Relation with iterated residues

Note thatZ and X_L have the same dimension.

Proposition 3.1.16. If the restriction of π_L to Z is not surjective, then the set Fl_L(Z) is empty.

Proof. If not, then dimπ_L(Z)< d. LetL={k₁, . . . , k_d}. LetZ_• ∈Fl_L(Z). Consider the corresponding flag of irreducible subvarieties ofX_L:

π_L(Z) = π_L(Z₀)⊃π_L(Z₁)⊃ · · · ⊃π_L(Z_d).

Because of the dimension reasoning there must be an indexiwithπ_L(Z_i) =π_L(Z_i−1).

This implies

π_ki(Z_i−1) = π_ki(Z_i) ⊂ S_ki.

ThereforeZ_i =Z_i−1, so the flag is not strict, which is a contradiction.

We may therefore suppose without loss of generality that π_L : Z → X_L is surjective. Hence the set of points on Z where this map is not ´etale is a proper closed subvariety. The irreducible components of this subvariety have dimension d−1 or less and are special. Thereforeπ_L⁻¹R_L∩Z belongs to its complement. This means the following is true.

Proposition 3.1.17. The projection

π_L:π_L⁻¹R_L∩Z→R_L is an unramified covering.

Let p = π_LZ_d ∈ S_L. Since for a flag Z_• ∈ Fl_L(Z) the set R_L,Z• is open and closed inπ⁻¹_L R_L∩Z we get

Proposition 3.1.18. The projection

π_L:R_L,Z• →R^p_L is an unramified covering.

Remark 3.1.5. By the proposition we see that sinceR^p_Lis a product of annuli,R_L,Z• is a disjoint union of products of annuli.

We are going to determine the cycle h_a(Z,U⁰)∩R_L,Z• ∈ H_d(R_L,Z•). The d-th homology group of a product of dannuli isZ. Hence there is a canonical generator h_cofH_d(R^p_L). In fact h_c can be defined as the productc₁× · · · ×c_d, wherec_i is the circle in R^p_k^ki_i going around p_ki counterclockwise.

Proposition 3.1.19. The cycle h_a(Z,U⁰)∩R_L,Z• is the pullback of(−1)^d(d−1)2 h_cvia the projection π_L:R_L,Z• →R^p_L.

Proof.

Remark 3.1.6. It is clear that we can decrease numbersε_k and increase numbersε⁰_k. The sequence obtained in this way will be also good. Moreover the open subsets of the new hypercover are contained in the corresponding open subsets of the old one. Therefore the residues computed with respect to these hypercovers are equal.

Therefore one can assume that the projection π_L : R_L,Z• → R^p_L extends to an unramified covering for the closures ofR_L,Z• in Z,R^p_L inX_L.

Consider the commutative diagram:

H_2d(X) −−−−→ H_2d(R_L,Z•, ∂R_L,Z•) ←−−−−^πL∗ H_2d(R_L^p, ∂R^p_L)

ha



y ^ha



y ^ha

 y

H_d(U_a⁰) −−−−→ H_d(R_L,Z•) ^π

∗

←−−−−L H_d(R^p_L)

We see that it is enough to prove that the image of the fundamental class of H_2d(R^p_L, ∂R^p_L) by the map h_a in H_d(R_L^p) is (−1)^d(d−1)2 h_c. There is a direct product decomposition R^p_L = ×_k∈LR^p_k^k. The hypercover on R^p_L is the product hypercover.

For k ∈ L let φ_k be the fundamental class of H₂(R^p_L, ∂R^p_L). Let us lift it to a hyperchainφf_k.

The hypercover of R_L^p is the ˇCech hypercover associated to the cover with two open sets:

V₀ = n(p_k, ε_k)\n(p_k, ε_k) corresponding to the cella(η_k), V₁ = n(p_k, ε_k)\n(p_k, ε_k) corresponding to the cella(p_k), V₀∩V₁ = n(p_k, ε_k)\n(p_k, ε_k) corresponding to the cella(η_k, p_k).

Letε⁰ < r < ε. Consider topological chains

c^k₀ = n(p_k, ε_k)\n(p_k, r)∈C₂(V₀), c^k₁ = n(p_k, r)\n(p_k, ε⁰_k)∈C₂(V₁), c^k₀₁ = ∂n(p_k, r)∈C₁(V₀∩V₁).

They define a hyperchain c^k. We have c₀ +c₁ = φ_k. The hyperchain is closed, thereforeh₀₁(φ_k) =c^k₀₁, which is the canonical generator ofH₁(R^p_k^k).

Since the product of φ_k is the fundamental class of H_2d(R_L^p, ∂R^p_L), the prod-uct of the hyperchains c = ×_k∈Lc^k lifts the fundamental class. The term of c at

×_k∈L(η_k, p_k) is, by the definition of the product for hyperchains, (−1)^d(d−1)2 ×_k∈Lc^k₀₁. This is exactly (−1)^d(d−1)2 h_c.

