INTEGRATION OF D-MODULES, BERNSTEIN-SATO POLYNOMIALS AND MILNOR FIBERS

EACA, VALLADOLID, SEPTEMBER 2002 — ULI WALTHER

1. Motivation. Consider the function

f(x, t) = \frac{1}{x^{2} − t} (1)

in two variables. The two linear differential operators

(2)  P_{1} = 2x\frac{d}{dt} + \frac{d}{dx} and P_{2} = x\frac{d}{dx} + 2t\frac{d}{dt} + 2

annihilate f(x, t). In fact, every element in the ℂ-algebra D_{x,t} (generated by the partial derivation operators \frac{d}{dx}, \frac{d}{dt} and the multiplication operators 1, x and t) that annihilates f(x, t) is a left linear combination of these two. (The ring D_{x,t} is also called the Weyl algebra in x, t.) The annihilator is hence a left ideal in D_{x,t}; we denote it by I.

Let γ be a loop within ℂ^{2} that does not meet the zero locus of x^{2} − t. It is of interest to ask what operators will annihilate

F (t) = ∮_{γ} f(x, t) dx. (3)

One such type of operators are of course left multiples of \frac{d}{dx}. Since no polynomial in x will annihilate F (t), the question reduces to finding operators in D_{t}, the ring of ℂ-linear differential operators on ℂ[t], that annihilate F (t). The annihilator is then a left ideal in D_{t}, which we call J. To simplify notation, we will use ∂_{x} and ∂_{t} instead of \frac{d}{dx} and \frac{d}{dt}.

Let us assume that P = P_{0} + ∂_{x}P_{x} ∈ D_{t} where P_{0} ∈ I and P_{x} ∈ D_{x,t}. Then

P • ∮_{γ} f dx = ∮_{γ} P • f dx
= ∮_{γ}(∂_{x}P_{x}) • f dx
= ∮_{γ} ∂_{x} • (P_{x} • (f)) dx = 0

by Stokes’ theorem. Conversely, if P • F (t) = 0 and P ∈ D_{t} then ∮_{γ} P • f dx = 0 so that P •f has an antiderivative and is hence a derivative. Thus P •f ∈ ∂_{x} ·D_{x,t} •(f) and P ∈ ∂_{x} · D_{x,t}.

The search for J = ann_{D_{t}}(F (t)) is therefore equivalent to finding

D_{t} ∩ (I + ∂_{x} · D_{x,t}).

2. Caveat. It is conceivable that F (t) is not the only function in t that satisfies all the equations in J. Consider the D_{t}-module generated by F (t):

D_{t} • F (t) = \frac{D_{t}}{D_{t} ∩ (I + ∂_{x} · D_{x,t})}.

It is a submodule of \frac{D_{x,t}}{I + ∂_{x} · D_{x,t}}. This latter D_{t}-module is called the integral

along ∂_{x} of the D_{x,t}-module generated by f(x, t), and is (over D_{t}) generated by the powers of x. It is coherent, so that a finite collection of powers suffices. The integral may be viewed as (part of) the D-module theoretic direct image of D_{x,t} • f(x, t) under the projection π_{x} : ℂ^{2} → ℂ sending (x, t) to (t) and one also writes it as

∫_{π_{x}}(D_{x,t}/I).

Under the inclusion D_{t}•F (t) ↪ D_{x,t}/(I +∂_{x}·D_{x,t}), F (t) corresponds to the coset of 1 on the right. Similarly, the integral of P • f(x, t) along a “generic γ” corresponds to the coset of P. This means that the annihilator of P + (I + ∂_{x} · D_{x,t}) consists of those operators that kill ∮_{γ} P • f dx irrespective of γ. In our example, there are two “interesting” integrals, that of f(x) and that of x · f(x). ∫_{π_{x}}(D_{x,t}/I) represents all D_{t}-linear combinations of these two integrals.

References: [2, 24]

3. The V -filtration. Let us put a(n increasing) filtration F_{∂_{x}}^{i} on D_{x,t} as follows:

F_{∂_{x}}^{i}(D_{x,t}) = ℂ − span{x^{α}∂_{x}^{β}t^{a}∂_{t}^{b} : α − β ≤ i}. (4)

This is a variation on the V -filtration introduced by Kashiwara and Malgrange for pairs of differentiable manifolds M ⊂ N, [9, 15]

Inside I one finds the following operator of filtration level zero:

−t∂_{x} · P_{1} + ∂_{x}x · P_{2} = s(s − 1) − t∂_{x}^{2}

where we write s for −∂_{x}x. Somewhat magically, this operator can be used to show that ∫_{π_{x}}(D_{x,t}/I) is as D_{t}-module generated by the cosets of 1 and x only. Let us see how this works. Suppose, for example, that P ∈ D_{x,t} equals P_{0} + xP_{1} + x^{2}P_{2} with P_{i} ∈ D_{t}. We want to show that there is an element in its coset modulo I + ∂_{x} · D_{x,t} in which x occurs only linearly.

