Holonomic Systems for Period Mappings

Period mappings were introduced in the sixties [G] to study variation of complex structures of families of algebraic varieties. The theory of tautological systems was introduced recently [LSY,LY] to understand period integrals of algebraic manifolds. In this paper, we give an explicit construction of a tautological system for each component of a period mapping.

1.1. Tautological systems. We will follow notations in [HLZ]. Let G be a connected algebraic group over C. Let X be a complex projective G-variety and let L be a very ample G-bundle over X which gives rise to a G-equivariant embedding X → P(V ), where V = Γ(X, L) ∨ . Let n = dim V . We assume that the action of G on X is locally effective, i.e. ker (G → Aut (X)) is finite. LetĜ := G × C × , whose Lie algebra isĝ = g ⊕ Ce, where e acts on V by identity. We denote by Z :Ĝ → GL(V ) the corresponding group representation, and by Z :ĝ → End (V ) the corresponding Lie algebra representation. Note that under our assumption, Z :ĝ → End (V ) is injective.
Letι :X ⊂ V be the cone of X, defined by the ideal I(X). Let β :ĝ → C be a Lie algebra homomorphism. Then a tautological system as defined in [LSY] [LY] is the cyclic D-module on V ∨ τ (G, X, L, β) where J(X) = { P | P ∈ I(X)} is the ideal of the commutative subalgebra C[∂] ⊂ D V ∨ obtained by the Fourier transform of I(X). Here P denotes the Fourier transform of P .
Given a basis {a 1 , . . . , a n } of V , we have Z(x) = ij x ij a i ∂ ∂a j , where (x ij ) is the matrix representing x in the basis. Since the a i are also linear coordinates on V ∨ , we can view Z(x) ∈ Der C[V ∨ ] ⊂ D V ∨ . In particular, the identity operator Z(e) ∈ End V becomes the Euler vector field on V ∨ .
Let X be a d-dimensional compact complex manifold such that its anticanonical line bundle L := ω −1 X is very ample. We shall regard the basis elements a i of V = Γ(X, ω −1 X ) ∨ as linear coordinates on V ∨ . Let B := Γ(X, ω −1 X ) sm ⊂ V ∨ , which is Zariski open. Let π : Y → B be the family of smooth CY hyperplane sections Y a ⊂ X, and let H top be the Hodge bundle over B whose fiber at a ∈ B is the line Γ(Y a , ω Ya ) ⊂ H d−1 (Y a ). In [LY] the period integrals of this family are constructed by giving a canonical trivialization of H top . Let Π be the period sheaf of this family, i.e. the locally constant sheaf generated by the period integrals.
Let G be a connected algebraic group acting on X.
In [LSY] and [LY], it is shown that if G acts on X by finitely many orbits, then τ is regular holonomic.
1.2. Period mapping. Now we consider the period mapping of the family Y: and Γ is the monodromy group acting on Gr(b p,k , H k (Y a 0 , C)).
Consider the local system R k π * C, its stalk at a ∈ B is H k (Y a , C). The Gauss-Manin connection on the vector bundle H k = R k π * C ⊗ O B has the property that Throughout this paper we shall consider the case k = d − 1. H top is the bundle on B whose fiber is Thus H top = F d−1 H d−1 . Theorem 1.1 tells us that its integral over a (d − 1)cycle on each fiber Y a 0 is governed by the tautological system τ .
We shall describe the Gauss-Manin connection explicitly below. Let (K −1 X ) × denote the complement of the zero section in the total space . Then there is a natural one-to-one correspondence between sections of ω −1 X and C × -equivariant morphisms f : We shall write f a the function that represents the section a. Since (K −1 X ) × is a CY bundle over X by [LY], it admits a global non-vanishing top formΩ. Let x 0 be the vector field generated by 1 ∈ C = Lie(C × ). Then Ω : Letπ ∨ : U → B be the restriction of π ∨ to B. Then there is a vector bundlê Then Ω f is a global section of F dĤd . And the Gauss-Manin connection on Ω f is where ∂ a i := ∂ ∂a i , i = 1, . . . , n. Consider the residue map Res : In [HLZ, Coro. 2.2 and Lemma 2.6], it is shown that Theorem 1.2. If β(g) = 0 and β(e) = 1, there is a canonical surjective map We now want to give an explicit description of each step of the Hodge filtration.
Let X be a projective variety of dimension d and Y a smooth hypersurface. We make the following hypothesis: Griffiths). Let X be a projective variety and Y a smooth hypersurface. Assume (*) holds. Then for every integer p between 1 and d, the image of the natural map which to a section α (viewed as a meromorphic form on X of degree d, holomorphic on X − Y and having a pole of order less than or equal to p along Y ) associates its de Rham cohomology class, is equal to Corollary 1.4. Assume (*) holds for smooth CY hypersurfaces Y a ⊂ X. Then the de Rham classes of Proof. By our assumption X is projective and O X (Y a ) = ω −1 X is very ample. By Theorem 1.3 it is sufficient to show that Since Y a are CY hypersurfaces, there is an isomorphism Let M(pY a ) be the sheaf of meromorphic functions with a pole along Y a of order less than or equal to p. Then there is an isomorphism

Thus we have an isomorphism
Since . Therefore the de Rham classes of Corollary 1.6. Assume (*) holds for smooth CY hypersurfaces Y a ⊂ X. Then the de Rham classes of is mapped surjectively onto the vanishing cohomology of H d−1 (Y a , C) under the residue map. Since the residue map preserves the Hodge filtration, by Corollary 1.4 the result follows.
The goal of this paper is to construct a regular holonomic differential system that governs the p-th derivative of period integrals for each p ∈ Z. By the preceding corollary, this provides a differential system for "each step" of the period mapping of the family Y.

