# Milnor Fiber Consistency via Flatness

Alex Hof

(  Abstract. We describe a new algebro-geometric perspective on the study of
the Milnor fibration and, as a first step toward putting it into practice,
prove powerful criteria for a deformation of a holomorphic function germ to
admit a Thom stratification and, more generally, to preserve the Milnor
fibration in an appropriate sense. As corollaries, we obtain a method of
partitioning the space of homogeneous polynomials of a fixed degree into
finitely many locally closed subsets such that the fiber diffeomorphism type
of the Milnor fibration is constant along each subset and a criterion under
which deformations of a function with critical locus a complete intersection
will be well-behaved. )

##  1 Introduction

Consider a nonzero holomorphic function germ
$f:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ for an integer $n\geq 0$. Then the
fiber over the origin in $\mathbb{C}$ of $f$ is well-defined as a complex-
analytic space germ $(V(f),0)$ at the origin in $\mathbb{C}^{n+1}$, which may
in general be singular. There is also a well-defined local smooth fiber of
$f$, given by the intersection of a sufficiently small ball around the origin
in $\mathbb{C}^{n+1}$ with the inverse image under $f$ of a sufficiently small
nonzero value in $\mathbb{C}$. Indeed, these nearby fibers fit together into a
well-defined smooth locally trivial fibration called the Milnor fibration,
which was introduced in [Mil68] and has been much studied in the subsequent
decades. For an introduction to the subject, see Chapter 3 of [Dim92], Chapter
9 of [Max19], the 50-year retrospective survey [Sea19], or the handbook entry
[LNS20].

There is much about the Milnor fibration which is not yet understood. In
particular, given $f$, it is far from clear in general how to determine the
structure of the Milnor fiber even at the level of homology, although there
are techniques rendering the question tractable in various special cases.
Perhaps the most compelling of these is one which has been known since the
beginning — Milnor showed in [Mil68] that, if $(V(f),0)$ has an isolated
singularity at the origin, then the Milnor fiber of $f$ has the homotopy type
of a bouquet of $n$-spheres, whose number, called the Milnor number, may be
computed as the length of the quotient of the local ring by the Jacobian ideal
$J_{f}$ of $f$.

Less powerfully but more generally, Kato and Matsumoto established in [KM75]
that, if $s$ is the complex dimension of the singular locus of $(V(f),0)$,
then the Milnor fiber of $f$ is at least $(n-s-1)$-connected — in particular,
the reduced homology of the fiber is zero outside the degrees $[n-s,n]$ (for a
perspective on this homological vanishing using the machinery of perverse
sheaves see, e.g., [Max20] or [MPT22]). Together with Milnor’s stronger result
in the case $s=0$, this control of the Milnor fiber by the singularities of
$(V(f),0)$ is suggestive — it now seems reasonable to wonder whether it is
possible to compute the Milnor fiber entirely from the critical locus of $f$
at $0$, endowed with the complex-analytic scheme structure given by
considering it as the vanishing of the Jacobian ideal $J_{f}$.

Note that, for there to be any hope of an affirmative answer to this question,
the scheme-theoretic perspective on the critical locus is essential — all
isolated singularities, for example, have the same singular locus set-
theoretically, but of course a variety of Milnor fibers are possible. The aim
of this paper is to give a first step toward the study of the Milnor fibration
from this viewpoint.

As is often the case in these sorts of situations, it is difficult to obtain
direct results of the kind we want about a given function germ on its own, but
studying how the objects of interest vary in families is more feasible.
Therefore we will consider holomorphic function germs
$F:(\mathbb{C}^{n+1}\times\mathbb{C}^{u},0\times 0)\to(\mathbb{C},0)$,
regarding the coordinates of $\mathbb{C}^{u}$ as parameters describing a germ
of a family of holomorphic maps $\mathbb{C}^{n+1}\to\mathbb{C}$. For the sake
of convenience, we denote by $f:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ the
germ given by restricting $F$ to $(\mathbb{C}^{n+1}\times 0,0\times 0)$.

We are interested in the following question, stated somewhat vaguely: When
does the Milnor fibration vary consistently in the family $F$? In the case
where $f$ defines an isolated singularity, it is not difficult to show that in
an appropriate sense the diffeomorphism type of the smooth fiber remains fixed
in the family. This gives rise to a method for computing the Milnor fiber of
$f$ known as Morsification, wherein $f$ is perturbed slightly to have only
Morse singularities, each of which corresponds to a separate cycle in the
homology of the smooth fiber — see, e.g., Chapter 2 of [AGV88] for the details
of this method, or see Chapter 7 of [Loo84] for a treatment in the more
general setting of isolated complete intersection singularities.

In the non-isolated case, the situation is much more complicated, and there
have been various attempts to establish conditions under which some sort of
consistency can be obtained. One example, of a somewhat different flavor than
our approach here, is the work of Massey on Lê numbers, coordinate-dependent
quantities defined for an arbitrary hypersurface singularity which in some
sense generalize the Milnor number in the isolated case. Massey’s book [Mas95]
gives an introduction and overview of major results, including the fact that
the constancy of these numbers at the origin in a family of holomorphic
functions guarantees the consistency of the Milnor fiber at least on the
homological level, and in stronger senses if the critical locus has small
enough dimension. The strength of these results is offset by the stringency of
the condition that the Lê numbers remain constant specifically at the origin —
this precludes, for example, splittings into simpler singularities as in the
Morsification method mentioned above.

The other main general method for finding perturbations which give useful
information about the Milnor fibration is to recover the finite determinacy
present in the isolated case by restricting to families within an ideal with
respect to which $f$ has finite extended codimension — geometrically, such
deformations keep the positive-dimensional parts of the critical locus more or
less fixed but allow us to manipulate zero-dimensional features (e.g., by
splitting off Morse points). This strain of research was initiated by Siersma
in [Sie83], which was the first of a series of papers [Sie87, Sie88, Jon88,
Pel89, Pel90, Zah94, Ném99, BM10, ST17] by various authors using these and
related techniques to obtain Milnor fiber information about functions with
critical loci satisfying various restrictions on dimension, structure,
transversal type, and so forth.

The theoretical underpinnings of this method were developed in the thesis
[Pel85] of Siersma’s student Pellikaan and the subsequent articles [Pel88,
Pel89, Pel90] based on its contents; much later, Bobadilla expanded on them in
[Bob04a, Bob04], proving a general theorem on Morsification in this context,
and used his results to provide answers to some open questions about the Lê
numbers and other equisingularity-related topics in [Bob05]. Connections to
other perspectives were also pursued around this time by Gaffney, who computed
invariants of some stable germs in this context in [Gaf08] and gave a related
condition via the notion of the multiplicity of a pair of modules in the case
of a function with critical locus a complete intersection in [Gaf07].

Despite these results, it seems unlikely that an approach based on finite
determinacy will be enough to solve the riddle of the Milnor fibration — as
mentioned, such methods allow for manipulations of the critical locus only at
isolated points, which puts most of the non-isolated phenomena we would like
to be able to address outside their purview. Fortunately, the perspective that
the Milnor fiber should be controlled by the scheme-theoretic critical locus
yields a much more general approach which does not suffer the same
limitations.

Specifically, if we believe that the critical scheme of $f$ should encode
information about the Milnor fiber, then we should try to get the Milnor
fibration to vary consistently in our family $F$ by imposing some kind of
control on the critical loci of the functions in the family. The most obvious
condition to require is flatness, ubiquitous in algebraic and complex-analytic
geometry as the correct notion of consistency for families of objects like
schemes or sheaves — for the critical locus to determine the Milnor fibration
intrinsically, we should hope that its flatness over the parameter space is
enough to force the consistency of the smooth fibers, and in nice situations
this does indeed turn out to be true. More generally, it seems that the
information we want involves not just the critical scheme but also its
inclusion into the ambient space, and so it is necessary to ask for
consistency in an embedded sense — that is, we should require the flatness
over the parameter space of all $k$th-order infinitesimal neighborhoods of the
critical locus in the ambient space, or equivalently of the normal cone to the
critical locus in the ambient space.

Such a requirement will in fact turn out to be enough to guarantee the
consistency of smooth fibers, as desired. Our first result in this direction,
which we will prove in Section 4, ties the flatness of the normal cone to the
theory of stratified maps:

######  Theorem 1.1.

Let $u\geq 0$ be an integer, $X^{\prime}$ a complex-analytic manifold,
$X\subseteq X^{\prime}\times\mathbb{C}^{u}$ an open subset,
$\pi:X\to\mathbb{C}^{u}$ the restriction to $X$ of the projection onto
$\mathbb{C}^{u}$, and $F:X\to\mathbb{C}$ a holomorphic function which is not
constant on any connected component of a fiber of $\pi$. Denote by
$C_{F\times\pi}$ the vanishing locus of the ideal sheaf
$\mathcal{J}_{F\times\pi}$ of maximal minors of the Jacobian matrix of
$F\times\pi$.

