_{j;l}¥rangle_{¥omega}=2¥pi¥sqrt{-1}$ $¥left(¥begin{array}{llll}z^{2} & z-a & 1 & 1¥¥ & 1 & 0 & ¥frac{-01}{a}¥end{array}¥right)$ .

(v) the Airy function$¥omega=-t^{2}dt+zdt$ ,

$D=4x_{1}$ , $X1$ $=¥infty$ .

$A_{1}=¥left(¥begin{array}{llll}1 & 0 & -z & 0¥¥0 & 1 & 0 & -z¥¥0 & 0 & 1 & 0¥¥0 & 0 & 0 & 1¥end{array}¥right)$ ,

$¥varphi_{1;1}=-t^{2}dt=¥frac{ds}{s^{4}}$ , $¥varphi_{1;2}=-tdt=¥frac{ds}{s^{3}}$ , $¥varphi_{1;3}=-dt=¥frac{ds}{s^{2}}$ ,

$¥langle¥varphi_{i;k}, ¥varphi_{j;l}¥rangle_{¥omega}=2¥pi¥sqrt{-1}$ $¥left(¥begin{array}{lll}0 & & 0¥¥Z & 0 & 1¥¥0 & 1 & 0¥end{array}¥right)$ .

References

[1] Aomoto, K. and Kita, M., Hypergeometric functions (in Japanese), Spring-Verlag, Tokyo,1994.

[2] Cho, K. and Matsumoto, K., Intersection theory for twisted cohomologies and twistedRiemann’s period relations I, Nagoya Math. J., 139 (1995), 67-86.

[3] Kimura, H., Haraoka, Y. and Takano, K., The generalized coIffiuent hypergeometricfunctions, Proc. Japan Acad., 68A (1992), 290-295.

[4] Kimura, H., On rational de Rham cohomology associated with the generalized confluenthypergeometric functions I, $P^{1}$ case, Proc. Roy. Soc. Edinburgh, 127A (1997), 145-155.

[5] Kita, M. and Matsumoto, K., Duality for hypergeometric functions and invariant Gauss-Manin systems, Compositio Math., 108 (1997), 77-106.

[6] Kita, M. and Yoshida, M., Intersection theory for twisted cycles I, II, Math. Nachr., 166(1994), 287-304, 168 (1994), 171-190.

[7] Haraoka, Y., Quadratic relations for coffiuent hypergeometric functions on $Z_{2,n+1}$ , preprint.[8] Matsumoto, K., Quadratic identities for hypergeometric series of type (k,$ $l), Kyushu J. of

Math., 2 (1994), 335-345.

308 Keiji MATSUMOTO

[9] Matsumoto, K., Intersection numbers for logarithmic k-forms, to appear in Osaka J. Math.[10] Ohara, K., Intersection numbers of twisted cohomology groups associated with Selberg-type

integrals, preprint.

nuna adreso:

Department of MathematicsFaculty of ScienceHiroshima UniversityHigashi-Hiroshima 739-8526JapanE-mail: matsucmath.sci.hiroshima-u.ac.jp

(Ricevita la 30-an de majo, 1997)

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