Scalar system for first derivative
We shall use τ to denote interchangeably both the D-module and its left defining ideal. Let P (ζ) ∈ I(X) where ζ ∈ V , then its Fourier transform P = P (∂ a ), a ∈ V ∨ . Then the tautological system τ = τ (G, X, ω −1 X , β 0 ) for Π γ (a) becomes the following system of differential equations: Then the period integral becomes Π γ (a) = γ ω a . Then by Theorem 1.1, Π γ (a) are solutions of τ . By [BHLSY] and [HLZ], if X is a projective homogeneous space, then τ is complete, meaning that the solution sheaf agrees with the period sheaf.
The goal of this section is to write a system of scalar valued partial differential equations whose solution contains all the information of first order partial derivatives of period integrals.
2.1. Vector valued system. Taking derivatives of equations in τ gives us a vector valued system of differential equations that involve all first order derivatives of period integrals.
2.2. Scalar valued system. Now let b 1 , . . . , b n be another copy of the basis of V ∨ . We now construct, by an elementary way, a system of differential equations over V ∨ × V ∨ that is equivalent to (2.2), but whose solutions are function germs on V ∨ × V ∨ . Consider the system Theorem 2.1. By setting φ(a, b) = k b k φ k (a) and φ k (a) = ∂ ∂b k φ(a, b), the systems (2.2) and (2.3) are equivalent.
Proof. First we show that if (φ 1 (a), . . . , φ n (a)) is a solution to system (2.2), let then φ(a, b) is a solution to system (2.3). Since thus equation (2.3a) holds. Equation (2.3b) can be shown as follows: Equation (2.2c) shows that φ k (a) is homogeneous of degree −2 in a, it implies that φ(a, b) is also homogeneous of degree −2 in a, which implies equation (2.3c). Since which is equation (2.3d). Since φ(a, b) is linear in b, equation (2.3e) holds. φ(a, b) is homogeneous of degree 1 in b, it implies equation (2.3f).
Next we show that if φ(a, b) is a solution to system (2.3), set φ k (a) = ∂ ∂b k φ(a, b), then (φ 1 (a), . . . , φ n (a)) is a solution to system (2.2). Equation (2.3e) tells us that φ(a, b) is linear in b, i.e. there exists functions h k (a) and g(a) on V ∨ such that Equation (2.3f) shows that φ(a, b) is homogeneous of degree 1 in b, which implies that g = 0 and

Equation (2.3a) shows that
Since the b i 's are linearly independent, this further shows that P (∂ a )φ k (a) = 0 ∀k, which coincides with equation (2.2a). From equation (2.3b) we can see that Since the b i 's are linearly independent, we have which is equation (2.2b). Equation (2.3c) shows that k b k φ k (a) is homogeneous of degree −2 in a, which implies that φ k (a) is homogeneous of degree −2 in a as well, which would imply equation (2.2c).
It's also clear that equation (2.3d) implies (2.2d). Therefore the two systems are equivalent in the above sense.
Therefore k b k ∂ ∂a k Π γ (a) are solutions to system (2.3).
2.3. Regular holonomicity of the new system. In this section we will show that system (2.3) is regular holonomic, by extending the proof for the original tautological system in paper [LSY].
Let M := D V ∨ ×V ∨ /J where J is the left ideal generated by the operators in system (2.3).
Theorem 2.2. Assume that the G-variety X has only a finite number of Gorbits. Then the D-module M is regular holonomic.
Proof. Consider the Fourier transform: Consider the G × C × × C × -action on V × V where G acts diagonally and each C × acts on V by scaling. Consider the ideal I generated by (2.4a), (2.4b) and (2.4c). Operator (2.4c) tells us that ξ i = 0 for all i, thus The ideal I also contains (2.4a), thus (V × V )/I =X × {0}, which is an algebraic variety. Operator (2.4d) comes from the G-action on V ×V , operators (2.4e) and (2.4f) come from each copy of C × -action on V . Now from our assumption the G-action on X has only a finite number of orbits, thus when lifting toX × {0} ⊂ V × V there are also finitely many G × C × × C × -orbits. Therefore M is a twisted G × C × × C × -equivariant coherent D V ∨ ×V ∨ -module in the sense of [Ho] whose support Supp M =X ×{0} consists of finitely many G × C × × C × -orbits. Thus the M is regular holonomic [Bo]. The D-module M = D V ∨ ×V ∨ /J is homogeneous since the ideal J is generated by homogeneous elements under the graduation deg ∂ ∂a i = deg ∂ ∂b i = −1 and deg a i = deg b i = 1. Thus M is regular holonomic since its Fourier transform M is regular holonomic [Br].