Then there exists a complex-analytic Whitney stratification of $X$ such that
$X\setminus C_{F\times\pi}$ is the ambient stratum, the non-flat locus of the
normal cone $C_{C_{F\times\pi}}X$ to $C_{F\times\pi}$ in $X$ over
$\mathbb{C}^{u}$ is a union of strata, and the Thom ($a_{F\times\pi}$)
condition with respect to the ambient stratum is satisfied on any stratum not
contained in this non-flat locus.

This theorem, as mentioned, will ultimately allow us to use the flatness of
the normal cone to the critical locus to control the behavior of smooth fibers
in a family of holomorphic function germs. Also as mentioned, it is not always
necessary to consider the full normal cone — the flatness of the critical
locus will be enough when there are, in some appropriate sense, vector fields
witnessing the existence of the Milnor fibration of the function being
deformed. We collect both of these results in the following theorem, which we
will prove in Section 5:

######  Theorem 1.2.

Let $n,u\geq 0$ be integers, $F:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C},0)$ a
holomorphic function germ, and
$\pi:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C}^{u},0)$ the projection onto the last
$u$ coordinates. Suppose $f:=F|_{\mathbb{C}^{n+1}\times 0}$ is non-constant
and denote by $(C_{F\times\pi},0)$ the vanishing locus of the ideal
$J_{F\times\pi}$ of maximal minors of the Jacobian matrix of $F\times\pi$.

Suppose that either of the following conditions holds:

  1. (i)

$f$ has a representative such that, for every critical point $p$ of $f$ in a
sufficiently small punctured neighborhood of the origin in
$\mathbb{C}^{n+1}\times 0$, there exists a germ $V$ of a holomorphic vector
field at $p$ such that $Vf=0$ as germs at $p$ and $V(p)$ is not tangent to the
sphere of radius $|p|$ centered at the origin. Moreover, $(C_{F\times\pi},0)$
is flat over $(\mathbb{C}^{u},0)$ everywhere on $(\mathbb{C}^{n+1}\times
0,0)$, except possibly at the origin.

  2. (ii)

The normal cone of $(C_{F\times\pi},0)$ in $(\mathbb{C}^{n+1+u},0)$ is flat
over $(\mathbb{C}^{u},0)$ everywhere on $(\mathbb{C}^{n+1}\times 0,0)$, except
possibly at the origin.

Then, if we fix a small enough $\varepsilon>0$, we can choose $\delta>0$ and
$\gamma>0$ small enough relative to $\varepsilon$ so that the following hold:
The representatives $F:B_{\varepsilon}\times B_{\gamma}\to\mathbb{C}$ and
$C_{F\times\pi}\subset B_{\varepsilon}\times B_{\gamma}$ exist and, if we let
$\Delta:=(F\times\pi)(C_{F\times\pi})\subset\mathbb{C}\times B_{\gamma}$ be
the discriminant, the restriction

$F\times\pi:(B_{\varepsilon}\times
B_{\gamma})\cap(F\times\pi)^{-1}((D_{\delta}\times
B_{\gamma})\setminus\bar{\Delta})\to(D_{\delta}\times
B_{\gamma})\setminus\bar{\Delta}$

defines a smooth locally trivial fibration, where
$B_{\varepsilon}\subset\mathbb{C}^{n+1}$, $D_{\delta}\subset\mathbb{C}$, and
$B_{\gamma}\subset\mathbb{C}^{u}$ are the open balls of the specified radii.

Thus, if the scheme-theoretic critical locus of the family defined by $F$ is
consistent over the parameter space in an appropriate sense, at least away
from the origin, then the local smooth fibers in the family fit together into
a smooth locally trivial fibration. By considering the restriction over the
origin in the parameter space, we see that the fiber of this fibration is
simply the Milnor fiber of $f$. Therefore, within such a family, it is
possible to study the smooth fiber by taking a slight perturbation, as in the
isolated case. As mentioned, this gives an important first step toward the
larger project of computing information about the Milnor fibration of $f$ from
the invariants of the Jacobian ideal of $f$ and lends credibility to the idea
that such a project will be feasible.

The rest of this paper is organized as follows. Section 2 gives short
introductions to a few concepts and results which will be important in the
remainder of the paper, with references to more complete treatments;
experienced readers may wish to skip over some or all of them, depending on
their areas of expertise. Section 3 lays the algebraic groundwork for the
proofs of the main theorems via a series of lemmas, which are then applied to
prove Theorem 1.1 in Section 4 and Theorem 1.2 in Section 5.

Section 6 explores consequences of the main theorems for the study of
homogeneous polynomials — in particular, Corollary 6.3 gives a partition of
the space of homogeneous polynomials of a fixed degree into finitely many
locally closed subsets along which the fiber diffeomorphism type of the Milnor
fibration is constant. Section 7 treats the special case, studied by various
prior authors, of functions with critical locus a complete intersection, for
which our theorems’ hypotheses become particularly simple.

Section 8 compares the theorems’ consequences to the existing techniques
described above and other past results. Finally, Section 9 collects concrete
examples not included elsewhere in the paper, including an instance of the
theorems’ use to compute homological data.

Acknowledgments.

The author is indebted to his advisor, Laurenţiu Maxim, for his guidance and
support throughout this project. He is also grateful to Javier Fernández de
Bobadilla, Dirk Siersma, and Terence Gaffney for interesting conversations.
Much of the work in this paper was carried out during a Research Assistantship
supported by funds from the University of Wisconsin-Madison’s 2021 Fall
Competition.

##  2 Preliminaries and References

###  2.1 Complex-Analytic Geometry and Cones

Here we recall some relevant parts of the theory of complex-analytic spaces;
for an introduction to these objects and their properties the reader is
encouraged to refer to both [GR65] and [Fis76]. Essentially, they are the
complex-analytic analogues to finite-type separated schemes over $\mathbb{C}$,
capturing holomorphic behavior instead of algebraic. In particular, germs of
complex-analytic spaces correspond directly to quotients of convergent power
series rings (see Section 0.21 of [Fis76]), so on a local level the study of
these spaces is nearly just the study of a certain kind of affine scheme
(although a certain amount of care must be taken as soon as any kind of non-
locality is introduced). For this reason, and for the sake of terminological
convenience, we will occasionally refer to nonreduced behavior and other
algebraic properties as “scheme-theoretic” rather than “complex-analytic-
space-theoretic”, as would be strictly correct.

In particular, some of the spaces we will be interested in are complex-
analytic cones (see [AM86]) over other complex-analytic spaces — roughly,
these are given by maps of complex-analytic spaces whose fibers are affine
cones over (possibly weighted) projective schemes, although the fully rigorous
definition expresses this requirement in terms of diagrams to properly account
for nonreduced behavior. In the spirit of Serre’s GAGA principle ([Ser56]), we
then have the following result, which will make these spaces easier to
manipulate in practice:

######  Proposition 2.1 ([AM86]).

Let $(X,\mathcal{O}_{X})$ be a complex-analytic space. Then there is an anti-
equivalence of categories, given by taking the analytic spectrum, between the
category of finitely-presented nonnegatively-graded sheaves of
$\mathcal{O}_{X}$-algebras with degree-zero part $\mathcal{O}_{X}$ and the
category of complex-analytic cones over $X$.

For the construction of the analytic spectrum in this case, see [Hou61].

There are two cones which will be of especial importance to us in what
follows. The first of these captures information about the embedding of a
closed subspace into a complex-analytic space:

######  Definition 2.2.

Let $(X,\mathcal{O}_{X})$ be a complex-analytic space and $Y$ the closed
subspace cut out by an ideal sheaf $\mathcal{I}$. Then the normal cone
$C_{Y}X$ to $Y$ in $X$ is the complex-analytic cone over $Y$ corresponding
under Proposition 2.1 to the associated graded sheaf of algebras

$\operatorname{gr}_{\mathcal{I}}\mathcal{O}_{X}:=\bigoplus_{i\geq
0}\mathcal{I}^{i}/\mathcal{I}^{i+1},$

which is finitely presented by Proposition 1.17 of [AM86] and the Oka
Coherence Theorem (see Section 0.12 of [Fis76]).

Geometrically, this can be thought of roughly as the “straightening-out of the
ambient space in directions normal to the subspace”; in the case where both
spaces are smooth, it is simply the normal bundle to the subspace in the usual
sense of differential geometry. There is also a corresponding notion of a
normal cone of a closed subscheme in algebraic geometry (see, e.g., [Ful98]),
in which, instead of the analytic spectrum, we have the typical relative
spectrum; these notions agree in the sense that, if we apply each to germ of
an inclusion of complex-analytic spaces, we get respectively the analytic
spectrum and the usual spectrum of the same finitely-presented algebra over
the appropriate local ring.