Let Z_• be a strict L, p-special flag, L = {k₁, . . . , k_d}, p = (p_k1, . . . , p_kd). Let Z⁰ =Z₁,k=k₁,L⁰={k₂, . . . , k_d}. LetZ_•⁰ be the flagZ₁ ⊃ · · · ⊃Z_d,Z_•⁰ ∈Fl_L⁰(Z⁰).

Lett_i be a local parameter onX_ki at the pointp_ki =π_kiZ_d. Let p⁰ =π_L⁰p. For Z⁰ we have the projection

π_L⁰ :R_L⁰_,Z•⁰ →R^p_L⁰0,

which is also an unramified covering. Let

Ze = Z⁰×_XL0 Z = (X_k1×Z⁰)×_XLZ.

We obtain the canonical diagrams

Ze −−−−→^ρ Z Ze −−−−→^ρ Z

τ

 y



y eτ

 y

 y Z⁰ −−−−→ X_L⁰ (X_k1×Z⁰) −−−−→ X_L

The diagonal embedding Z⁰ →Z⁰×_XL0 Z⁰ induces a morphism ∆ :Z⁰ →Z, whiche is a section to the natural projectionτ :Ze→Z⁰. Let

R_Z = Z ∩ π⁻¹_L (U_k^pk×R^p_L⁰0) ∩ n(Z_•, ~δ).

Consider the fiber products overU_k^pk×R^p_L⁰0:

Re −−−−→^ρ R_Z

e τ



y ^πL

 y U_k^pk ×R_L⁰_,Z•⁰ −−−−→ U_k^pk×R^p_L⁰0

Again, we have a section ∆ : R_L0,Z•⁰ → R. Lete Re⁰ be the union of connected components of Re which intersect the image of ∆. Let ρ⁰ be the restriction of ρ to Re⁰.

Remark 3.1.7. The setR_Z is open and closed inZ∩π_L⁻¹(U_k^pk×R^p_L⁰0) because Propo-sition 3.1.11 and the first sentence of Corollary 3.1.12 work if we replace R_L by U_k^pk ×R_L^p00.

Remark 3.1.8. All special subsets of Z which intersect R_Z are contained in Z⁰. Therefore the mapπ_Lon the diagram is an unramified covering outside {p_k} ×R^p_L⁰0

and π_k⁻¹p_k∩R_Z =R_L⁰_,Z⁰_•.

Proposition 3.1.20. The mapρ⁰ is an analytic isomorphism.

Proof. Since ρ⁰ is a base change of the unramified covering π_L⁰ :R_L⁰_,Z•⁰ → R^p_L⁰0, it is an unramified covering itself. Therefore it is enough to construct a continuous section s⁰ : R_Z → Re⁰ to ρ⁰ which extends the diagonal map. This is equivalent to constructing a retractions:R_Z →R_L⁰_,Z⁰_• which respects the projection π_L⁰.

Take a compact connected setA⊂R^p_L⁰0 such thatπ⁻¹_L0 (A)∩R_L0,Z•⁰ =A₁∪· · ·∪A_m is a disjoint union of spaces isomorphic to A.

One can choose disjoint open subsetsV₁, . . . , V_m inπ_L⁻¹0 (A)∩R_Z such thatA_i ⊂ V_i. The setC =π_L⁻¹0 (A)∩R_Z\(V₁∪ · · · ∪V_m) is closed. Thereforeπ_L(C) is closed.

Take α >0 such that n(p_k, α)×A does not intersect π_L(C). This means that the open setπ_L⁻¹(n(p_k, α)×A)∩R_Z is contained in the union of the setsV_i.

LetV_i⁰=π_L⁻¹(n(p_k, α)×A)∩V_i. Let us prove that V_i⁰ is connected for eachi. If not, then V_i⁰ =B₁tB₂ withB_j open, closed and nonempty. Since A_i is connected,

for somej B_j does not intersectA_i. Thereforeπ_LB_j is open, closed and nonempty.

Hence it must be the wholeU_k^pk×A. This contradicts the assumption thatB_j does not intersectA_i.

One can construct a deformation retract retractingπ_L⁻¹0 (A)∩R_Zinsideπ_L⁻¹(n(p_k, α)×

A)∩R_Z. Therefore there are exactlymconnected components ofπ_L⁻¹0 (A)∩R_Z, each containing exactly one A_i. Therefore there is a unique map s_A : π_L⁻¹0 (A)∩R_Z → π⁻¹_L0(A)∩R_L⁰_,Z•⁰ which is identity on π⁻¹_L0(A)∩R_L⁰_,Z•⁰ and makes the diagram below commutative.

π_L⁻¹0 (A)∩R_Z −−−−→ U_k^pk×A

sA

 y

 y π_L⁻¹0 (A)∩R_L0,Z•⁰ −−−−→ A

Patching these s_A together gives s as required proving the first statement of the proposition.

