Hironaka's conditions on boundaries and orders, for a finite succession S of an analytic manifold M with centres C_i; they are the analytic forms of those of the algebraic theorems (Definition 4.2.1.1, Definition 4.2.3.1):

• S.HasSncBoundaries B: the boundaries E_i started at B (Definition 3.2.1.3) have simple normal crossings with the centres, and the last one has simple normal crossings (Definition 3.1.5.1) [Hir64, Main Theorem II′(N) (iii) and the first half of (iv)]; [Wlo09, Theorem 2.0.3 (2)]. Started at the unit ideal ⊤, the empty boundary, the E_i are the exceptional divisors;

• S.HasConstantPositiveOrderAlongCenters J: for every i the order of the weak transform J_i is a positive constant along C_i [Hir64, Main Theorem II′(N) (ii)].

I.IsLocallyFinitelyPrincipalizedBy F, for an ideal sheaf I on M and a compatible family F with map \mathrm{prin}_I, says that F is a locally finite principalization of I in Włodarczyk's sense. Write F.seq K for the succession over the neighbourhood U_K of a compact K, with composite \sigma_K and exceptional divisors E_i. Then \mathrm{prin}_I is proper; every U_K has compact closure; the centres are smooth; (F.seq K).HasSncBoundaries ⊤ (Definition 4.3.1.1); the total transform \sigma_K^*(I|_{U_K}) is near every point the ideal of an effective combination of the components of E_r (IsMulBoundaryMonomial with the unit ideal as its first factor, Definition 3.1.5.2); the total transform \mathrm{prin}_I^*(I) is a normal-crossings divisor; and \mathrm{prin}_I is an isomorphism off the support of I [Wlo09, Theorem 2.0.3 (1)–(3); Lemma 4.0.3]. It is the analogue on manifolds of Triple.IsPrincipalizedBy on schemes (Definition 4.2.4.2), with properness and the compact closures of the U_K added and each condition stated over every compact.

A pair (M, I) is an analytic manifold M over \mathbb{K}, modelled on a finite-dimensional space E, with an ideal sheaf I on M with nonzero stalks: the input of Włodarczyk's principalization, Kollár's triple (X, I, \emptyset) with an empty boundary (Definition 4.2.4.1). The model space is part of the input, so that one assignment covers the manifolds of every dimension. T.pullback g hg is the pair (N, g^* I) over a local analytic isomorphism g \colon N \to M of manifolds modelled on one space, the source of the morphism of pairs over g [Wlo09, Theorem 3.5.1].

A type C of inputs has underlying manifolds when every input has an underlying analytic manifold over \mathbb{K} on a finite-dimensional model space (HasUnderlyingManifold, the counterpart of HasUnderlyingScheme, Definition 2.4.4.1); the pairs are such a type. A compatible family assignment on C (CompatibleFamilyAssignment) is a compatible family of finite successions of blow-ups (Definition 3.2.2.2) of the underlying manifold of every input: Włodarczyk's \mathrm{prin} and \mathrm{res} as rules on all pairs, the counterpart on manifolds of a blow-up sequence functor [Kol07, Definition 31]. Its properties are the clauses of the functorial theorems.

I.IsPushforwardAlong J τ, for ideal sheaves I on M and J on N and an analytic map \tau \colon N \to M between analytic manifolds modelled on E' and on E: \tau is a closed embedding of analytic manifolds, that is an analytic embedding (Mathlib's IsSmoothEmbedding at \omega: an analytic immersion, of the form u \mapsto (u, 0) in suitable charts near every point, which is a homeomorphism onto its image) with closed image; the ideal of its image is contained in I (at every point, the vanishing ideal of \tau(N), Definition 3.1.3.4, lies in the stalk of I); and J = \tau^* I. So N carries the analytic structure induced on the closed subset \tau(N) (every analytic map with values in \tau(N) factors analytically through \tau; for a finite-dimensional model of M, \tau(N) is a closed submanifold, Definition 3.1.4.1), and the closed subspace of I lies in \tau(N) and is the image of the closed subspace of J: Kollár's "closed embedding of smooth schemes" and his hypothesis \mathcal{O}_X / I_X = j_*(\mathcal{O}_Y / I_Y) on the ideal sheaves in his commutation with closed embeddings, for an empty boundary; the embedding of Włodarczyk's "embeddings of ambient varieties".