In dealing in particular with complete intersections, we will make use of the
following fact, which appears as Exercise 17.16 of [Eis04] and is originally
due to Rees:

######  Proposition 2.3 ([Ree57]).

An inclusion $Y\hookrightarrow X$ of locally Noetherian schemes is a regular
embedding — that is, is locally cut out by a regular sequence — if and only if
$C_{Y}X$ is a vector bundle over $Y$. Moreover, a given choice of local
generators for the ideal sheaf of $Y$ in $X$ will form a regular sequence
exactly when the generators’ initial forms give a local basis of sections for
$C_{Y}X$.

The second cone we will be interested in is in some senses similar, but
captures instead the cotangent directions perpendicular to the smooth points
of fibers of a map between complex manifolds:

######  Definition 2.4.

Let $\Phi:(X,\mathcal{O}_{X})\to(Y,\mathcal{O}_{Y})$ be a map between smooth
complex-analytic spaces which is a submersion on an open dense subset $U$ of
$X$. Then the relative conormal space of $\Phi$ is the closed subset of the
cotangent bundle $T^{*}X$ given by

$T_{\Phi}^{*}X:=\overline{\\{(x,\eta)\in T^{*}U\mid\eta(\ker
D\Phi|_{x})=0\\}}.$

For our purposes, the following characterization will be more useful. It is
straightforward to see that, under the anti-equivalence of Proposition 2.1,
the cotangent bundle of $X$ corresponds to the symmetric algebra on the sheaf
$\Theta_{X}$ of vector fields on $X$, and the pullback along $\Phi$ of the
cotangent bundle of $Y$ is likewise given by the symmetric algebra on
$\Phi^{*}\Theta_{Y}$. Then we have:

######  Proposition 2.5.

$T_{\Phi}^{*}X$ corresponds under Proposition 2.1 to the sheaf of
$\mathcal{O}_{X}$-algebras which is the image of the map
$\operatorname{Sym}(D\Phi):\operatorname{Sym}(\Theta_{X})\to\operatorname{Sym}(\Phi^{*}\Theta_{Y})$
given by the Jacobian of $\Phi$.

Proof

Around points of $X$ where $\Phi$ is a submersion, working in local
coordinates makes this immediate. To prove it in general, we localize around
each point of $T^{*}X$ — which we regard as the analytic spectrum of
$\operatorname{Sym}(\Theta_{X})$ — and then use the fact that the germ of the
closed subcone of the cotangent bundle corresponding to the algebra described
above is cut out by a prime ideal in the local ring and apply Theorem III.C.16
of [GR65] to show that it does in fact give the closure of the part over the
submersive locus.

For both of these cones, the primary questions we will be interested in
revolve around the notion of flatness, which was introduced by Serre in
[Ser56]. Expositions in the algebraic setting can be found in most textbooks
on commutative algebra and algebraic geometry — see, e.g., [Mat89], [Eis04],
or [Har77]. The complex-analytic perspective on the subject is covered in
Chapter 3 of [Fis76] and in [Hir77], particularly the Appendix.

######  Remark 2.6.

As mentioned in Section 1, flatness is generally accepted as the correct
notion of consistency for algebro-geometric or complex-analytic objects
parameterized over a base. It is common to speak of the geometric meaning of
this consistency as being somewhat mysterious and unintuitive — however, this
need not be the case. As Hironaka points out in the Appendix to [Hir77] (or
see conditions $(4),(4^{\prime})$ of Theorem 22.3 of [Mat89]), in the
Noetherian setting flatness at a point is equivalent to the triviality of the
normal cone to the fiber over the tangent cone to the base, a
straightforwardly geometric condition.

We conclude with the following remark about the interplay between cones and
their corresponding sheaves of algebras, and between schemes over $\mathbb{C}$
and their analytifications (for which see [Ser56]):

######  Remark 2.7.

In what follows, we will very often be interested in verifying the flatness of
complex-analytic spaces and cones over other complex-analytic spaces. To use
these notions freely, we would like to be able to say the following:

  * •

Given a map $(X,\mathcal{O}_{X})\to(Y,\mathcal{O}_{Y})$ of complex-analytic
spaces and a complex-analytic cone over $X$, the cone’s flatness over $Y$ as a
complex-analytic space is everywhere equivalent to that of the corresponding
sheaf of $\mathcal{O}_{X}$-algebras over $\mathcal{O}_{Y}$.

  * •

Given a map $(X,\mathcal{O}_{X})\to(Y,\mathcal{O}_{Y})$ of separated finite-
type schemes over $\mathbb{C}$, the flatness of $X$ over $Y$ is everywhere
equivalent to that of $X$’s analytification over $Y$’s analytification.

Both of these turn out to be true, essentially because of the faithful
flatness of all maps of the form

$\mathbb{C}\\{u_{1},\ldots,u_{a}\\}[v_{1},\ldots,v_{b}]_{(u_{1},\ldots,u_{a},v_{1},\ldots,v_{b})}\to\mathbb{C}\\{u_{1},\ldots,u_{a},v_{1},\ldots,v_{b}\\},$

where $\mathbb{C}\\{u_{1},\ldots,u_{a}\\}$ and
$\mathbb{C}\\{u_{1},\ldots,u_{a},v_{1},\ldots,v_{b}\\}$ are the appropriate
convergent power series rings. Thus we can address questions of flatness in
the algebraic or partially algebraic setting without needing to work
explicitly with the analytifications of the objects involved.

###  2.2 Stratifications and the Milnor Fibration

In dealing with singular spaces from a set-theoretic perspective, we often
wish to apply the machinery of differential geometry, either by working with a
smooth ambient space or by decomposing the singular one into smooth pieces.
This latter approach gives rise to the theory of stratifications, and, in
particular, when equisingularity along the strata is required, Whitney
stratifications. Brief introductions to these topics can be found in Chapter 1
of [Dim92] and Chapter 1 of Part I of [GM88]; for a much more comprehensive
overview, see [Tro20] or [Tei82]. Here we recall a few facts and definitions
from stratification theory which will be particularly relevant in what follows
and use them to briefly discuss the Milnor fibration, introductory references
for which have already been provided in Section 1.

Since it is important to the statement of Theorem 1.1, we recall explicitly
the following relative version of the Whitney ($a$) condition:

######  Definition 2.8.

Let $X$ be a closed subset of a smooth manifold $M$ and $f$ the restriction to
$X$ of a differentiable map from $M$ to another smooth manifold $N$. We say
that a stratification of $X$ satisfies the Thom ($a_{f}$) condition on a
stratum $S$ with respect to another stratum $S^{\prime}$ such that
$S\subset\overline{S^{\prime}}$ if $f$ has constant rank on both strata and
the following condition holds: If $s$ is a point of $S$ and $s_{i}$ is a
sequence of points of $S^{\prime}$ approaching $s$ such that the limiting
tangent space $T_{f}:=\lim_{i\to\infty}T_{s_{i}}(S^{\prime}\cap
f^{-1}(f(s_{i})))$ to fibers of $f$ exists, then $T_{s}(S\cap
f^{-1}(f(s)))\subseteq T_{f}$. We say that the stratification satisfies this
condition if it does so on each stratum, and that a stratification of a map of
real- or complex-analytic spaces does so if it does so locally with respect to
the local embeddings into Euclidean space.

Some authors prefer to write this as “$A_{f}$” rather than “($a_{f}$)”. Unlike
Whitney stratifications, stratifications satisfying the Thom condition are not
guaranteed to exist. In [Hir77], particularly Section 5, Hironaka posits and
explores a nebulous relationship between the flatness of $f$ and the existence
of such a stratification, obtaining, among other things, the following result:

######  Proposition 2.9 ([Hir77], Section 5, Corollary 1).

Let $f:X\to Y$ be a complex-analytic map with $Y$ smooth of complex dimension
1. Then there exists a complex-analytic Whitney stratification of $f$ which
satisfies the Thom ($a_{f}$) condition.

Hironaka’s proof of this fact, which uses Nash-style modifications, is quite
involved — in other works, such as [BMM94], [HMS84], [LT88], [Gaf07], and
[GR19], it is more common to establish Thom conditions using relative conormal
spaces, which will be our approach for Theorem 1.1.

The utility of Thom stratifications for proving fibration theorems such as
Theorem 1.2 lies in the following well-known result, also due to Thom:

######  Lemma 2.10 (Thom’s first isotopy lemma [Mat70, Mat12]).

Let $X$ be a closed subset of a smooth manifold $M$ and $f:M\to N$ a map to
another smooth manifold $N$ whose restriction to $X$ is proper. Then, if $X$
has a Whitney stratification such that the restriction of $f$ to each stratum
is a smooth submersion, $f$ is a stratified-homeomorphically locally trivial
fibration whose trivializations are moreover diffeomorphic along each stratum.