Take a meromorphic form ω on Z which is holomorphic outside the special subvarieties ofZ. The formω can be written as

ω = f dt₁∧dt₂∧ · · · ∧dt_n,

where f is a rational function on Z. Let K be the field of fractions of Z⁰. Then Z×SpecK is a curve and ∆(SpecK) is a point. Therefore the algebraic residue res_∆(SpecK)ρ^∗f dt₁ is defined. We have

Proposition 3.1.21. Put

ω⁰ = (res_∆(SpecK)ρ^∗f dt₁)dt₂∧ · · · ∧dt_n. Then

res^int_L,Z•ω = (−1)^d−12πi res^int_L0,Z•⁰ ω⁰. Proof. By the definition

res_U,L,Z•ω = Z

ha∩RL,Z•

ω.

Sinceρ⁰ is an isomorphism, Z

ha∩RL,Z•

ω = Z

ρ^0∗(ha∩RL,Z•)

ρ^∗ω.

We have

ρ^0∗(h_a∩R_L,Z•) = (−1)^d(d−1)2 Re⁰ ∩ ρ^∗π^∗_Lh_c

= (−1)^d(d−1)2 Re⁰ ∩ τe^∗(h_ck1×(R_L⁰_,Z•⁰ ∩π_L^∗0h⁰_c)),

whereh_ck1 is the circle inR^p_k^k₁¹ and h⁰_c is the product of circles inR^p_L⁰0. By Fubini’s theorem

Z

Re⁰∩fτ^∗(hck1×(R_L0,Z0

•∩π^∗_L0h⁰c))

ρ^∗ω = Z

R_L0,Z0

•∩π^∗_L0h⁰c

g(z⁰)dt₂∧ · · · ∧dt_d,

where forz⁰ ∈R_L⁰_,Z•⁰

g(z⁰) = Z

Re⁰∩τ⁻¹z⁰∩π⁻¹_k1hck1

ρ^∗f dt₁. The last integral is nothing else than

2πi res_∆(z⁰₎ρ^∗f dt₁. Therefore

g(z⁰)dt₂∧ · · · ∧dt_d = 2πiω⁰ Taking into account that

R_L0,Z•⁰ ∩π^∗_L0h⁰_c = (−1)^{(d−1)(d−2)2} R_L0,Z•⁰ ∩h_a⁰_,Z⁰,

wherea⁰ is the cell obtained fromaby replacing the componenta(η_k1, p_k1) with the component a(p_k1), we obtain the statement.

Let us denote by res_L,Z•ω the iterated algebraic residue ofω with respect to the flag Z_•. This is defined by induction on the dimension of Z by the formula

res_L,Z•f dt₁∧ · · · ∧dt_d = res_L⁰_,Z⁰_•(res_∆(SpecK)ρ^∗f dt₁)dt₂∧ · · · ∧dt_d. Thus we obtain a formula for our residue.

Corollary 3.1.22.

res^int_L,Z•ω = (−1)^d(d−1)2 (2πi)^dres_L,Z•ω.

For any subvariety Z ⊂ X and any hyperform using the decomposition of the residue according to flags we also can state a formula for the trace:

Corollary 3.1.23. For a meromorphic hyperformω of degree 2donZ⊂X₁× · · · × X_n which is holomorphic outside the special subvarieties of Z, dimZ =d,

Tr^int_Z ω= (−1)^d(d−1)2 (2πi)^d X

L⊂{1,...,n},|L|=dimZ

X

Z•∈FlL(Z)

res_L,Z•ω_aL(Z•).

Dropping the coefficient (−1)^d(d−1)2 (2πi)^dwe may also define the algebraic version of the trace:

Tr_Zω := X

L⊂{1,...,n},|L|=dimZ

X

Z•∈FlL(Z)

res_L,Z•ω_aL(Z•). Then the latter corollary can be reformulated as follows:

Corollary 3.1.24. For a meromorphic hyperformω of degree 2donZ⊂X₁× · · · × X_n which is holomorphic outside the special subvarieties of Z, dimZ =d,

Tr^int_Z ω = (−1)^d(d−1)2 (2πi)^dTr_Zω.

Now we make a summary of the computation of Tr_Zω. The necessary notation is explained in Section 3.1.10.

(i) List all the subsetsL⊂ {1, . . . , n}of size dimZ.

(ii) For each Llist all the flags Z_• ∈Fl_L(Z).

(iii) For each flag find the cell a_L(Z_•) of ∆_m1 × · · · ×∆_mn, which is a product of several vertices and edges (vertices for k∈L^c and edges fork∈L).

(iv) Compute the iterated residue corresponding to the flag Z_• of the form which is the component of ω on the open setU_aL(Z•) =×_kU_k, where U_k is an open set from the cover of X_k fork∈L^c, or an intersection of two open sets of the cover ofX_k fork∈L.

(v) Add these residues.

Im Dokument Higher Green’s functions for modular forms (Seite 69-74)