Since

(1) · (2) · (P_{0} + xP_{1} + x^{2}P_{2}) = (s + 1)(s + 2)(P_{0} + xP_{1} + x^{2}P_{2})
= (s + 1)(s + 2)(P_{0} + xP_{1}) + P_{2}(s + 1)(s + 2)x^{2}

inside D_{x,t}/(I + ∂_{x} · D_{x,t}), we only need to show that P_{2}(s + 1)(s + 2)x^{2} can be rewritten. But in the quotient,

P_{2}(s + 1)(s + 2)x^{2} = P_{2}x^{2}(s − 2 + 1)(s − 2 + 2)
= P_{2}x^{2} · (t∂_{x}^{2})
= P_{2}t(x∂_{x}x∂_{x} − x∂_{x})
= tP_{2}((−1)(−1) − (−1)) = 2tP_{2}.

Similarly, for k > 1, (k)(k − 1)x^{k}P_{k} = P_{k}x^{k}(t∂_{x}^{2}) = P_{k}tx^{k−2}((x∂_{x})^{2} − x∂_{x}) = ((x∂_{x} − k + 2)^{2} − x∂_{x} + k − 2)P_{k}tx^{k−2} = k(k − 1)P_{k}tx^{k−2} so that really all higher powers of x are superfluous.

References: [9, 15, 17, 24]

4. b-function basics. We allow x and t to stand for several variables. Unless expressly indicated otherwise, all modules are left modules. If one filters D_{x,t} by assigning x, t, ∂_{x} and ∂_{t} degree 1, the associated graded ring is a polynomial ring of dimension 2n where n is the total number of variables. A holonomic module over D_{x,t} is one whose associated graded module is of dimension n. There is a general theorem to the effect that if M = D_{x,t}/I is any holonomic D_{x,t}-module then there is a b-function for integration along ∂_{x} which can be used to determine the powers of x that are required to generate (D_{x,t}/∂_{x} · D_{x,t}) ⊗_{D_{x,t}} M over D_{t}. This b-function b(s) is defined via the requirement that it be the polynomial of smallest degree such that

b(−∂_{x}x) ∈ I + F_{∂_{x}}^{−1}(D_{x,t}).

In our example, b(s) = s(s − 1) as t∂_{x}^{2} ∈ F_{∂_{x}}^{−2}(D_{x,t}).

In our case, the integral is the module (D_{t})^{2}/D_{t} · ⟨(0, 2t∂_{t} + 1), (∂_{t}, 0)⟩ (the program Macaulay 2 [6] by Grayson and Stillman contains a D-module package [12] by Leykin, Stillman and Tsai that can do these and various other computations). By good fortune, the relation matrix is diagonal. The coset of (0, 1) stands for the integral of f(x, t) itself, and is annihilated by ⟨2t∂_{t} + 1⟩. On the other hand, the integral of x · f(x, t) is as one easily sees a constant, equal to 2πi times the winding number of the path around Var(x^{2} − t). Hence it is annihilated by ⟨∂_{t}⟩ and that is exactly what the coset of (1, 0) suggests. The ideal J annihilating F (t) is easy to find here; in general one needs to compute the intersection of the module of relations of the integral with the last component.

Even if x and t have several components, b-functions still exist and are defined via the appropriate generalization of the V -filtration. In the multivariate case, −∂_{x}x means − ∑_{i=1}^{n} ∂_{x_{i}}x_{i}. The largest integral root of this b-function bounds the largest power of x required to generate the integral over D_{t}. (These integrals are now to be pictured along homology cycles of (potentially) higher dimension.) This was proved by T. Oaku and has since been used to design various algorithms that manipulate holonomic modules. These include the computation of integrals along projections with higher dimensional fibers, the restriction of a system or module to a linear subspace, the computation of holonomic extensions of a module from an open set in ℂ^{n} to the entire space (and more generally the corresponding local cohomology groups with coefficients in that module), Hom- and Ext-functors between D-modules, and others. We would like to point out that the computation of de Rham cohomology groups with coefficients in a holonomic module is a special case of integration (the map being projection to a point) and its computation a highlight of the various applications of Oaku’s work.