For a long time the canonical reference on this result was Mather’s lecture
notes [Mat70]; these were eventually published with slight revisions as
[Mat12]. Among the lemma’s uses are the proof of a topological local
triviality statement for Whitney stratifications — see, e.g., Theorem 4.2.17
of [Tro20]. We mention it explicitly mainly to make the following note about
the exact nature of the trivializations involved:

######  Remark 2.11.

It is common to omit the statement about stratum-wise diffeomorphism entirely,
and state the lemma in terms of topological triviality alone. However, as
Goresky and MacPherson note in their statement of the lemma in Part I, Chapter
1, Section 5 of [GM88], the local trivialization homeomorphisms are indeed
smooth along each stratum — in fact, the proof in Mather’s notes shows that
the inverses are smooth as well, so they are even stratum-wise
diffeomorphisms. This is because the trivializations and their inverses arise
from the integration of controlled vector fields, which are by definition
smooth along each stratum and hence yield flows which are likewise stratum-
wise smooth. Thus the proof of Thom’s first isotopy lemma gives a stronger
result than is usually explicitly stated, even in [Mat70, Mat12].

In particular, Proposition 2.9 and Lemma 2.10 can be applied as in [Lê77] to
prove the existence of the following form of the Milnor fibration, sometimes
called the Milnor tube fibration or Milnor-Lê fibration:

######  Proposition 2.12 ([Mil68, Lê77]).

Let $f:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ be a germ of a holomorphic
function. Then, if we fix a small enough $\varepsilon>0$, we can choose
$\delta>0$ small enough relative to $\varepsilon$ so that the restriction

$f:B_{\varepsilon}\cap f^{-1}(D_{\delta}^{*})\to D_{\delta}^{*}$

defines a diffeomorphically locally trivial fibration, where
$B_{\varepsilon}\subset\mathbb{C}^{n+1}$ denotes the open ball of radius
$\varepsilon$ and $D_{\delta}^{*}\subset\mathbb{C}$ denotes the punctured open
disk of radius $\delta$. Moreover, this is independent of the $\varepsilon$
and $\delta$ involved, provided they are indeed chosen sufficiently small.

Lê’s proof, which we will generalize to prove Theorem 1.2, is essentially an
application of Proposition 2.9 to obtain the transversality of smooth fibers
with the boundary sphere of $B_{\varepsilon}$ necessary for the use of Lemma
2.10. Lemma 2.10 can likewise be used to give the independence of the
resulting fibration from the chosen $\varepsilon,\delta$; for a proof that it
is independent of the chosen coordinate system, see, e.g., Proposition 6.2.12
of [LNS20].

######  Remark 2.13.

The method of proof described above gives us a precise criterion for which
values of $\varepsilon$ are sufficiently small to be used to define the Milnor
fibration. Specifically, we require that $S_{\varepsilon}$ and all smaller
spheres centered at the origin are transverse to the strata of a Whitney
stratification of $f$ satisfying the ($a_{f}$) condition; Whitney’s condition
($b$) guarantees that this is indeed possible.

##  3 Algebraic Lemmas

Here we state and prove various results, essentially algebraic in nature,
which will be necessary for the proofs of Theorems 1.1 and 1.2.

###  3.1 Normal Cones and Infinitesimal Neighborhoods

To begin with, we discuss in more detail the properties of normal cones. We
claimed in Section 1 that the two notions of consistency of a closed embedding
given by considering the infinitesimal neighborhoods and normal cone
respectively were equivalent. We now prove this:

######  Lemma 3.1.

Let $\pi:X\to Y$ be a map of schemes (resp., complex-analytic spaces) and
$Z\subseteq X$ a closed subscheme (resp., complex-analytic subspace). Denote
by $\mathcal{I}$ the sheaf of ideals corresponding to $Z$. Then the following
are equivalent:

  * •

For each $k\geq 0$, the $k$th-order infinitesimal neighborhood of $Z$ in $X$
(that is, the closed subspace of $X$ given by the vanishing of
$\mathcal{I}^{k+1}$) is flat over $Y$.

  * •

The normal cone $C_{Z}X$ to $Z$ in $X$ is flat over $Y$.

Proof

As is noted in Remark 2.7 in the complex-analytic case and is immediate in the
algebraic case, the flatness of the normal cone is equivalent to that of the
associated graded sheaf of algebras
$\operatorname{gr}_{\mathcal{I}}\mathcal{O}_{X}$. Since the tensor product
distributes over direct sums, the flatness of the normal cone over $Y$ is
hence equivalent to that of $\mathcal{I}^{k}/\mathcal{I}^{k+1}$ over
$\mathcal{O}_{Y}$ for all $k\geq 0$, which is to say of the flatness of
$(\mathcal{I}^{k}/\mathcal{I}^{k+1})_{p}$ over $R:=\mathcal{O}_{Y,\pi(p)}$ for
all $k\geq 0$ at each point $p$ of $X$. Hence we must show that this is
equivalent to the flatness of $(\mathcal{O}_{X}/\mathcal{I}^{k+1})_{p}$ over
$R$ for each $k\geq 0$ at each such point; hence we fix such a $p$ for the
remainder of the proof.

In general, the flatness of a module $M$ over $R$ is equivalent to the
vanishing of $\operatorname{Tor}_{1}^{R}(M,N)$ for all $R$-modules $N$ and
hence of $\operatorname{Tor}_{i}^{R}(M,N)$ for all integers $i\geq 1$ and all
$R$-modules $N$. We can then prove each direction of the equivalence by
applying the long exact sequence in Tor to the short exact sequences

$0\to(\mathcal{I}^{k}/\mathcal{I}^{k+1})_{p}\to(\mathcal{O}_{X}/I^{k+1})_{p}\to(\mathcal{O}_{X}/\mathcal{I}^{k})_{p}\to
0$

for $k\geq 0$; this gives us the exactness of the sequences

$\operatorname{Tor}_{2}^{R}\left(\left(\tfrac{\mathcal{O}_{X}}{\mathcal{I}^{k}}\right)_{p},N\right)\to\operatorname{Tor}_{1}^{R}\left(\left(\tfrac{\mathcal{I}^{k}}{\mathcal{I}^{k+1}}\right)_{p},N\right)\to\operatorname{Tor}_{1}^{R}\left(\left(\tfrac{\mathcal{O}_{X}}{\mathcal{I}^{k+1}}\right)_{p},N\right)\to\operatorname{Tor}_{1}^{R}\left(\left(\tfrac{\mathcal{O}_{X}}{\mathcal{I}^{k}}\right)_{p},N\right)$

for any $R$-module $N$. It is then immediate that the flatness of the
infinitesimal neighborhoods implies that of the normal cone, and the reverse
implication can be proven by induction on $k$.

Hence we can pass freely back and forth between the two notions — we will see
that the flatness of the infinitesimal neighborhoods seems to arise more
naturally as a desirable condition in proofs of other facts, whereas the
flatness of the normal cone is much easier to establish by computations
insofar as it involves a single finitely-presented algebra instead of a
countably infinite family of modules.

In dealing with the normal cone, it is often useful to know that it will
behave nicely under pullback. The following result uses the prior one to give
sufficient conditions for this to occur:

######  Lemma 3.2.

Let $X\to Y$ be a map of schemes (resp., complex-analytic spaces) and
$Z\subseteq X$ a closed subscheme (resp., complex-analytic subspace). Let
$\phi:Y^{\prime}\to Y$ be a map of schemes (resp., complex-analytic spaces)
and suppose that either of the following holds:

  1. (i)

$\phi$ is a flat morphism.

  2. (ii)

$C_{Z}X$ is flat over $Y$.

Then $\phi^{*}(C_{Z}X)=C_{\phi^{*}Z}(\phi^{*}X)$.

Proof

Let $\mathcal{I}$ be the sheaf of ideals on $X$ corresponding to $Z$ and
$\mathcal{I}^{\prime}$ the one on $X^{\prime}:=\phi^{*}X$ corresponding to
$Z^{\prime}:=\phi^{*}Z$. By the anti-equivalence of Proposition 2.1, it is
enough to show that the natural map
$\phi^{*}(\operatorname{gr}_{\mathcal{I}}\mathcal{O}_{X})\to\operatorname{gr}_{\mathcal{I}^{\prime}}\mathcal{O}_{X^{\prime}}$
of sheaves of algebras on $X^{\prime}$ is an isomorphism. This can be checked
separately on each graded piece — that is, it is enough to show that the
natural map
$\phi^{*}(\mathcal{I}^{k}/\mathcal{I}^{k+1})\to(\mathcal{I}^{\prime})^{k}/(\mathcal{I}^{\prime})^{k+1}$
is an isomorphism.

