Functorial affinization of Nash's manifold

TL;DR

This paper explores the functorial properties of Nash's manifold, introducing specific divisors and their relations to singularities, and investigates conditions for liftings, contractions, and resolutions in complex geometry.

Contribution

It introduces a new framework for analyzing Nash's manifold using specific divisors and their properties related to singularities and resolutions.

Findings

Divisors D[-1], D[0], ..., D[n+1] have specific ampleness and basepoint free properties.

Liftings of arcs relate to the finiteness of a generating number involving divisors.

Conditions for contractions and resolutions are characterized through divisor intersections and properties.

Abstract

Let M be a singular irreducible complex manifold of dimension n. There are Q divisors D[-1], D[0], D[1],...,D[n+1] on Nash's manifold U -> M such that D[n+1] is relatively ample on bounded sets, D[n] is relatively eventually basepoint free on bounded sets, and D[-1] is canonical with the same relative plurigenera as a resolution of M. The divisor D=D[n] is the supremum of divisors (1/i)D_i. An arc g containing one singular point of M lifts to U if and only if the generating number of oplus_i O_g(D_i) is finite. When it is finite it equals 1+(K_U-K) .g where O_U(K) is the pullback mod torsion of Lambda^n Omega_M. If C is a complete curve in U then (-1/(n+1))K_U .C=D_1 .C + D_n+2 .C + D_(n+2)^2 .C +..... When there are infinitely many nonzero terms the sum should be taken formally or p-adically for a prime divisor p of n+2. There are finitely many nonzero terms if and only if C. D=0. The natural holomorphic map U -> M factorizes through the contracting map U -> Y_0. If M is bounded, the Grauert-Riemenschneider sheaf of M is Hom(O_M(D_{(n+2)^i - 1}), O_M(D_{(n+2)^i})) for large i. If M is projective, singular foliations on M such that K+(n+1)H is a finitely-generated divisor of Iitaka dimension one are completely resolvable, where K is the canonical divisor of the foliation and H is a hyperplane. There are some precise open questions in the article. According to a question of [7] it is not known whether Y_0 has canonical singularities.