References: [17, 19, 18, 20, 21, 24, 25, 29, 28].

5. Bernstein-Sato polynomials. We now switch gears to consider another Dmodule theoretic quantity, the Bernstein-Sato polynomial b_{f}(s) associated to a hypersurface Var(f) ∈ ℂ^{n} defined by the polynomial f. This polynomial is defined by the functional equation

P (s) • f^{s+1} = b(s) · f^{s}. (5)

Here s is a new variable, P (s) ∈ D_{x}[s], D_{x} is the Weyl algebra in the variables x_{1}, …, x_{n}, and we differentiate f^{s} formally by the power rule (or consider just a germ of the complex multi-valued function f^{s} and differentiate locally).

Of all polynomials b(s) above, the monic one of minimal degree is b_{f}(s). This polynomial contains numerical information about the singularity structure of Var(f). The roots of b_{f}(s) are negative, rational, and related to other invariants of the singularity:

• if f is an isolated singularity then the roots of b_{f}(s) are the logarithms of the eigenvalues of the monodromy of f;

• If f is a normal crossing divisor with k divisors meeting but no more, b_{f}(s) = (s + 1)^{k};

• in the case of curves, the roots of the Bernstein-Sato polynomial are related to numerical data of the embedded resolution of singularities;

• if f is (quasi-)homogeneous with only isolated singularities, b_{f}(s) can be determined from the Jacobian ring of f – roots are essentially the degrees of nonzero elements in that quotient;

• the roots are in nice cases (e.g., curves) related to the Newton polygon and the poles of the topological Zeta function of Denef and Loeser;

• via [b_{f}(r) = 0] → (D_{n} • f^{r}) the roots correspond to certain nontrivial local

systems on the open set ℂ^{n} \ Var(f). The precise structure of the root set remains mysterious in general.

References: [14, 13, 5, 1, 31, 11, 7, 8, 27, 26, 10]

6. b_{f}(s) d’après Malgrange. Identifying s with −∂_{t}t, D_{x}[s] is a subring of D_{x,t} = D_{x} ⊗_{ℂ} D_{t}. Then D_{x}[s] • f^{s} = D_{x}[s]/J(f^{s}) where J(f^{s}) = ann_{D_{x}[s]}(f^{s}) is a submodule of D_{x,t} • f^{s} where the actions of t and ∂_{t} are as follows:

t • (\frac{g(x, s)}{f^{k}} ⊗ f^{s}) = \frac{g(x, s + 1)f}{f^{k}} ⊗ f^{s},
∂_{t} • (\frac{g(x, s)}{f^{k}} ⊗ f^{s}) = \frac{−sg(x, s − 1)}{f^{k+1}} ⊗ f^{s}.

This extends the D_{x}[s]-structure because −∂_{t}t acts as multiplication by s.

As Malgrange pointed out, it is easy to determine the elements of D_{x,t} that annihilate f^{s}: they form the (maximal) left ideal generated by t − f and all operators {∂_{i} + ∂_{i} • (f) · ∂_{t}}_{1≤i≤n}. We write J_{n+1}(f^{s}) for this ideal. Using the module D_{x,t}/J_{n+1}(f^{s}) we are going to establish a connection between b_{f}(s) and the Milnor fiber of f.

References: [14, 15]

7. Milnor fibers. Let us recall the construction of the Milnor fiber at the point x ∈ ℂ^{n} for which f : ℂ^{n} → ℂ is interpreted as a polynomial mapping. Let ε ∈ ℝ^{+} be sufficiently small and let τ ∈ ℝ^{+} sufficiently small compared to ε. Milnor has proved that f^{−1}(t) ∩ B(x, ε) is (up to diffeomorphism) independent of τ, ε and t provided that 0 < |t − f(x)| < τ and B(x, ε) denotes the open ball around x of radius ε. This gives a locally trivial bundle over B(f(x), τ) \ {f(x)}. If f is homogeneous, ε may be taken to be infinity. The Milnor fiber is then “the same” in every point x and given by f^{−1}(t) for any t ≠ 0.

References: [16] 8. The relative de Rham complex. The mapping f can be broken into two steps, an inclusion ℂ^{n} ↪ ℂ^{n+1} as the graph of f, and the projection ℂ^{n+1} → → ℂ onto the t-coordinate. The image of ℂ^{n} in ℂ^{n+1} is the hypersurface f(x) = t. The D_{x,t}-module associated to J_{n+1}(f^{s}) is supported on this hypersurface and it turns out that it is the D-module theoretic direct image of the D_{x}-module R_{x} = D_{x}/D_{x}·⟨∂_{1}, …, ∂_{n}⟩. Let us now investigate an interpretation of the Bernstein-Sato polynomial in the homogeneous situation.

