March 25, 2017
1 2 Standard definitions The map ord S The map ac S 3 Toroidal constructible functions Main Theorem Some conjectures 4 Toroidalization Key Theorem Proof of Key Theorem
Igusa fibre integrals Outline Let K be any n.a. local field and f : X = A n K Y = A1 K K-morphism. Consider the fibre integral I (y) = Φ(x) dx df, K x K n,f (x)=y a for any y K which is not a critical value of f, where Φ is a locally constant function on K n with compact support. Assume 0 is a critical value of f. Then on a small neighborhood of 0, I (y) is locally constant on the complement of {0}. Igusa s Theorem describes how I (y) behaves for y 0.
Notation for local fields Let K be any n.a. local field. We will use the following notation. O K is the valuation ring of K. ord K : K \ {0} Z the valuation of K. K the residue field of K. q := ( K) the cardinality of the residue field of K. y K := q ord K (y), for any y K. The Haar measure on K is normalized such that O K has measure 1. For any y O K, denote by ȳ the reduction modulo the maximal ideal. Let π be a fixed element of K with ord K (π) = 1. For any y K \ {0}, the angular component ac(y) := yπ ord K (y). And moreover ac(y) := ac(y).
Igusa s Theorem Outline Igusa s Theorem (1975) Assume char(k) = 0. For y 0 in a small enough neighborhood of 0, the fibre integral I (y) is a finite C-linear combination of functions of the form χ(ac(y)) C(ord K (y)) q E(ord K (y)) with χ a character (continuous and complex valued) of the group of units in O K, and C, E numeric quasi-polynomials on Z, deg(e) 1. A quasi-polynomial on Z (or more generally on a lattice) is a complex valued function that is given by a polynomial on each congruence class modulo a suitable modulus N 0. It is called numeric if it takes values in Z.
Igusa s question Outline Igusa s proof is based on embedded resolution of singularities. Can also be proved by cell decomposition (D. 1984) Igusa posed the question how this would generalize when the dimension of the target of f is > 1 (the multivariate case). First results for the multivariate case were obtained by Loeser (1988), Lichtin (1997), and D. (1998). This was for a fixed local field. The present talk is about an analog of Igusa s Theorem, yielding a description of the fibre integral in the multivariate case, which is uniform in all local fields of residue characteristic 0. Cluckers and Loeser (2005) proved, using cell decomposition, that these integrals are specializations of motivic constructible functions. In this talk I will use a different method (flat toroidalization of morphisms) to get a related result. Local toroidalization was already used by Lichtin (1997) to get bounds for some fibre integrals.
Strict toroidal embeddings Standard definitions The map ord S The map ac S Let R be a normal noetherian integral domain. A variety over R is a separated integral scheme of finite type over R. An inclusion U X of varieties over R is a strict toroidal embedding over R if U is smooth /R, U is dense open in X, and locally for the Zariski topology on X there is an étale morphism from X to a toric variety Spec (R[x e e ˇσ]), with U = inverse image of the big torus. D := X \ U is called the toroidal divisor. A strict toroidal embedding U X over R is called smooth if X is smooth over R. Equivalent with X smooth over R and X \ U strict normal crossings divisor on X over R. Denote the irreducible components of D by E i, i I. For J I let E J := j J E j \ i I \J E i. This is smooth over R. The irreducible components of the E J are called the strata of U X. For S a stratum, put Star(S) := {strata whose closure S}.
The natural map ord S Standard definitions The map ord S The map ac S Let S be stratum of the toroidal embedding U X over R, D := X \ U. Let M S = group of Cartier divisors on Star(S) with support D. Note that M S = Z codim S. If X smooth/r, then M S has standard basis {[E i ] E i S}. Let ˇσ S = cone in M S R generated by the effective Cartier divisors. Let N S = Hom(N S, Z) = Z codim S. Let σ S = dual of ˇσ S. Is a full dimensional polyhedral cone in N S R. If X smooth/r, then σ S = R codim S 0. Let R = O F, with F a number field. For any n.a. local field K F, we have a natural map: ord S : {a X (O K ) \ D(O K ) ā S} σ S N S a linear map : c ˇσ S M S ord K (x c (a)) > 0 with x c any element in O X,ā with div(x c ) = c locally at ā.
