Motivic integration, the nonarchimedean Milnor fiber, and a Thom-Sebastiani formlua Yimu Yin A conference in honour of Thomas C. Hales on the occasion of his 60th birthday University of Pittsburgh, June 18-22, 2018 1
What is motivic integration? Starting with the question: what is counting? This (only) makes sense for (discrete) sets, so it is cardinality, following Cantor. But Euler s idea on how to count extended bodies is much more important. What is the length lh(l) of a line segment L in the Euclidean straight line? There is this issue with the endpoints that is often ignored. lh(l) = lh(l ) if and only if L, L match (via shifting ). So an open interval and a closed interval should not have the same length. If L is the (disjoint) union of L 1 and L 2 then lh(l) = lh(l 1 ) + lh(l 2 ). So the length of a closed interval cannot be the sum of the lengths of two closed intervals. This suggests: The length of a half-closed line segment is a real number, but the length of a closed line segment is a real number plus a point.
If we allow stretching in matching line segments, then the lengths of half-closed, closed, open line segments are forced to be 0, 1, 1, respectively. This is of course the Euler characteristic. Making a leap towards abstraction: An integral is an invariant (numerical or not) up to certain likeness. What is the area of a polygon in the Euclidean plane? The stuff: planar polygons, written as [P ], identified up to congruence and rotation. The operation cut (ignoring boundary). We think of [P 1 ] + [P 2 ] as the polygonal complex that is the disjoint union of [P 1 ], [P 2 ]. So our stuff gives rise to a free semigroup (no subtraction, yet), denoted by G poly. If [P ] can be cut into polygons [P 1 ], [P 2 ] then we write [P ] = [P 1 ] + [P 2 ]. This gives an equivalence relation, actually a semigroup congruence relation, on G poly, called the scissor relation. The scissor semigroup S poly is the semigroup G poly modulo the scissor relation.
The Wallace-Bolyai-Gerwien theorem: S poly is isomorphic to R + 0, and the isomorphism is constructed by assigning, to each [P ], its area. Hilbert s third problem: Does this still hold in three-dimension (polyhedra)? Conjectured No and proved by Dehn via Dehn invariant. In a sense, we want to put a positive spin on Hilbert s problem by way of vast generalizations of the Wallace-Bolyai-Gerwien theorem. In its crudest form, we propose the following definition of motivic integration (or rather additive invariant ). The stuff we are interested in is collected in a category C. We only care about the isomorphisms in C, so may take C to be a groupoid. For an object A of C, denote its isomorphism class by [A]. Denote by GC the free abelian group generated by the set {[A] : A C}. Usually there are two operations in C: product A B (example: cartesian product) and cut A B (example: set subtraction), and the product makes GC a commutative ring: [A] [B] := [A B].
An integral is a ring homomorphism E : GC R, for some ring R, that is compatible with the scissor relation, i.e., where B is a subobject of A. E([A]) = E([B]) + E([A B]), In particular, we can take R to be the quotient K C of GC by the ideal generated by the scissor relation, called the Grothendieck ring of C. Let (G, ) be a locally compact Hausdorff group, such as R, C, Q p, F p (t ), and the Lie groups over them. Let B(G) be the collection of the Borel subsets of G. Then, up to scalar, there is a unique translation-invariant measure µ : B(G) R (countably additive, finite on compact sets, etc.). For R, C this is the Lebesgue measure, for Q p, F p (t ) this is the Haar measure. If (G, ) is not locally compact then such a measure is impossible. In geometry, groups (or rather fields) such as C (t ) (field of Puiseux series over C) do occur naturally.
For n N, the field C (t 1/n ) consists of power series of the form i=k c it i/n, where c i C and k is an integer. C (t ) = n C (t1/n ). It is the algebraic closure of C (t ) (known to Newton, essentially). We will describe motivic integrals on categories of definable sets in C (t ).
Milnor fiber Motivation: In geometry, much can be said when the object in question is smooth. To study singular objects, one may consider two general strategies: make the singularities go away or investigate what happens nearby. Milnor fiber is a line of inquiry following the latter strategy. It draws on nearby smooth replicas of the singular locus in order to understand the singular locus itself. There is a dual procedure of passing from smooth to singular which is perhaps more intuitive to grasp (equidistant deformation). The statement: Given the following data: X C n a smooth complex algebraic variety of dimension d, f : X C a nonconstant polynomial map,
x X a singular point of f 1 (0), i.e. df(x) = 0, w.l.o.g. x = 0, closed ball B(0, ɛ) C n and closed disc D(0, η) C, if 0 < η ɛ 1, there are the Milnor fibration B(0, ɛ) f 1 (D η 0) D η 0 and the Milnor fiber F f = B(0, ɛ) f 1 (η). F f is a smooth manifold with boundary and has a diffeomorphism type that does not depend on η and ɛ, and is endowed with an automorphism (monodromy). There are many questions concerning the topology and geometry of F f, its boundary (the link), the monodromy, etc.
