DIRECT IMAGE OF THE DE RHAM SYSTEM ASSOCIATED WITH A RATIONAL DOUBLE POINT

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1997

DIRECT IMAGE OF THE DE RHAM SYSTEM ASSOCIATED WITH A RATIONAL DOUBLE POINT

—A FIVE FINGERS EXERCISE

S H I N I C H I T A J I M A

Department of Information Engineering, Faculty of Engineering, Niigata University 8050 Ikarashi Niigata 950-21, Japan

E-mail: tajima@geb.ge.niigata-u.ac.jp

1. Introduction. In 1976, M. Kashiwara [6] introduced the notion of direct image of D-modules in his study of b-functions. The notion of direct image enjoys nice functorial properties, and the structure of direct image of D-modules arouses great interest in various problems. In this paper we study the direct image of the de Rham system associated with a resolution of a rational double point singularity. In Section 2, we briefly recall some basic notions which are used later. In Section 3, we consider the surface with a rational double point of the type A m . We give some explicit integral representation formulae for the Dirac delta function.

2. The de Rham system and the direct image functor.

de Rham system. Let X be a complex manifold of dimension n, O _X the sheaf of holomorphic functions. Let D X be the sheaf on X of rings of partial differential operators with holomorphic coefficients. The sheaf O _X is naturally endowed with a structure of left D X -Module by differentiation. For instance, let (x 1 , x 2 , . . . , x n ) be a system of local coordinates of X. For any germ h of holomorphic function, we have ∂

∂x j

h = ∂h

∂x j

. But if we regard h as a section of D X , i.e. as a linear partial differential operator of order zero, we have

∂

∂x _j h = ∂h

∂x _j + h ∂

∂x _j , j = 1, 2, . . . , n.

Hence we have

O X ∼ = D X /D X

∂

∂x ₁ , ∂

∂x ₂ , . . . , ∂

∂x _n
.

1991 Mathematics Subject Classification: 35A27, 35N10.

The paper is in final form and no version of it will be published elsewhere.

[155]

In fact, the sheaf O X is generated by the constant function 1 over the sheaf of rings D X and the annihilating ideal of the function 1 is locally equal to the following ideal:

D X

∂

∂x 1

, ∂

∂x 2

, . . . , ∂

∂x n

.

The coherent left D X -Module O X is called the de Rham system.

Algebraic local cohomology. Let Y be a closed analytic subset of X, J Y the defining ideal of Y . For each positive integer k, we set

H ^k _{[Y ]} (O X ) = lim

m→∞ Ext ^k _O

(O X /J _Y ^m , O X ).

Since the sheaf O _X is a left D _X -Module, the algebraic local cohomology group H ^k _{[Y ]} (O X ) is endowed with the structure of left D X -Module. Moreover, Z. Mebkhout [8] and M. Kashiwara [6] proved the following facts:

(i) H ^k _{[Y ]} (O X ) is a coherent D X -Module, (ii) H ^k _{[Y ]} O X
is a regular holonomic system.

When Y is a complex submanifold, we have the following result.

Proposition (Kashiwara [4].) If Y is defined by x ¹ = . . . = x d = 0 for a local coordinate system (x ₁ , . . . , x _n ) of X, then:

(i) H ^k _{[Y ]} (O X ) = 0 for k 6= d, (ii) H ^d _{[Y ]} (O X ) ∼ = D X /D X (x 1 , . . . , x d , ∂

∂x d+1

, . . . , ∂

∂x n

).

Direct image. Let us recall briefly the notion of the direct image of D-Modules.

Let X, Z be complex manifolds, f : Z → X a proper holomorphic map. We set D X←Z = f ⁻¹ (D X ⊗ O

Ω ^⊗−1 _X ) ⊗ _f

_O

Ω Z ,

where Ω _Z and Ω _X are the sheaves of the highest degree holomorphic forms on Z and X respectively. Note that D X←Z is a (f ⁻¹ D X , D Z )-bi-Module.

For any coherent left D Z -Module M, we set Z

f

M = Rf _∗ (D X←Z ⊗ ^L _D

M)

in the derived category D ^b (D X ) of D X -Modules (we refer to [3], [6] and [9]).

We have the following fundamental result.