By Euler’s Lemma,

(x_{1}∂_{1} + … + x_{n}∂_{n}) • f^{s} = sdf^{s},

where d is the degree of f. We call the operator on the left the Euler operator and denote it by E. Thus E − sd annihilates f^{s}. With the help of this relation we will interpret the Bernstein-Sato polynomial in a new way.

To be a Bernstein polynomial means that b_{f}(s) ∈ J_{n+1}(f^{s}) ∩ D_{x}[s] + D_{x}[s] · f. Write b_{f}(s) = j + P (s)f with j ∈ J_{n+1}(f^{s}) ∩ D_{x}[s], P (s) ∈ D_{x}[s].

The ideal J_{n+1}(f^{s}) is homogeneous if we set deg(x_{i}) = 1, deg(∂_{i}) = −1, deg(t) = k, deg(∂_{t}) = −k. Since b_{f}(s) is homogeneous of degree 0, we may assume that P (s)· f is also homogeneous, of degree 0. Writing P (s) = ∑_{i=0}^{l} P_{i}s^{i} with P_{i} ∈ D_{x}, we see that P_{i} is of degree −k. This implies, as P_{i} ∈ D_{x}, that P_{i} ∈ F_{∂_{x}}^{−k}(D_{x,t}) ∩ D_{x}, so P (−∂_{t}t) ∈ F_{∂_{x}}^{−k}(D_{x,t}). As t − f ∈ J_{n+1}(f^{s}), ∑_{i=1}^{l} P_{i}(−∂_{t}t)^{i}f = ∑_{i=1}^{n} P_{i}(−∂_{t}t)^{i}t ∈ F_{∂_{x}}^{−k}(D_{x,t}) modulo J_{n+1}(f^{s}).

Also, b_{f}(−∂_{t}t) is modulo J_{n+1}(f^{s}) equivalent to b_{f}(E/k) because E + k∂_{t}t ∈ J_{n+1}(f^{s}). Thus

b_{f}(\frac{E}{k}) ∈ J_{n+1}(f^{s}) + F_{∂_{x}}^{−k}(D_{x,t}),

proving that b_{f}(−(s + n)/k) is a multiple of the b-function for integration along ∂_{1}, …, ∂_{n}.

The integration along ∂_{1}, …, ∂_{n} is the D-module theoretic direct image for the projection ℂ^{n+1} → ℂ. Following the module R_{x} through the map f we see that its direct image is the relative de Rham complex

Ω_{f}^{•} = (D_{x,t}/⟨∂_{1}, …, ∂_{n}⟩ · D_{x,t}) ⊗_{D_{x,t}}^{L} (D_{x,t}/J_{n+1}(f^{s})).

It has finitely generated cohomology modules (over D_{t}) and the top cohomology group is the integral ∮_{π_{x}}(H_{t−f}^{1}(R_{x,t})).

What is the fiber of this complex over the various t ∈ ℂ? If a ≠ 0, results of Deligne imply that the restriction Ω_{f}^{•} ⊗_{D_{t}}^{L} (D_{t}/D_{t} · ⟨t − a⟩) is a representative of the de Rham complex of f^{−1}(a), that is, of the Milnor fiber.

Since the degrees of integration cohomology classes are governed by the b-function for integration, and since in turn an upper bound of this b-function can be determined from b_{f}(s), the Bernstein-Sato polynomial bounds the possible degrees of the top cohomology of the Milnor fiber: a degree k can only belong to a de Rham cohomology class if b_{f}((−k − n)/d) = 0.

The nonzero cohomology elements in the restriction to t = a of the relative de Rham complex “are” differential forms on ℂ^{n}. In order to recognize them as de Rham cohomology elements on f^{−1}(a), one needs to use the relation deg(f)f = ∑_{i=1}^{n} x_{i}∂_{i} • (f). For simplicity, let a = 1 and consider the top cohomology only. If g dx is such a form, g dx = g ∑_{i=1}^{n} x_{i}∂_{i} • (f)/ deg(f) along f^{−1}(a) so that one can write g dx as (∑_{i=1}^{n}(−1)^{i}x_{i}g dx_{1} ∧ … ∧ d̂x_{i} ∧ … ∧ dx_{n}) ∧ (df). This division by df sets up the equivalence of the top cohomology of the restricted relative de Rham complex with the (n − 1)-st de Rham cohomology of the Milnor fiber.