Now note that the formation of the infinitesimal neighborhoods of $Z$ in $X$
commutes with pullback — that is, for each $k\geq 0$,
$\phi^{*}(\mathcal{O}_{X}/\mathcal{I}^{k})\cong\mathcal{O}_{X^{\prime}}/(\mathcal{I}^{\prime})^{k}$.
Hence, by considering for each $k\geq 0$ the exact sequence

$0\to\mathcal{I}^{k}/\mathcal{I}^{k+1}\to\mathcal{O}_{X}/\mathcal{I}^{k+1}\to\mathcal{O}_{X}/\mathcal{I}^{k}\to
0$

and applying $\phi^{*}$, we see that it is enough to verify that the map
$\mathcal{I}^{k}/\mathcal{I}^{k+1}\to\mathcal{O}_{X}/\mathcal{I}^{k+1}$
remains injective on pullback. This is immediate if our Condition (i) holds;
if Condition (ii) holds, it follows from the long exact sequence in Tor by
Lemma 3.1.

######  Remark 3.3.

In particular, Lemma 3.2 allows us to compute normal cones algebraically when
we are dealing with complex-analytic spaces defined by polynomials — by
applying the result together with the faithful flatness of the appropriate
maps of local rings as in Remark 2.7, we find that the formation of the normal
cone will commute with analytification.

This lemma is, of course, also useful in the complex-analytic context, where
it allows us to manipulate normal cones more freely in general.

As one example of the aforementioned tendency of the flatness of the
infinitesimal neighborhoods to show up more readily in theoretical situations,
we give the following result, which links the flatness of infinitesimal
neighborhoods of a family’s critical locus to the study of the partial
differential equations satisfied by the functions in the family and plays a
role in the proof of Theorem 1.2:

######  Lemma 3.4.

Let $n,u\geq 0$ be integers, $F:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C},0)$ a
holomorphic function germ, and
$\pi:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C}^{u},0)$ the projection onto the last
$u$ coordinates. Suppose $f:=F|_{\mathbb{C}^{n+1}\times 0}$ is non-constant
and denote by $(C_{F\times\pi},0)$ the vanishing locus of the ideal
$J_{F\times\pi}$ of maximal minors of the Jacobian matrix of $F\times\pi$.

Then, for each integer $d\geq 1$, the $(d-1)$st-order infinitesimal
neighborhood of $(C_{F\times\pi},0)$ in $(\mathbb{C}^{n+1+u},0)$ is flat over
$(\mathbb{C}^{u},0)$ if and only if every homogeneously first-order
homogeneous degree-$d$ partial differential equation

$\sum_{\alpha_{0}+\cdots+\alpha_{n}=d}c_{\alpha_{0},\ldots,\alpha_{n}}\left(\frac{\partial
f}{\partial x_{0}}\right)^{\alpha_{0}}\cdots\left(\frac{\partial f}{\partial
x_{n}}\right)^{\alpha_{n}}=0$

satisfied by $f$ (with
$c_{\alpha_{0},\ldots,\alpha_{n}}:(\mathbb{C}^{n+1},0)\to\mathbb{C}$
holomorphic function germs) can be extended to a family

$\sum_{\alpha_{0}+\cdots+\alpha_{n}=d}C_{\alpha_{0},\ldots,\alpha_{n}}\left(\frac{\partial
F}{\partial x_{0}}\right)^{\alpha_{0}}\cdots\left(\frac{\partial F}{\partial
x_{n}}\right)^{\alpha_{n}}=0$

of such equations satisfied by $F$ (with
$C_{\alpha_{0},\ldots,\alpha_{n}}:(\mathbb{C}^{n+1+u},0)\to\mathbb{C}$
holomorphic function germs which restrict to the
$c_{\alpha_{0},\ldots,\alpha_{n}}$ on $(\mathbb{C}^{n+1}\times 0,0)$).

Proof

We can see by considering the Jacobian of $F\times\pi$ that the flatness
statement can be written more explicitly as the requirement that the map

$\mathbb{C}\\{y_{1},\ldots,y_{u}\\}\to
S:=\frac{\mathbb{C}\\{x_{0},\ldots,x_{n},y_{1},\ldots,y_{u}\\}}{(\tfrac{\partial
F}{\partial x_{0}},\ldots,\tfrac{\partial F}{\partial x_{n}})^{d}}$

of convergent power series rings be flat. By the local criterion for flatness
(e.g., Theorem 6.8 of [Eis04]), this is equivalent to the vanishing of
$\operatorname{Tor}_{1}^{\mathbb{C}\\{y_{1},\ldots,y_{u}\\}}(S,\mathbb{C})$.
Since the ring $R:=\mathbb{C}\\{x_{0},\ldots,x_{n},y_{1},\ldots,y_{u}\\}$ is
flat over $\mathbb{C}\\{y_{1},\ldots,y_{u}\\}$, we can compute this using a
free resolution

$\cdots\to R^{\oplus\ell}\to R^{\oplus\binom{n+d}{d}}\to R\to S\to 0$

over $R$ of $S$, where each summand of $R^{\oplus\binom{n+d}{d}}$ corresponds
to a degree-$d$ monomial in the $n+1$ variables $\xi_{0},\ldots,\xi_{n}$ and
the map $R^{\oplus\binom{n+d}{d}}\to R$ is given by evaluation of each
monomial at $\left(\tfrac{\partial F}{\partial x_{0}},\ldots,\tfrac{\partial
F}{\partial x_{n}}\right)$. Letting $\mathfrak{m}=(y_{1},\ldots,y_{u})$ be the
maximal ideal of $\mathbb{C}\\{y_{1},\ldots,y_{u}\\}$, we then see that the
flatness statement is equivalent to the exactness of the bottom row of the
diagram

${\cdots}$${R^{\oplus\ell}}$${R^{\oplus\binom{n+d}{d}}}$${R}$${0}$${\cdots}$${(R/\mathfrak{m}R)^{\oplus\ell}}$${(R/\mathfrak{m}R)^{\oplus\binom{n+d}{d}}}$${R/\mathfrak{m}R}$${0}$

at the term $(R/\mathfrak{m}R)^{\oplus\binom{n+d}{d}}$.

Now note that we can identify the kernel $E$ of the map
$(R/\mathfrak{m}R)^{\oplus\binom{n+d}{d}}\to R/\mathfrak{m}R$ with the set of
equations of the sort we are considering satisfied by $f$. Likewise, we can
interpret the kernel $\tilde{E}$ of $R^{\oplus\binom{n+d}{d}}\to R$ as the set
of families of equations of the sort described in the statement of the lemma
satisfied by $F$.

Hence we can see that our flatness condition is equivalent to the requirement
that the bottom arrow of the commutative square

${R^{\oplus\ell}}$${\tilde{E}}$${(R/\mathfrak{m}R)^{\oplus\ell}}$${E}$

be surjective, where the top arrow is surjective by the exactness of the
resolution. However, a simple diagram chase then tells us that the bottom
arrow and the rightmost vertical arrow have exactly the same image, so in
particular one will be surjective if and only if the other is. This proves the
result.

We will not require the full power of this lemma — the $d=1$ case will be
sufficient to demonstrate the facts about vector fields necessary for
Condition (i) of Theorem 1.2 — but the perspective it offers is intriguing and
potentially worthy of further study in the future.

###  3.2 Relative Conormal Spaces

To prove Theorem 1.1, we will relate the flatness of the normal cone to
properties of the relative conormal space, which can be used to track the
limiting tangent planes relevant to the Thom condition. Recall by Proposition
2.5 that the relative conormal of a map $\Phi:X\to Y$ of complex manifolds is
given by the image of the map
$\operatorname{Sym}(D\Phi):\operatorname{Sym}(\Theta_{X})\to\operatorname{Sym}(\Phi^{*}\Theta_{Y})$
of sheaves of algebras, and note in particular that
$\operatorname{Sym}(\Theta_{X})$ and $\operatorname{Sym}(\Phi^{*}\Theta_{Y})$
are flat over $X$.

It then behooves us to establish the following general result, which says
essentially that the behavior on pullback of a map of flat objects should be
controlled by its cokernel:

######  Lemma 3.5.

Let $R$ be a ring, $0\to A_{3}\to A_{2}\to A_{1}\to A_{0}\to 0$ an exact
sequence of $R$-modules such that $A_{1}$ and $A_{2}$ are flat over $R$, and
$B$ another $R$-module. Then the following hold:

  * •

$H_{2}(A_{\bullet}\otimes_{R}B)\cong\operatorname{Tor}_{1}^{R}(A_{0},B)$.

  * •

$H_{3}(A_{\bullet}\otimes_{R}B)\cong\operatorname{Tor}_{2}^{R}(A_{0},B)$.