The natural map ac S Standard definitions The map ord S The map ac S Choose c 1,, c r ˇσ S which form a basis for the lattice M S. If X smooth/r, then choose the standard basis. The normal torus bundle N S X of S X is the restriction to S of i=1,...,r L (c i ) where L (c i ) is the line bundle of c i on X minus 0. If X smooth/r, then N S X is the normal bundle of S X minus the classes of tangent vectors to X that are tangent to some E i S. For any n.a. local field K F, we have a natural map: ac S : {a X (O K ) \ D(O K ) ā S} N S X (K) a ( ā, r i=1 ac(x i(a)) s i (ā) ) with locally at ā, div(x i ) = c i, and s i the local section 1/x i of L(c i ). If X smooth/r, then a ( ā, ( r i=1 ac(x i(a)) x i )ā ).
Toroidal constructible functions (1) Toroidal constructible functions Main Theorem Some conjectures Let A := Z[q, q 1, ( 1 1 q j ) j N0 ]. Let R = O F, with F a number field. Let U F X F a strict toroidal embedding over F. Let Φ be a family of functions Φ K : U F (K) C, where K runs over all n.a. local fields F, with char K 0. Definition Φ is toroidal constructible on U F w.r.t. U F X F, if for some (hence each) model U X over R of U F X F we have (uniformly for char( K) 0): 1 The support of Φ K is contained in {x X (O K ) \ D(O K )} U F (K). 2 For each stratum S of U X, the restriction of Φ K to {x X (O K ) \ D(O K ) x S} is a finite A-linear combination of functions of the form given in next slide.
Toroidal constructible functions (2) Toroidal constructible functions Main Theorem Some conjectures Functions of the form with (λ ac S (x) ( K)) C(ord S (x)) q E(ordS (x)) λ K 0 (Var N ) S X C a numerical quasi-polynomial on N S E a numerical quasi-polynomial on N S, with deg(e) 1
Toroidal constructible functions Main Theorem Some conjectures Schwartz-Bruhat constructible functions Let U F nonsing. variety over number field F. Let Φ be family of functions Φ K : U F (K) C, where K runs over all n.a. local fields F, char K 0. Definition Φ is Schwartz-Bruhat constructible on U F (S-B-constructible on U F ) if it is toroidal constructible w.r.t. the trivial toroidal embedding U F U F. Then Φ K has support U(O K ), with U a model over O F of U F. On U(O K ), Φ K is a finite A-linear combination of functions x (λ x ( K)), with λ K 0 (Var U ). Hence it is locally constant with compact support.
Main Theorem Outline Toroidal constructible functions Main Theorem Some conjectures Main Theorem Let f : X Y be a dominant morphism of nonsingular varieties over a number field F, and ω X, ω Y volume forms on X, Y. Let U Y Y be a dense open subset over which f is smooth. For K any n.a. local field F with char( K) 0, consider I K (y) = Φ K (x) ω X f (ω Y ), for y U Y (K), K x X (K), f (x)=y where the family (Φ K ) K is toroidal constructible on X w.r.t. some strict toroidal embedding of X. Then there exists a toroidal constructible function on some open dense subset Ǔ Y U Y, w.r.t. a smooth strict toroidal embedding of Ǔ Y, which equals I K (y) for any y Ǔ Y (K) with I K (y) absolutely convergent.
Conjectures Outline Toroidal constructible functions Main Theorem Some conjectures Note that the integral I K (y) always converges absolutely when Φ is S-B constructible on X. Conjecture 1. In the Main Theorem we can take Ǔ Y = U Y, if Φ is S-B constructible on X. Theorem 2. Conjecture 1 is true if the Strong Toroidalization Conjecture is true. Thus true when dim X 3 and F big enough, by Cutkosky (2007). Conjecture 2. (± Loeser s conjecture 1988) In the Main Theorem assume moreover that Φ is S-B constructible on X and that f is without blow-up. If Y \ U Y is a strict normal crossings divisor, then (I K ) K is toroidal constructible on U Y w.r.t. U Y Y.
Toroidalization of Morphisms, I Toroidalization Key Theorem Proof of Key Theorem Theorem: Flat Weak Toroidalization (Abramovich-Karu 2000, + ɛ) Let k be a field, char(k) = 0. Let f : X Y be a dominant morphism of k-varieties, and Z X X, Z Y Y closed subsets. Then there exist strict toroidal embeddings U X X, U Y Y, and a commutative diagram U X X m X f U Y Y m Y X f Y with m X, m Y modifications, Y nonsingular, and 1 f is flat and toroidal w.r.t. U X X, U Y Y, 2 m 1 X (Z X ) X \ U X, and m 1 Y (Z Y ) Y \ U Y, 3 the restrictions of m X, m Y to U X, U Y are open embeddings.