References: [4, 30]

9. Arrangements. We now turn our attention to hyperplane arrangements. An arrangement A is the collection of irreducible components of a product of linear polynomials. We can assume that no factor has multiplicity greater than 1. We are interested in the Bernstein-Sato polynomial of A. Since b_{f}(s) is the least common multiple of all local Bernstein-Sato polynomials (where the operator P (s) of (5) is entitled to be in the local ring at x rather than a global quantity) we can restrict our attention to the homogeneous case where all factors are forms. Such arrangements are called central.

For isolated homogeneous singularities, b_{f}(s)/(s + 1) is the minimal polynomial of (−n − E)/d on the Jacobian ring. Arrangements in dimension greater than 2 are generally not isolated singularities and so the formula involving the Jacobian ring fails. Nonetheless, the isolated case inspires the following definition. Let q(s) ∈ ℂ[s]. For a fixed f ∈ R_{x} we define the ideal 𝔞_{q(s)} ⊆ R_{x} as the set of elements g ∈ R_{x}

[g ∈ 𝔞_{q(s)}] ⟺ [∃P (s) ∈ D_{x}[s] : P (s) • f^{s+1} = q(s)gf^{s}] .

We remark that 𝔞_{q(s)} ⊆ 𝔞_{q(s)q′(s)}, and if q′(s)g ∈ 𝔞_{q(s)} · R_{x}[s] then g ∈ 𝔞_{q(s)q′(s)}. The Jacobian ideal 𝔄 of R_{x} · f is contained in 𝔞_{(s+1)}, and of course f ∈ 𝔞_{1}.

The Bernstein polynomial of f is evidently the polynomial b_{f}(s) of smallest degree such that 1 ∈ 𝔞_{b_{f} (s)}.

The ideals 𝔞_{q(s)} determine certain subvarieties of Var(f) that seem to have significant geometric meaning. In the case of generic arrangements (in which every collection of n distinct participating hyperplanes meet only in the origin) the ideals 𝔞_{(s+1)^{i}} define the iterated i-th singular loci of Var(f). In particular, 𝔞_{(s+1)^{n−1}} defines the origin. Using an argument generalizing the isolated singularity case one can show that q(s) = (s + 1)^{n−1} · ∏_{i=0}^{l}(s + (i + n)/d) satisfies 𝔞_{q(s)} ∋ 1 (where l is the largest degree of a nonzero element in R_{x}/𝔞_{(s+1)^{n−1}}. Hence this q(s) must be a multiple of b_{f}(s). Using a combinatorial argument, one can show that l = 2d−n−2. This establishes an upper bound for b_{f}(s) for generic arrangements.

A lower bound is derived from knowledge about the cohomology generators of the corresponding Milnor fiber. Combining work of Orlik and Randell with the upper bound for the Bernstein-Sato polynomial one can see that the degrees of nonvanishing classes in H_{dR}^{n−1}(f^{−1}(1); ℂ) are the elements of the set {0, …, 2d−n−2}.

If the arrangement is not generic, b_{f}(s) has roots that do not come from cohomology of the Milnor fiber with ℂ-coefficients. Consider the non-generic arrangement given by f = xyz(x + y)(x + z). Its Bernstein-Sato polynomial equals

(s + 1)(s + \frac{2}{3})(s + \frac{3}{3})(s + \frac{4}{3})(s + \frac{3}{5})(s + \frac{4}{5})(s + \frac{5}{5})(s + \frac{6}{5})(s + \frac{7}{5}).

Therefore the b-function for integration of J_{n+1} along ∂_{1}, …, ∂_{n} is a divisor of

(s − 2)(s − \frac{1}{3})(s − 2)(s − \frac{11}{3})(s − 0)(s − 1)(s − 2)(s − 3)(s − 4).

This indicates that the degrees of the top cohomology of the Milnor fiber f^{−1}(1) are at most 5. It also shows that in this case these cohomology degrees do not suffice to determine the roots of b_{f}(s). In fact, the degrees of no class in any H_{dR}^{i}(f^{−1}(1), ℂ) will explain the roots −2/3 and −4/3 in b_{f}(s).

We believe that there are relations to Cohen and Orlik’s theory of finite covers on one side, and the local systems induced by f^{a} mentioned above on the other that would explain the roots −2/3 and −4/3.

References: [3, 22, 23, 30]

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