  * •

$\operatorname{Tor}_{i}^{R}(A_{3},B)\cong\operatorname{Tor}_{i+2}(A_{0},B)$
for all integers $i\geq 1$.

Proof

Let $\cdots\to F_{2}\to F_{1}\to F_{0}\to B\to 0$ be a flat resolution of $B$.
We can then form a first-quadrant double complex
$A_{\bullet}\otimes_{R}F_{\bullet}$ whose $(i,j)$th entry is $A_{i}\otimes
F_{j}$ using the usual sign trick (see, e.g., Section 2.7 of [Wei94]) and
consider both of the corresponding spectral sequences (e.g., Section 5.6 of
[Wei94]). By the flatness of the $F_{j}$ and the exactness of $A_{\bullet}$,
the rows of this double complex are exact, so the convergence of the spectral
sequence given by the filtration by rows tells us that the homology of the
associated total complex vanishes.

On the other hand, if we take the spectral sequence given by the filtration by
columns, we find that the $E^{1}$ page has the following form:

${\vdots}$${\vdots}$${\vdots}$${\vdots}$${0}$${\operatorname{Tor}_{2}(A_{0},B)}$${0}$${0}$${\operatorname{Tor}_{2}(A_{3},B)}$${0}$${0}$${\operatorname{Tor}_{1}(A_{0},B)}$${0}$${0}$${\operatorname{Tor}_{1}(A_{3},B)}$${0}$${0}$${\operatorname{Tor}_{0}(A_{0},B)}$${\operatorname{Tor}_{0}(A_{1},B)}$${\operatorname{Tor}_{0}(A_{2},B)}$${\operatorname{Tor}_{0}(A_{3},B)}$${0}$

By considering the next few pages and using the convergence of this spectral
sequence to the homology of the associated total complex, which we have
already demonstrated is trivial, we obtain the desired isomorphisms.

There has been a fair amount of interest in the literature in results which
characterize the behavior of the relative conormal space upon specialization,
for example to the fiber over a point in the base — see, e.g., Corollary 4.2.1
of [HMS84], Proposition 1.2.6 of [LT88], Theorem 1.2.1 of [HM96], and Lemma
1.1 of [GM18]. Lemma 3.5 gives us control of this behavior on specialization
along maps to the parameter space of a family using the flatness of the normal
cone of the critical locus:

######  Lemma 3.6.

Let $n,u\geq 0$ be integers, $F:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C},0)$ a
holomorphic function germ, and
$\pi:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C}^{u},0)$ the projection onto the last
$u$ coordinates. Suppose $f:=F|_{\mathbb{C}^{n+1}\times 0}$ is non-constant
and denote by $(C_{F\times\pi},0)$ the vanishing locus of the ideal
$J_{F\times\pi}$ of maximal minors of the Jacobian matrix of $F\times\pi$.

Suppose that the normal cone to $(C_{F\times\pi},0)$ in
$(\mathbb{C}^{n+1+u},0)$ is flat over $(\mathbb{C}^{u},0)$. Then the
restriction over $0\in\mathbb{C}^{u}$ of the relative conormal space of
$F\times\pi$ satisfies
$T_{F\times\pi}^{*}\mathbb{C}^{n+1+u}|_{\pi^{-1}(0)}\cong
T_{f}^{*}\mathbb{C}^{n+1}\times\mathbb{C}^{u}$ compatibly with the
identification $T^{*}\mathbb{C}^{n+1+u}|_{\pi^{-1}(0)}\cong
T^{*}\mathbb{C}^{n+1}\times\mathbb{C}^{u}$.

Proof

By using Proposition 2.5 and considering the structure of the Jacobian matrix
in this case, we find that the relative conormal space in this case is given
as a cone over the space germ $(\mathbb{C}^{n+1+u},0)$ by the graded algebra
which is the image of the map

$\tilde{\phi}:\mathcal{O}_{\mathbb{C}^{n+1+u},0}[\xi_{0},\ldots,\xi_{n},\upsilon_{1},\ldots,\upsilon_{u}]\to\mathcal{O}_{\mathbb{C}^{n+1+u},0}[\tau,\upsilon_{1},\ldots,\upsilon_{u}]$

which sends each $\xi_{i}$ to $\tfrac{\partial F}{\partial x_{i}}\tau$.

Now, we can see that this map is in fact given by applying the functor
$-\otimes_{\mathcal{O}_{\mathbb{C}^{n+1+u},0}}\mathcal{O}_{\mathbb{C}^{n+1+u},0}[\upsilon_{1},\ldots,\upsilon_{u}]$,
which corresponds under Proposition 2.1 to the product over
$(\mathbb{C}^{n+1+u},0)$ with a trivial rank-$u$ vector bundle, to the
corresponding map

$\phi:\mathcal{O}_{\mathbb{C}^{n+1+u},0}[\xi_{0},\ldots,\xi_{n}]\to\mathcal{O}_{\mathbb{C}^{n+1+u},0}[\tau].$

Likewise, if we regard $\mathcal{O}_{\mathbb{C}^{n+1},0}$ as a quotient of
$\mathcal{O}_{\mathbb{C}^{n+1+u},0}$ by the coordinate functions of
$(\mathbb{C}^{u},0)$, we find that $T_{f}^{*}\mathbb{C}^{n+1}$ is given by the
image of the induced map
$\phi\otimes_{\mathcal{O}_{\mathbb{C}^{n+1+u},0}}\mathcal{O}_{\mathbb{C}^{n+1},0}$
in the quotient ring.

It thus suffices to show that the formation of the image in this case commutes
with passage to the quotient ring. Regarding $\phi$’s domain and codomain as
graded $\mathcal{O}_{\mathbb{C}^{n+1+u},0}$-modules, we see that

$\operatorname{coker}\phi\cong\bigoplus_{i\geq
0}\frac{\mathcal{O}_{\mathbb{C}^{n+1+u},0}}{{J_{F\times\pi}}^{i}}\tau^{i}.$

By Lemma 3.1, we can then conclude that the flatness of the normal cone over
$(\mathbb{C}^{u},0)$ implies that of $\operatorname{coker}\phi$. This is to
say that, for every $\mathcal{O}_{\mathbb{C}^{u},0}$-module $M$ and $i\geq 1$,
$\operatorname{Tor}_{i}^{\mathcal{O}_{\mathbb{C}^{u},0}}(\operatorname{coker}\phi,M)=0$.

Observing that
$\mathcal{O}_{\mathbb{C}^{u},0}\to\mathcal{O}_{\mathbb{C}^{n+1+u},0}$ is flat
and hence that $\mathcal{O}_{\mathbb{C}^{n+1+u},0}[\xi_{0},\ldots,\xi_{n}]$
and $\mathcal{O}_{\mathbb{C}^{n+1+u},0}[\tau]$ are flat
$\mathcal{O}_{\mathbb{C}^{u},0}$-modules, we can then apply Lemma 3.5 to the
exact sequence

$0\to\ker\phi\to\mathcal{O}_{\mathbb{C}^{n+1+u},0}[\xi_{0},\ldots,\xi_{n}]\xrightarrow{\phi}\mathcal{O}_{\mathbb{C}^{n+1+u},0}[\tau]\to\operatorname{coker}\phi\to
0$

to find that the sequence remains exact upon tensor over
$\mathcal{O}_{\mathbb{C}^{u},0}$ by any module. In particular, tensor with the
residue field of $\mathcal{O}_{\mathbb{C}^{u},0}$ gives us
$\phi\otimes_{\mathcal{O}_{\mathbb{C}^{n+1+u},0}}\mathcal{O}_{\mathbb{C}^{n+1},0}$
in place of $\phi$; the exactness of the whole sequence then tells us that the
desired result holds.

This result will form the crux of our proof of Theorem 1.1.

##  4 Proof of the Stratification Theorem

We are now in a position to prove Theorem 1.1; once we have applied Lemma 3.6,
our approach will follow the proof of the existence of Thom stratifications in
[BMM94] — see also the proof of Proposition 8.3.10 of [KS94], Proposition
10.6.5 of [KS79], or Proposition 1.3.5 of [LT88].

Proof

We first observe that the non-flat locus of $C_{C_{F\times\pi}}X$ over
$\mathbb{C}^{u}$ in $X$ is indeed a complex-analytic subset of $X$. By, e.g.,
Theorem IV.9 of [Fri67], the non-flat locus in $C_{C_{F\times\pi}}X$ itself is
analytic. Moreover, since the complement of the zero section in
$C_{C_{F\times\pi}}X$ is a $\mathbb{C}^{*}$-bundle, and hence faithfully flat,
over the projectivized normal cone $\mathbb{P}C_{C_{F\times\pi}}X$, we can see
that any failure of flatness at a point off the zero section of the normal
cone will induce similar failures at every point in the same fiber of the
bundle map and hence at the corresponding point of the zero section itself by
the aforementioned analyticity. Thus, if we identify $C_{F\times\pi}$ with the
zero section of the normal cone, we can see that the non-flat locus in $X$ we
seek is simply its intersection with the non-flat locus in the normal cone
itself, hence a complex-analytic subset of $X$ as claimed.

