Hsiang-Pati coordinates

Transcription

1 Hsiang-Pati coordinates Edward Bierstone University of Toronto Cortona, 20 June 2014 Collaborators Andre Belotto Vincent Grandjean Pierre Milman Franklin Vera Pacheco

2 Hsiang Pati problem Given a complex variety X 0 Z 0 (Z 0 smooth, e.g., P N (C)), can we find a resolution of singularities σ : X X 0 Z 0 such that the pull-back cotangent sheaf is locally generated by d(u α i ), i = 1,..., s, d(u β j v j ), j = 1,..., n s where n = dim X 0 (u, v) = (u 1,..., u s, v 1,..., v n s )) local coordinates on X, E = (u 1 u s = 0) exceptional divisor α i linearly independent over Q {α i, β j } totally ordered? Consequence Pull-back to X of induced Fubini-Study metric on X 0 \Sing X 0 locally quasi-isometric to s i=1 n s d(u α i ) d(u α i ) + j=1 d(u β j v j ) d(u β j v j ) Proved in case X 0 surface with isolated singularities by Wu-Chung Hsiang, Vishwambkar Pati (1985) William Pardon, Mark Stern (2001) Formulation of HP problem due to Boris Youssin (1998) (u, v) called Hsiang-Pati coordinates

3 Interest? Applications to L 2 -cohomology (following Cheeger) Hsiang-Pati: Intersection cohomology (with middle perversity) of a surface X 0 = L 2 -cohomology of X 0 \Sing X 0 (cf. Cheeger-Goresky-Macpherson conjecture) Melrose: Extension of b-calculus to singular varieties Local HP problem. Is there a semiproper locally finite covering {σ j : X j X 0 } of X 0 such that each σ j is a finite composite of local blowings-up satisfying HP? Theorem HP holds for X 0 of dimension 3 (at least locally) Exercises (1) y 1 = x α 11 1 x α 1n n δ 1. y n = x α n1 1 x α nn n δ n (δ i units). We can absorb units, i.e., y i = x α i after coordinate change x j = δ ɛ j x j, where ɛ j Q n, provided that {α i } linearly independent (2) HP coordinates induce HP coordinates at nearby points Problem. Does HP toroidalization (monomialization) of morphisms? (cf. Cutkosky)

4 Regularization of the Gauss mapping We can reduce HP problem to the case that σ (Ω 1 M 0 ) is locally free of rank n (i.e., defines a vector bundle) by regularization of the Gauss mapping G X0 : X 0 \Sing X 0 Grass(n, TM 0 ) a T a X 0 T a M 0 A Gauss-regular resolution of singularities of X 0 can be obtained by taking the Nash blow-up of X 0, followed by resolution of singularities. The Nash blow-up is the closure in X 0 Grass(n, TM 0 ) of the graph of G X0. Log Fitting ideals Given (X, E), where X smooth, E exceptional divisor Ω 1 X (log E) denotes sheaf of log 1-forms on X i.e., in local coordinates (u, v) = (u 1,..., u s, v 1,..., v n s ) such that E = (u 1 u s = 0), generated by du i u i, dv j Given resolution σ : (X, E) (M 0, X 0, Sing X 0 ), consider σ (Ω 1 M 0 ) Σ Ω 1 X (log E) Coker Σ 0

5 If σ Gauss-regular, then Σ has a presentation given by log Jac σ = σ 1 σ 1 σ 1 σ 1 u 1 u s u 1 u s v σ n σ n σ n σ n u 1 u s u 1 u s v 1 v n s v n s Fitting ideal F n k = F n k (σ) O X generated by k k minors of log Jac σ (independent of presentation of Σ) Log rank log rk a σ := rk a log Jac σ = rk a σ E(a), where E(a) = stratum of a in E Let p := max log rk σ (at points of E) = dim Sing X 0 Σ k := {a E : log rk a σ p k} Y k := σ(σ k ) (so Y 0 = Sing X 0 ) Clearly Σ p Σ p 1 Σ 0

6 Theorem HP is equivalent to the following conditions: (1) σ I Yj principal (generated by monomial in E), j = 0,..., p (2) fitting ideal F n k (σ) principal, k = 1,..., n Neither condition behaves well with respect to blowing up: (1) is not stable after an admissible blowing up β but, given σ, we can principalize the σ I Yj by further blowings up (2) F n k (σ β) β F n k (σ), though F 0 (σ β) = exc l β F 0 (σ) If log rk a σ = r and σ I Yp r is principle, then we can assume σ 1 = v 1,..., σ r = v r and σ r+1 = u α 1 Corollary. HP in two-dimensional case Proof of (local) HP in three dimensions In general (dimension n), we can begin with σ 1 = u α 1 (e.g., at a point of log rank 0) and write σ j = g j (u) + u δ T j (u, v), j 2, where u1 α divides σ j, and g j (u) comprises all monomials u γ of σ j, with γ linearly dependent on α 1 over Q Say T j = ɛ N n s c jɛ (u) v ɛ Let d denote smallest ɛ such that c jɛ is a unit, for some j (maybe d = ) Then d is a local invariant of the Fitting ideal F n 2

7 We can reduce to the case d finite, by resolution of singularities of the ideal generated by the coefficients c jɛ At an n-point (i.e., u = (u 1,..., u n )) this means we get u δ T j (u) = u α2 unit, for some j, say j = 2, and can absorb the unit to put σ 2 in HP form In three dimensions (n = 3): we have 3-points: coordinates (u 1, u 2, u 3 ) 2-points: (u 1, u 2, w) 1-points: (u, v, w) I. We reduce to the following normal forms: at a 3-point: u δ T 2 = u α 2, α 2 independent of α 1 (HP) at a 2-point: at a 1-point: T j = a jd w d + d 1 i=0 a ji u γ ji w i T j = a jd w d + d 1 i=0 a ji u q ji v s ji w i where a jd unit and a j,d 1 0 (j = 2 or 3), a ji unit or 0, i d 1, there is a term of order 1 in (v, w), we can assume a 20 = 1, and s 20 = 1 (1-point case) by HP in dimension 2, all s ji = 0 or 1

8 II. We principalize the ideal generated by w d, u γ ji w i (or u q ji w i ) to achieve order reduction over every point Example u 2 u 3 (w 3 + (v 2 + u x 2 ) w + u 2 y) u 6 w u 6 v u 6 x Thank you for your attention!