Motivic zeta function and motivic Milnor fiber Recall the field C((t)) of complex Laurent series and the subring C [ t ]. ] The space of C [ t ]-points ] in X is referred to as (formal) arcs on X. Let ˆµ be the profinite group of roots of unity. The category of complex varieties with ˆµ-actions is denoted by Var C. Its Grothendieck ring is denoted by Kˆµ Var C. Let L(X) be the space of arcs γ(t) with γ(0) = 0. Let L m (X) = L(X)/t m+1 and X m = {γ(t) L m (X) : f(γ(t)) = t m mod t m+1 }. Denef and Loeser define the motivic zeta function of f as the following power series in T : Z f (T ) = n 1[X n ][A] nd T n, where the coefficients are in Kˆµ Var C [[A] 1 ]. By Denef and Loeser, the power series Z f (T ) is indeed rational and lim T Z f (T ) exists (evaluated via a formal process). Set S f = lim T Z f(t ).
This is an element in Kˆµ Var C [[A] 1 ] and is called the motivic Milnor fiber. There is a priori no connection between the construction of the topological Milnor fiber F f and that of the motivic Milnor fiber S f. Nevertheless, by Denef and Loeser, the Euler characteristic of F f coincides with the Euler characteristic of S f (the latter is defined via a natural specialization map Kˆµ Var C [[A] 1 ] Z), justifying the terminology. The proofs make heavy use of resolution of singularities.
Resolution-free: the Hrushovski-Loeser method Using Hrushovski-Kazhdan style motivic integration, Hrushovski and Loeser give a resolutionfree construction of the motivic Milnor fiber S f as follows. The motivic zeta function is now expressed as Z f (T ) = n 1 H n (X )T n, where H n (X ) is a gadget from the Hrushovski-Kazhdan integration theory that takes values in Kˆµ Var C [[A] 1 ] and X is the nonarchimedean Milnor fiber of f (to be defined below). Rationality of Z f (T ) now follows from the equality lim T Z f(t ) = (Θ E )([X ]) and certain computation rules for (convergent) geometric series, where K VF K RV[ ]/([1] [RV + ]) E!K RES Θ Kˆµ Var C
is a sequence of canonical homomorphisms constructed in the Hrushovski-Kazhdan integration theory (more details below). Hrushovski and Loeser also give a resolution-free proof that the Euler characteristics of F f and S f. The key is to prove an A Campo-Denef-Loeser formula that expresses χ(x m ) as the trace of the corresponding monodomy action. In doing so, they are very careful in choosing tools from algebraic geometry that do not involve resolution of singularities. This can also be done using a variant of the Hrushovski-Kazhdan integral over o-minimal fields, free of sophisticated algebro-geometric machineries. One lesson learned: the nonarchimedean Milnor fiber is a richer object to study.
Hrushovski-Kazhdan style integration We work in C := C (t ), the field of complex Puiseux series, considered as a model of the first-order theory ACVF of algebraically closed valued fields. ACVF may be expressed in the traditional language with three sorts VF = C, Γ = Q, k = C, and valuation map val : VF Γ and residue map res : O k. But the language L RV we use is more subtle. It has two sorts (VF, RV), where RV = VF /(1 + M) and rv : VF RV is the quotient map. We have a short exact sequence 1 k RV vrv Γ 0. RV may be thought of as the set of leading terms of the power series in C and rv the leading term map. In fact, RV = C Q in C, since there is a natural cross-section Q C. Let γ = (γ 1,..., γ n ) Γ n. Then a definable subset of vrv 1 (γ) RV n is a twisted copy of a constructible set in k n = C n, called a generalized constructible set over k n = C n.
The categories VF[n]. Objects: definable subsets of n-dimensional varieties over C (t ). Morphisms: definable bijections. RES[n]. Objects: finite unions of generalized constructible sets over k n = C n. Morphism: definable bijections. Γ[n]. Objects: finite disjoint unions of definable sub-polytopes of Γ n = Q n. Morphism: piecewise GL n (Z)-transformations. Here definable is equivalent to the condition that the equalities and inequalities defining the polytope are over Z. Set VF = n VF[n], RES[ ] = n RES[n], Γ[ ] = n Γ[n].
The main construction of the HK theory There is a surjective lifting map L : RES[ ] Γ[ ] VF. L induces a surjective homomorphism of rings, also denoted by L K RES[ ] K Γ[ ] K VF. The kernel P of L is principal, being generated by the element 1 1 + 1 [Q + ] [1] 1. Inverting, we obtain an isomorphism : K VF K RES[ ] K Γ[ ]/P.