Scalar systems for higher derivatives
Now we take derivative of system (2.2) and get a new scalar valued system whose solution consists of n 2 functions φ lk := ∂ ∂a l ∂ ∂a k φ(a), 1 ≤ l, k ≤ n. (3.1a) And considering φ(a, b) := l,k b l b k φ l,k (a), we get a new system: Similar to the previous case, we have: Proposition 3.1. By setting φ(a, b) = l,k b l b k φ l,k (a) and φ lk (a) = ∂ ∂b l ∂ ∂b k φ(a, b), the systems (3.1) and (3.2) are equivalent.
Proof. Here we check for (3.2b) and the rest is clear. Let φ(a, b) = l,k b l b k φ l,k (a), then the left-hand side of (3.2b) becomes Therefore the left-hand side of (3.2b) equals 0. The reverse direction is also clear.
Proposition 3.2. Assume that the G-variety X has only a finite number of G-orbits, then system (3.2) is regular holonomic.
The proof of Theorem 2.2 follows here.
In general, for p-th derivatives of φ(a) satisfying τ , we can construct a new system as follows: where 1 ≤ k i ≤ n and {l 1 , . . . , l p } is any permutation of {1, . . . , p}.
The relationship between φ(a, b) and the derivatives of φ(a) is: From our previous argument it is clear that if G acts on X with finitely many orbits, this system is regular holonomic.

Differential relations
Theorem 4.1. [BHLSY], [HLZ] There are isomorphisms between following Dmodules: These D-module isomorphisms allow us to extract some explicit information regarding vanishing of periods and their derivatives as follows.
The isomorphisms in the Theorem 4.1 induce, for each f a ∈ B, the isomorphism [BHLSY,Theorem 2.9]  Then it is a closed subset of B. Moreover, for any f a ∈ N (p) we have a differential relation: If X = P d , then ω −1 X = O(d + 1) and V = Γ(X, O(d + 1)) ∨ . We can identify R with the subring of C[x] := C[x 0 , . . . , x d ] consisting of polynomials spanned by monomials of degree divisible by d + 1. By Lemma 2.12 in [BHLSY], meaning that elements of the form ∂ ∂x i x j p(x)e f are in Z * (ĝ)(Re f ) for p(x) ∈ R. Now we specialize to the Fermat case f F = x d+1 0 + · · · + x d+1 d and look at two examples.
it implies that f F ∈ N (x 0 x 1 )and by Corollary 4.2 we have a differential relation ∂ We know that when |a 0 | >> 0, Γ ) k , so the above equality means that any analytic continuation of this power series function must have a vanishing derivative with respect to a 0 at the Fermat point, which we can verify easily since the analytic continuation is given explicitly by (a 2 0 − 4a 1 a 2 ) − 1 2 .

Concluding remarks
We conclude this paper with some remarks about differential zeros of period integrals of differential systems for period mappings.
For a given a ∈ B, what can we say about the the function space {p(ζ) ∈ R | p(ζ)e fa ∈ Z * (ĝ)(Re fa )}? This space is unfortunately neither an ideal nor aĝ-submodule of R in general, unless f a isĝ-invariant in which case it is equal to Z * (ĝ)R. However, it seems that it is more interesting to consider the closed subset N (p) ⊂ B, for each function p(ζ) ∈ R that we defined above. For this is an algebraic set that gives us the vanishing locus of certain derivatives (corresponding to p) of all period integrals. For example, it is well known that period integrals of certain CY hypersurfaces can be represented by (generalized) hypergeometric functions. In this case, the vanishing locus above therefore translates into a monodromy invariant statement about differential zeros of the hypergeometric functions in question.
A remark about our new differential systems for period mappings is in order. For the family of CYs Y, since the period mapping is given by higher derivatives of the periods of (d − 1, 0) forms, any information about the period mapping can in principle be derived from period integrals, albeit somewhat indirectly. However, the point here is that an explicit regular holonomic system for the full period mapping would give us a way to study the structure of this mapping by D-module techniques directly. Thanks to the Riemann-Hilbert correspondence, these techniqus have proven to be a very fruitful approach to geometric questions about the family Y (e.g. degenerations, monodromy, differential zeros, etc) [BHLSY, HLZ] when applied to τ . The new differential system that we have constructed for the period mapping is in fact nothing but a tautological system. Namely, it is a regular holonomic D-module defined by a polynomial ideal together with a set of first order symmetry operatorsconceptually of the same type as τ . It is therefore directly amenable to the same tools (Fourier transforms, Riemann-Hilbert, Lie algebra homology, etc) we applied to investigate τ itself. The hope is that understanding the structure of the new D-module will shed new light on Hodge-theoretic questions about the family Y. For example, there is an analogue Theorem 4.1 which allows us to construct differential zeros of solution sheaves for this class of D-modules. It would be interesting to understand their implications about period mappings. We would like to return to these questions in a future paper.