Proposition (Kashiwara, cf. [3]) Let Y be a complex d-codimensional submanifold of X. Let i be the natural embedding map. Then we have

Z

i

O Y = H ^d _{[Y ]} (O _X ).

Example ([10], [11]). As an illustration of the direct image, let us examine the de Rham system associated with the resolution of a plane curve singularity.

Let X = C ² with coordinates (x, y). Let Y = {(x, y) | x ⁵ − y ³ = 0}. Let T = C with

coordinate t, π : T → X with π(t) = (t ³ , t ⁵ ). Let i : T → Z be the natural embedding

map, where Z = X × T . We have the following commutative diagram:

T −−−−−−→ ⁱ Z



 y



 y ^proj

Y −−−−−−→ X

here proj is the natural projection map proj : X × T → X.

Now we set

u = Z

π

1

where 1 stands for the constant function, which is a generator over D T of the de Rham system O _T . We have

u = Z

proj

Z

i

1 = Z

proj

δ(x − t ³ )δ(y − t ⁵ ).

Then u satisfies the following system of linear partial differential equations:

P 1 u = P 2 u = P 3 u = 0, where

P ₁ = x ⁵ − y ³ , P 2 = 3x ∂

∂x + 5y ∂

∂y + 7, P 3 = 3y ² ∂ ³

∂x ² ∂y + 5x ⁴ ∂ ³

∂x∂y ² + 25x ³ ∂ ²

∂y ² + 9y ∂ ²

∂x ² . Furthermore we have

D X u = D X /D X (P 1 , P 2 , P 3 ) and u is equal to xyδ(x ⁵ − y ³ ) up to non-zero constant.

3. Calculation and a result. In this section we take a resolution of a surface with a rational double point and consider the de Rham system on the resolution. One of our aims is to calculate the D _X -Module structure of the direct image of the de Rham system.

We present here the key point of our calculation.

Resolution. Let X = C ³ with coordinates (x, y, z). Let S be the surface with a rational double point at the origin defined by

S = {(x, y, z) ∈ X | z ^m+1 = xy}.

We resolve the singularity of the surface S as follows. Let W 0 , W 1 , . . . , W m be copies of C ² with coordinates (u ₀ , v ₀ ), (u ₁ , v ₁ ), . . . , (u _m , v _m ) respectively. Following a standard ar- gument, we patch them up and construct a non-singular surface M by using the following transition functions:

u _k+1 = 1/v _k , v _k+1 = u _k v _k ² , for k = 0, 1, 2, . . . , m − 1.

We introduce a holomorphic map π : M → X by







x = u ^k+1 _k v ^k _k y = u ^m−k _k v _k ^m−k+1

z = u _k v _k on W _k , k = 0, . . . , m.

It is easy to see that π : M → X is well defined and π is a resolution of the singularity of the surface S. The exceptional set of the resolution consists of curves C 1 , . . . , C m , where C _k = {u _k−1 = 0} ∪ {v _k = 0}.

Set Z = X × P ¹ × P ¹ × . . . × P ¹ . Let ([ξ 1 , η 1 ], [ξ 2 , η 2 ], . . . , [ξ m , η m ]) be the standard homogeneous coordinates in the product P ¹ × P ¹ × . . . × P ¹ . Set

p _k = ξ _k /η _k , q _k = η _k /ξ _k , k = 1, 2, . . . , m and

p 1 = u ^m−k−1 _k v _k ^m−k , p 2 = u ^m−k−2 _k v _k ^m−k−1 , . . . , p m−k = v k ,

q m−k+1 = u k , q m−k+2 = u ² _k v k , . . . , q m = u ^k _k v ^m−k−1 _k for k = 0, . . . , m − 1.

This defines a holomorphic embedding map i : M → Z. Note that we have i(C k ) = [0, 1] × . . . × [0, 1] × P ¹ × [1, 0] × . . . × [1, 0]. We have the following diagram:

M −−−−−−→ ⁱ Z



 y



 y ^proj

S −−−−−−→ X

here proj is the natural projection map proj : X × P ¹ × P ¹ × . . . × P ¹ → X.

Calculation. Let us examine the integrals along π of the de Rham system O M . We use the following fact:

Z

π

O M = Z

proj

Z

i

O M = Z

proj

N . where N = H _{[i(M )]} ^m+1 (O Z ).