Constructing the motivic Milnor fiber Forgetting the gradation of RES[ ], we obtain a category RES. Formally identifying all twisted n-dimensional affine spaces with A n, we obtain a quotient of K RES, denoted by!k RES. Applying the bounded Euler characteristic map on polytopes, we construct a canonical homomorphism E : K RES[ ] K Γ[ ]/P!K RES. Choosing a (reduced) cross-section csn : Γ RV and untwist, we construct a canonical isomorphism Θ :!K RES Kˆµ Var C. Note that the choice of csn is not natural, but different choices will yield the same Θ, and the ˆµ-actions come from these difference choices, which correspond to the Galois group ˆµ = Gal( C/C (t )). Recall X and f. A key object in VF to consider: X = {y X(M) : f(y) = t}.
This is closely related to the analytic Milnor fiber defined by Nicaise and Sebag. The nonarchimedean Milnor fiber is even more important: X = {y X(M) : rv(f(y)) = rv(t)}. We have X = X (1 + M). The Hrushovski-Loeser method shows that the motivic Milnor fiber S f of f equals (Θ E )(X ).
Thom-Sebastiani formalism It is a common idea in mathematics to construct complicated objects from simpler ones. In singularity theory, this is embodied by the Thom-Sebastiani formalism: Suppose that f : C n+1 C and g : C m+1 C are germs of holomorphic functions with vanishing isolated singularities at the origin. Let f + g be the germ of the function on C n+m+2 given by (f + g)(x, y) = f(x) + g(y). Then F f+g is homotopy equivalent to F f F g (compatible with the monodromies). Here F f F g is the topological join of F f and F g. The original result of Sebastiani and Thom is about cohomological groups, which can be recovered from the statement above. Denef-Loeser and Looijenga also obtained a Thom-Sebastiani formula for motivic Milnor fibers: S f+g = S f + S g S f S g. where S f S g is a convolution product on Kˆµ Var C [[A] 1 ].
Later, Guibert, Loeser, and Merle generalized this to pairs of functions on the same variety: Let f, g : X C be nonconstant polynomial maps on a smooth complex algebraic variety X. Then, for sufficiently large N N, Here S g N +f = S f + S g N([Z f ]) Ψ Σ (S g N(S f )). Z f = f 1 (0) and S g N([f 1 (0)]) is the restriction of S g N to the class [f 1 (0)], S g N(S f ) is an iterated motivic Milnor fiber, Ψ Σ is a certain convolution operator. Again, heavy dosage of resolution of singularities is used in the proof.
In recent joint work with Fichou, we generalized the Guibert-Loeser-Merle formula, using the Hrushovski-Loeser method. Let f, g, X be as above. Let h(x, y) be a polynomial of the form y N + x m ı, m 2 N m 3... m l. For each 2 ı l, let 2 ı l f (ı) = 2 i ı f m i and f (ı) = 2 i ı f m i ; Then S g N +f (l) = S g N([Z f(l) ]) + S f m 2 + S f mı([z g N +f (ı 1) ]) Ψ ı (S g N f (ı) ), 2<ı l 2 ı l where again, for each ı, Ψ ı is a certain convolution operator.
For each 2 ı l, let α ı = 1/m ı and β ı = m 2 /Nm ı. Let L ı (Q + ) 2 be the open line segment between the two points (α ı, β ı ) and (α ı+1, β ı+1 ). The four types of terms on the right-hand side may be read off from the following illustration: y H 2 1 N H 1 H 3 (α l, β l ) (α 4, β 4 ) (0, 0) 1 x (0, 0)... (α m 2 ( 1 m 2, 1 3, β 3 ) N ) L l L l 1... L 4 L 3 L 2 Figure 1. The tropical curve H of h(x, y) and the vertical rectangular pane P of height 1 on the line segment H 3. the terms S g N([Z f(l) ]) and S f m 2 correspond to the rays H 1 and H 2, respectively,
The restricted motivic Milnor fibers S f mı([z g N +f (ı 1) ]) correspond to the points (α ı, β ı ), The points above (α ı, β ı ) that lie on the sloped line segments contribute another term, so does the corresponding open line segment L ı, and the two of them are jointly referred to as a term of convolution product.
Some remarks: In the condition m 2 N m 3... m l, means sufficiently large relative to the data, not necessarily in terms of magnitude. For instance, h(x, y) = y 2 x 3 + x works for certain choice of f, g. The Guibert-Loeser-Merle formula is a special case, but their method offers a geometric interpretation of sufficiently large N in terms of log-resolutions. Our interpretation depends on compactness and is not as informative. Thus, for sufficiently large N, we have Ψ 2 (S g N f) = Ψ Σ (S g N(S f )). The left-hand side of this equality is commutative: S g N f = S f g N, and perhaps this can be translated into an expression on the right-hand side?