Now consider the projectivization $\mathbb{P}T_{F\times\pi}^{*}X$ of the
relative conormal space. The projection $\mathbb{P}T_{F\times\pi}^{*}X\to X$,
being a proper complex-analytic map of complex-analytic spaces, admits a
complex-analytic Whitney stratification by, e.g., Theorem 1 of Section 4 of
[Hir77]; the method of proof of that theorem shows that this can be taken such
that $F\times\pi$ has constant rank on each stratum of $X$, the aforementioned
non-flat locus is a union of strata, and $X\setminus C_{F\times\pi}$ is the
ambient stratum — this last condition can be met since
$\mathbb{P}T_{F\times\pi}^{*}X\to X$ is a $\mathbb{P}^{u}$-bundle, and hence a
surjective submersion, over $X\setminus C_{F\times\pi}$. We then claim that
each stratum outside the non-flat locus satisfies the Thom ($a_{F\times\pi}$)
condition with respect to this ambient stratum.

To show this, consider such a stratum $S$ and observe that, since $S$ is
contained in the critical locus of $F\times\pi$ and $\pi$ itself has full rank
everywhere on $X$, the partial derivatives of $F$ with respect to the
directions tangent to the fibers of $\pi$ must all vanish identically on $S$.
Hence the fibers of $(F\times\pi)|_{S}$ are exactly those of $\pi|_{S}$ — as
such, we consider a point $q\in\mathbb{C}^{u}$ and let $M=S\cap\pi^{-1}(q)$.
This is a manifold because of our stipulation that $F\times\pi$ have constant
rank on $S$; it remains to show that $M$’s tangent space at each point is
contained in the intersection of all limiting tangent spaces to smooth points
of fibers of $F\times\pi$. This is equivalent to the requirement that the
conormal bundle $T_{M}^{*}X$ of $M$ in $X$ set-theoretically contain the
restriction of the relative conormal space $T_{F\times\pi}^{*}X$ over $M$.

Since $M$ is contained in $\pi^{-1}(q)$, we have a short exact sequence of
vector bundles

$0\to T_{\pi^{-1}(q)}^{*}X|_{M}\to T_{M}^{*}X\to T_{M}^{*}\pi^{-1}(q)\to 0,$

which has a splitting given by the natural product decomposition of the
cotangent bundle of $X^{\prime}\times\mathbb{C}^{u}$. Hence, since
$T_{\pi^{-1}(q)}^{*}X|_{M}$ is a trivial vector bundle of rank $u$, we find
that $T_{M}^{*}X\cong T_{M}^{*}\pi^{-1}(q)\times\mathbb{C}^{u}$ compatibly
with the identification $T^{*}X|_{\pi^{-1}(q)}\cong
T^{*}\pi^{-1}(q)\times\mathbb{C}^{u}$. Applying Lemma 3.6 at each point, which
is possible since we are outside the non-flat locus, then tells us that it is
enough to show $T_{f}^{*}\pi^{-1}(q)|_{M}\subseteq T_{M}^{*}\pi^{-1}(q)$ set-
theoretically, where $f:=F|_{\pi^{-1}(q)}$.

Let $T^{\prime}$ be any stratum of $\mathbb{P}T_{F\times\pi}^{*}X$ mapped
(surjectively and submersively) to $S$. Denote by $X\times 0$ the zero section
of the cotangent bundle, and define $T$ to be the inverse image of
$T^{\prime}$ under the $\mathbb{C}^{*}$-bundle map
$(T_{F\times\pi}^{*}X)\setminus(X\times 0)\to\mathbb{P}T_{F\times\pi}^{*}X$.
Since the projection $\mathbb{P}T_{F\times\pi}^{*}X\to X$ is a proper
stratified submersion, Thom’s first isotopy lemma (2.10) tells us that it is
smoothly locally trivial along each stratum. In particular, we can see that
the inverse image $N$ of $M\subseteq S$ in $T$ is a complex-analytic manifold
mapped surjectively and submersively to $M$.

Now, since $M$ is contained in $C_{F\times\pi}\cap\pi^{-1}(q)$, we can see in
particular that it is contained in a single fiber of $f$. By Theorem 3.4.2 of
[BMM94] (see also Theorem 3.1 of [GM18]), the restriction of
$T_{f}^{*}\pi^{-1}(q)$ over this fiber of $f$ is set-theoretically the
characteristic variety of a regular holonomic $D$-module and hence Lagrangian.
In particular, it is isotropic. Thus, if we work around a point of $M$ and
choose local coordinates $x_{0},\ldots,x_{n}$ for $\pi^{-1}(q)$ such that $M$
is locally cut out by the vanishing of the coordinates $x_{0},\ldots,x_{c-1}$,
the vanishing of the canonical 1-form $\xi_{0}dx_{0}+\ldots+\xi_{n}dx_{n}$ on
$T_{f}^{*}\pi^{-1}(q)|_{M}$ gives us the vanishing of the 1-form
$\xi_{c}dx_{c}+\ldots+\xi_{n}dx_{n}$ on $N\subset T_{f}^{*}\pi^{-1}(q)|_{M}$.
Since $dx_{c},\ldots,dx_{n}$ are linearly independent on $M$ and hence on $N$,
this implies that $\xi_{c},\ldots,\xi_{n}$ vanish on $N$, which is to say that
$N\subseteq T_{M}^{*}\pi^{-1}(q)$.

Repeating this reasoning for each such $T^{\prime}$ — and hence each such $N$
— demonstrates that the complement of the zero section in
$T_{f}^{*}\pi^{-1}(q)|_{M}$ is set-theoretically contained in
$T_{M}^{*}\pi^{-1}(q)$. Since this complement is set-theoretically dense in
$T_{f}^{*}\pi^{-1}(q)|_{M}$ and $T_{M}^{*}\pi^{-1}(q)$ is closed in
$T^{*}\pi^{-1}(q)|_{M}$, the result follows.

##  5 Proof of the Fibration Theorem

We can now prove Theorem 1.2 by applying either Lemma 3.4 or Theorem 1.1 to
obtain the transversality necessary for Lemma 2.10:

Proof

Let $\Gamma,E>0$ be small enough that $F$ is defined on $B_{E}\times
B_{\Gamma}\subset\mathbb{C}^{n+1}\times\mathbb{C}^{u}$ and note that
$C_{F\times\pi}$ thus has a well-defined representative as a closed complex-
analytic subspace of $B_{E}\times B_{\Gamma}$. For any $0<r<E$, let $S_{r}$
denote the sphere of radius $r$ centered at the origin in $\mathbb{C}^{n+1}$.
We will begin by arguing that, for small enough $0<\varepsilon<E$, we can
produce for each $p\in C_{F\times\pi}\cap(S_{\varepsilon}\times 0)$ a
neighborhood $U_{p}$ of $p$ in $B_{E}\times B_{\Gamma}$ such that, for each
$q\in U_{p}\setminus C_{F\times\pi}$, the fiber
$(F\times\pi)^{-1}((F\times\pi)(q))$ meets $S_{|q|}\times B_{\Gamma}$
transversally at $q$.

Suppose first that Condition (i) holds. Then, if we make $\varepsilon$ small
enough, we have for each such $p$ a holomorphic vector field germ $V_{p}$ on
$\mathbb{C}^{n+1}\times 0$ at $p$ satisfying the stated hypotheses. Moreover,
the condition tells us that $0\times 0$ is either not contained in the non-
flat locus of $C_{F\times\pi}$ over $B_{\Gamma}$ or is an isolated point of
its intersection with $B_{E}\times 0$, so we can suppose by further shrinking
$\varepsilon$ that $C_{F\times\pi}$ is flat over $B_{\Gamma}$ at each $p$. We
can then apply Lemma 3.4 in the case $d=1$ to the differential equation
$V_{p}f=0$ to obtain a germ at $p$ of a holomorphic vector field
$\tilde{V}_{p}$ on $B_{E}\times B_{\Gamma}$ such that $\tilde{V}_{p}F=0$,
$\tilde{V}_{p}$ is everywhere tangent to the fibers of the projection $\pi$,
and $\tilde{V}_{p}|_{B_{E}\times 0}=V_{p}$.

By hypothesis, the tangent vector given by $V_{p}$ at $p$ is not contained in
the tangent space $T_{p}S_{\varepsilon}$. By the openness of non-containment,
we can then let $U_{p}$ be a neighborhood of $p$ in $B_{E}\times B_{\Gamma}$
small enough that $\tilde{V}_{p}$ has a representative defined on $U_{p}$ and
the tangent vector given by this representative at each point $q$ of $U_{p}$
is not contained in the corresponding tangent space to $S_{|q|}\times
B_{\Gamma}$. Since $\tilde{V}_{p}$ is everywhere tangent to the smooth fibers
of $F\times\pi$, this $U_{p}$ thus has the desired transversality property.