Toroidalization of Morphisms, II Toroidalization Key Theorem Proof of Key Theorem The source X of f might have singularities. Such a morphism f is called toroidal w.r.t. the given toroidal embeddings, if it is dominant, maps U X into U Y, and is locally for the étale topology (on source and target) isomorphic to a toric morphism (i.e. given by monomials) of toric k-varieties, respecting the toroidal divisors. Strong Toroidalization Conjecture (weakened) In the Weak Toroidalization Theorem assume moreover: X,Y are nonsingular, f is smooth over Y \ Z Y, and f 1 (Y \ Z Y ) = X \ Z X. Then we can choose f such that the open embeddings of U X, U Y induced by m X, m Y are surjections onto X \ Z X, Y \ Z Y.
Key Theorem Outline Toroidalization Key Theorem Proof of Key Theorem Key Theorem Let f : X Y be a flat toroidal morphism over a number field F, w.r.t. strict toroidal embeddings U X X, U Y Y, and ω X, ω Y volume forms on U X, U Y. Note that f UX is smooth over U Y. Assume Y is nonsingular. For K any n.a. local field F with char( K) 0, consider I K (y) = Φ K (x) ω X f (ω Y ), for y U Y (K), K x U X (K), f (x)=y where the family (Φ K ) K is toroidal constructible on U X w.r.t. U X X. Then there exists a toroidal constructible function on U Y, w.r.t. U Y Y, which equals I K (y) for any y U Y (K) with I K (y) absolutely convergent.
Consequences Outline Toroidalization Key Theorem Proof of Key Theorem The Main Theorem is an easy consequence of the Key Theorem, using Flat Weak Toroidalization. Alternatively it can be proved using the tame cell decomposition of Pas. In about the same way one obtains an alternative proof that the (uniform p-adic) constructible functions of Cluckers-Loeser are closed under integration, in the special case that there are no additive characters involved (by using toroidalization instead of cell decomposition). Corollary(of the Key Theorem). Loeser s Conjecture (Conjecture 2) is true if f : X Y is a toroidal morphism w.r.t. strict toroidal embeddings of an open in X and U Y Y. Indeed, a morphism without blow-up is flat.
The proof: preparations Toroidalization Key Theorem Proof of Key Theorem Set D X := X \ U X, D Y := Y \ U Y, the toroidal divisors. Denote by same letters models for X, Y, f,, over R = O F. Let µ tor X /Y be the measure on the fibers given locally by ωx tor K f (ωy tor) with ωx tor, ωtor Y local generators for Ωdim R X X (log D X ), Ω dim R Y Y (log D Y ). It suffices to prove the Key Theorem for the measure ω X f (ω Y ) K replaced by the toroidal measure µ tor X /Y. Let S be any stratum for U X X, and S a stratum for U Y Y with f (S) S. Consider the following commutative diagram.
The proof: the diagram Toroidalization Key Theorem Proof of Key Theorem {x X (O K ) \ D X (O K ) x S} f (ord S, ac S ) ( σ S N S ) N S X ( K) f {y Y (O K ) \ D Y (O K ) ȳ S } (ord S, ac S ) ( σ S N S ) N S Y ( K) f induced by morphism of lattices, and smooth morphism of torus bundles. The two vertical arrows are surjections. Toroidal Hensel s Lemma. Given y bottom-left, and α top-right, with y and α mapping to same element in bottom-right, then there exists an x top-left which maps to y and α. The µ tor X /Y -measure of the set of these x equals qdim Y dim X.
The proof: finishing Outline Toroidalization Key Theorem Proof of Key Theorem Summing over all strata S with f (S) S, and summing over all α, yields a very explicit formula for the fibre integral I K (y). This formula contains expressions of the form B(w) := C(v) q E(v) v σ S N S, f (v)=w for w σ S N S, where C(v), E(v) are numeric quasi-polynomials on the lattice N S, deg(e) 1. Because f is flat, a Lemma of Abramovich and Karu (2000) implies that f maps each one-dimensional face of the cone σ S into a one-dimensional face of σ S. This combinatorial fact implies that B(w) is an A-linear combination of functions of the form C (w) q E (w), with C, E numeric quasi-polynomials, deg(e ) 1 (whenever absolutely convergent).