IdealSheafPair.CommutesWithLocalAnalyticIsomorphisms P, for a compatible family assignment P on the pairs: the last sentence of [Wlo09, Theorem 2.0.3] for local analytic isomorphisms, in two forms. Over every pair of compact sets: for a local analytic isomorphism g \colon N \to M of manifolds modelled on one space, the pair (N, g^* I) and compacts K' \subseteq N, K \subseteq M with g(U_{K'}) \subseteq U_K, the succession of P(N, g^* I) over U_{K'} is the pull-back along g of the succession of P(M, I) over U_K, up to blow-ups with empty centre (Definition 3.2.4.1), the form of [Wlo09, Theorem 3.5.1 (2)]. Globally: writing \mathrm{prin}_I \colon \tilde M \to M and \mathrm{prin}_J \colon \tilde N \to N for the maps of P(M, I) and P(N, J), J = g^* I, there is an analytic map \tilde g \colon \tilde N \to \tilde M with \mathrm{prin}_I \circ \tilde g = g \circ \mathrm{prin}_J which is a local analytic isomorphism, maps the fibre of \mathrm{prin}_J over every point y bijectively onto the fibre of \mathrm{prin}_I over g(y), and is the only continuous map \tilde N \to \tilde M over g: the "natural lifting" of [Wlo09, Theorem 2.0.1 (3)], whose fibre bijection is the identification of \tilde N with the fibre product N \times_M \tilde M stated pointwise, and whose uniqueness makes the lift of a composite the composite of the lifts. The two forms are bundled as the two conditions of Kollár's 34.1 are in CommutesWithSmoothMorphisms (Definition 4.2.4.3).

IdealSheafPair.CommutesWithClosedEmbeddings P, for a compatible family assignment P on the pairs: for pairs (M, I) and (N, J) and a closed embedding of analytic manifolds \tau \colon N \to M along which I is the push-forward of J (Definition 4.3.3.4), and compacts K' \subseteq N, K \subseteq M with \tau(U_{K'}) \subseteq U_K, the succession of P(M, I) over U_K is the push-forward along \tau of the succession of P(N, J) over U_{K'}, up to blow-ups whose centres lie outside the part over \tau(U_{K'}) (Definition 3.2.5.1). This is the rest of the last sentence of [Wlo09, Theorem 2.0.3], "commutes with … embeddings of ambient varieties", read as Kollár's commutation with closed embeddings [Kol07, Theorem 35 (5)] in the form of [Kol07, 34.3], B(X, I_X, E) = j_* B(Y, I_Y, E|_Y) for an empty E, over every pair of compact sets; the counterpart on manifolds of CommutesWithClosedEmbeddingsOfEmptyBoundary (Definition 4.2.4.5).

For a finite succession S of an analytic manifold M with centres C_i, and the closed subspace Y with ideal sheaf I, I.IsDesingularizedBy S says, with composite \sigma, exceptional divisors E_i and strict transforms Y_i, \tilde Y = Y_r: the centres are smooth; S.HasSncBoundaries ⊤ (Definition 4.3.1.1); no point of a centre lies over a point of \operatorname{Reg}(Y); the support of E_r is the preimage of \operatorname{Sing}(Y) = Y \setminus \operatorname{Reg}(Y); \tilde Y is smooth and IsSncBoundaryTransversalTo holds for E_r and \tilde Y; and \sigma^*(I_Y) = I_{\tilde Y} \cdot I_{\tilde E} (IsMulBoundaryMonomial, Definition 3.1.5.2) [Wlo09, Theorem 2.0.2 (1)–(3), (6)]. It is the analogue on manifolds of EmbeddedPair.IsDesingularizedBy on schemes (Definition 4.2.10.2).

I.IsLocallyFinitelyDesingularizedBy F, for the ideal sheaf I of a closed subspace Y of M and a compatible family F with map \mathrm{res}_{Y,M}, says that F is a locally finite embedded desingularization of Y in Włodarczyk's sense: \mathrm{res}_{Y,M} is proper, it is an isomorphism over the complement of the singular locus \operatorname{Sing}(Y) = Y \setminus \operatorname{Reg}(Y), and for every compact K the succession F.seq K over the neighbourhood U_K desingularizes Y \cap U_K (I.IsDesingularizedBy, Definition 4.3.4.1), has no empty centre and has every centre of codimension at least two, or of codimension one inside the exceptional divisor of its stage, and \tilde M carries an exceptional divisor E and a strict transform \tilde Y gluing the last exceptional divisors and strict transforms of the successions: over every compact K, read on the last stage U_r of F.seq K through its identification F.toSpace K with \mathrm{res}_{Y,M}^{-1}(U_K), E is E_r and \tilde Y the strict transform Y_r of Y \cap U_K (Definition 3.2.1.3). Then E is the reduced ideal sheaf of a simple normal crossings family of hypersurfaces, with which \tilde Y has simple normal crossings, transversally (Definition 3.1.5.2), and \mathrm{res}_{Y,M}^*(I_Y) = I_{\tilde Y} \cdot I_{\tilde E} with \tilde E near every point an effective combination of the components of E. His clause (5) is the compatibility of the family (Definition 3.2.2.2).

An embedded pair (M, Y) is an analytic manifold M over \mathbb{K}, modelled on a finite-dimensional space, with a reduced closed analytic subspace Y \subseteq M, given by its ideal sheaf: the input of Włodarczyk's embedded desingularization, the counterpart on manifolds of the algebraic embedded pair (Definition 4.2.10.1), with the model space part of the input as for Definition 4.3.3.2; the embedded pairs have underlying manifolds (Definition 4.3.3.3). T.pullback g hg is the embedded pair (N, g^{-1}(Y)) over a local analytic isomorphism g \colon N \to M of manifolds modelled on one space; its subspace is reduced since the germ maps of g are ring isomorphisms.