We set, for instance on η ₁ 6= 0, η ₂ 6= 0, . . ., η _m 6= 0 g m = −p m δ(y − x ^m p ^m+1 _m )δ(z − xp m )δ(p 1 − x ^m−1 p ^m _m ) ·

δ(p 2 − x ^m−2 p ^m−1 _m ) · · · δ(p m−2 − x ² p ³ _m )δ ⁰ (p m−1 − xp ² _m )dp 1 ∧ dp 2 ∧ · · · ∧ dp m . It is easy to verify that the differential form g m is globally well-defined on Z as a relative differential form supported on i(M ):

g _m ∈ Γ(Z, N ⊗ Ω _P

×...×P

),

and that g m is not exact, but the differential forms xg m , yg m and zg m are relatively exact. In fact, if we set

f = δ(y − x ^m p ^m+1 _m )δ(z − xp _m )δ(p ₁ − x ^m−1 p ^m _m )δ(p ₂ − x ^m−2 p ^m−1 _m ) ·

· · · δ(p _m−2 − x ² p ³ _m )δ(p _m−1 − xp ² _m )dp ₁ ∧ dp ₂ ∧ · · · ∧ dp _m−2 ∧ dp _m ,

then the differential forms f , p m−1 f and p ² _m−1 f are globally well-defined. Furthermore we have

d(zf ) = xg _m , d(p ² _m−1 z ^m−2 f ) = yg _m and d(p _m−1 f ) = zg _m ,

where d is the relative exterior differentiation. These equalities hold globally. This implies that R

proj g _m is equal to a constant multiple of the delta-function on X supported at the origin (0, 0, 0). In particular, we have

Z

proj

g _m ∈ H ³ _[0,0,0] (O _X ).

Similarly, on η 1 6= 0, η 2 6= 0, . . ., η m 6= 0, we set

g _k = −p _k δ(y − x ^m p ^m+1 _m )δ(z − xp _m )δ(p ₁ − x ^m−1 p ^m _m )δ(p ₂ − x ^m−2 p ^m−1 _m ) ·

· · · δ ⁰ (p _k−1 − x ^m−k+1 p ^m−k+2 _m ) · · · δ(p _m−1 − xp ² _m )dp ₁ ∧ dp 2 ∧ · · · ∧ dp m . for k = 2, . . . , m and

g ₁ = [(m + 1)p ₁ δ ⁰ (y − x ^m p ^m+1 _m )δ(z − xp _m )δ(p ₁ − x ^m−1 p ^m _m ) · · · δ(p _m−1 − xp ² _m ) +δ(y − x ^m p ^m+1 _m )δ ⁰ (z − xp m )δ(p 1 − x ^m−1 p ^m _m ) · · · δ(p m−1 − xp ² _m )

+mp 2 δ(y − x ^m p ^m+1 _m )δ(z − xp m )δ ⁰ (p 1 − x ^m−1 p ^m _m ) · · · δ(p m−1 − xp ² _m ) +(m − 1)p 3 δ(y − x ^m p ^m+1 _m )δ(z − xp m )δ(p 1 − x ^m−1 p ^m _m ) ·

δ ⁰ (p 2 − x ^m−2 p ^m−1 _m ) · · · δ(p m−1 − xp ² _m ) + . . . .

+2p m δ(y − x ^m p ^m+1 _m )δ(z − xp m )δ(p 1 − x ^m−1 p ^m _m ) · · · δ ⁰ (p m−1 − xp ² _m )] · dp 1 ∧ dp 2 ∧ · · · ∧ dp m .

The differential forms g 1 , . . . , g m are globally well-defined on Z as relative differential form supported on i(M ) and the integrals along the fibers of these differential forms are equal to the Dirac delta function up to non-zero constant factors. We can summarize the results of our calculation in the following form:

Theorem. The integrals along the fibers of the map proj : X × P ¹ × . . . × P ¹ → X of the relative differential forms g ₁ , g ₂ , . . . , g _m are equal to the delta-function supported at the origin (0, 0, 0) up to non-zero constant:

Z

proj

g _k = const · δ(x)δ(y)δ(z) k = 1, . . . , m.

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