Now suppose instead that Condition (ii) holds. As before, the flatness
hypothesis tells us that $0\times 0$ is either not contained in the non-flat
locus of the normal cone $C_{C_{F\times\pi}}(B_{E}\times B_{\Gamma})$ over
$B_{\Gamma}$ or is an isolated point of its intersection with $B_{E}\times 0$.
We can then apply Theorem 1.1 to obtain a Whitney stratification of
$B_{E}\times B_{\Gamma}$ such that the non-flat locus is a union of strata and
the Thom ($a_{F\times\pi}$) condition with respect to the ambient stratum
$(B_{E}\times B_{\Gamma})\setminus C_{F\times\pi}$ holds for the strata
outside of it.

Applying Whitney’s condition ($b$) to lines through the origin as in Remark
2.13, we can then see that, for sufficiently small $0<\varepsilon<E$, each
point $p$ of $C_{F\times\pi}\cap(S_{\varepsilon}\times 0)$ lies on a stratum
outside the non-flat locus which is transverse to $S_{\varepsilon}\times
B_{\Gamma}$ at $p$. Using the fact that this stratum satisfies the Thom
condition with respect to the ambient stratum, we find by the openness of non-
containment that the smooth points $q$ of fibers of $F\times\pi$ in a
sufficiently small neighborhood $U_{p}$ of $p$ are transverse to the
corresponding submanifolds $S_{|q|}\times B_{\Gamma}$, as desired.

Hence our hypotheses guarantee in all cases the existence of neighborhoods
$U_{p}$ with the properties claimed. Now suppose toward a contradiction that
we have a sequence of points of $S_{\varepsilon}\times B_{\Gamma}$ whose
images under $F\times\pi$ are contained in the complement of the discriminant
closure $\bar{\Delta}$ and approach $0\times 0\in\mathbb{C}\times B_{\Gamma}$
such that, at each point of the sequence, the fiber of $F\times\pi$ fails to
be transverse to $S_{\varepsilon}\times B_{\Gamma}$. By the compactness of
$S_{\varepsilon}\times\bar{B}_{\Gamma}$, this sequence has a convergent
subsequence, and we can see that the point $p$ to which it converges must lie
in $S_{\varepsilon}\times 0$. Indeed, if $\varepsilon$ is small enough, we
must have $p\in C_{F\times\pi}$ as well, since the spheres $S_{\varepsilon}$
will be eventually transverse to the regular locus of
$(F\times\pi)^{-1}(0\times 0)$ by, e.g., the Generic Whitney Lemma applied
with Whitney’s condition ($b$) as before and non-containment, as previously
noted, is an open condition. Thus the definition of convergence implies that
the sequence must be eventually contained in the corresponding neighborhood
$U_{p}$. This gives the desired contradiction by the construction of $U_{p}$.

Hence we can pick $\delta>0$ and $0<\gamma<\Gamma$ such that no such points
have image in $D_{\delta}\times B_{\gamma}$. Then the restriction

$F\times\pi:(\bar{B}_{\varepsilon}\times
B_{\gamma})\cap(F\times\pi)^{-1}((D_{\delta}\times
B_{\gamma})\setminus\bar{\Delta})\to(D_{\delta}\times
B_{\gamma})\setminus\bar{\Delta}$

is a stratified submersion when we give the domain the stratification with
strata $(B_{\varepsilon}\times
B_{\gamma})\cap(F\times\pi)^{-1}((D_{\delta}\times
B_{\gamma})\setminus\bar{\Delta})$ and $(S_{\varepsilon}\times
B_{\gamma})\cap(F\times\pi)^{-1}((D_{\delta}\times
B_{\gamma})\setminus\bar{\Delta})$. Moreover, it is proper, since it is a
restriction over a subset of the target of the proper map
$F\times\pi:\bar{B}_{\varepsilon}\times\bar{B}_{\gamma}\to\mathbb{C}\times\bar{B}_{\gamma}$.
As such, we can apply Thom’s first isotopy lemma (2.10) to show that it is a
locally trivial fibration both topologically and diffeomorphically along each
stratum. Discarding the boundary stratum gives us the original result.

##  6 Global Consequences

If $f:\mathbb{C}^{n+1}\to\mathbb{C}$ is a homogeneous polynomial, the Milnor
fibration has a global interpretation as well (see, e.g., Chapter 10 of
[Max19] or Example 3.1.9ff. of [Dim92]). Specifically, the restriction

$f:\mathbb{C}^{n+1}\setminus V(f)\to\mathbb{C}^{*}$

defines a smooth locally trivial fibration, called the global or affine Milnor
fibration of $f$. It is not difficult to show that this is fiber diffeomorphic
to the usual local Milnor fibration of $f$ at the origin.

Hence any homogeneous polynomial of degree $d$ has a well-defined smooth
fiber, which comes with a monodromy action given by the action of a primitive
$d$th root of unity on the polynomial’s level sets, and these are together
equivalent to the corresponding Milnor fibration information at the origin. We
can then use Theorem 1.2 to obtain results about these global objects:

######  Corollary 6.1.

Let $n\geq 0$ and $d\geq 1$ be integers, $Y$ a connected complex-analytic
space, and

$F:\mathbb{C}^{n+1}\times Y\to\mathbb{C}$

a holomorphic map such that, for each point $y\in Y$, the map
$f_{y}:=F|_{\mathbb{C}^{n+1}\times y}$ is a nonzero homogeneous degree-$d$
polynomial. Let $\mathcal{J}_{F\times\pi}$ be the sheaf of ideals of
$\mathcal{O}_{\mathbb{C}^{n+1}\times Y}$ generated by the functions
$\tfrac{\partial F}{\partial x_{0}},\ldots,\tfrac{\partial F}{\partial
x_{n}}$, and denote by $\Sigma_{F\times\pi}$ its fiberwise-projective
vanishing in $\mathbb{P}^{n}\times Y$.

Let $C_{\Sigma_{F\times\pi}}(\mathbb{P}^{n}\times Y)$ be the normal cone and
suppose the natural projection $C_{\Sigma_{F\times\pi}}(\mathbb{P}^{n}\times
Y)\to Y$ is flat. Then the fibrations over $\mathbb{C}^{*}$ induced by the
$f_{y}$ as $y$ varies throughout $Y$ are all fiber diffeomorphic to one
another.

Proof

We first observe that, by pulling back along a resolution of singularities
(see [Wło09]) $\tilde{Y}\to Y$, we may assume $Y$ is smooth — this may result
in a parameter space $\tilde{Y}$ with multiple connected components, but,
since the fibration over any point of $\tilde{Y}$ is the same as the one over
the corresponding point of the connected space $Y$, proving the constancy
along each connected component will be sufficient to get consistency
everywhere. Since the formation of the ideal $\mathcal{J}_{F\times\pi}$
commutes with pullback along maps into $Y$ in general and the flatness
hypothesis guarantees that the formation of the normal cone will do so as well
by Lemma 3.2, the passage from $Y$ to $\tilde{Y}$ does not affect our
hypotheses.

Let $C_{F\times\pi}:=V(\mathcal{J}_{F\times\pi})$ be the critical locus. Then
$C_{F\times\pi}\setminus(0\times Y)$ is pullback of $\Sigma_{F\times\pi}$
along the natural $\mathbb{C}^{*}$-bundle map $(\mathbb{C}^{n+1}\times
Y)\setminus(0\times Y)\to\mathbb{P}^{n}\times Y$ given by sending each point
of $\mathbb{C}^{n+1}\setminus 0$ to the point of $\mathbb{P}^{n}$
corresponding to the line through the origin it lies on. In particular, since
the bundle map is flat, the formation of the normal cone commutes with this
pullback by Lemma 3.2, and hence the flatness of
$C_{\Sigma_{F\times\pi}}(\mathbb{P}^{n}\times Y)$ over $Y$ implies that of
$C_{C_{F\times\pi}\setminus(Y\times 0)}((\mathbb{C}^{n+1}\times
Y)\setminus(0\times Y))$. Since flatness is a local property and the formation
of the normal cone is a local process, this is exactly to say that
$C_{C_{F\times\pi}}(\mathbb{C}^{n+1}\times Y)$ is flat over $Y$ everywhere on
$\mathbb{C}^{n+1}\times Y$ except possibly at $0\times Y$.

Then, since the discriminant of $F\times\pi$ (for $\pi$ the projection to $Y$)
in this case will simply be $0\times Y\subset\mathbb{C}\times Y$, we can apply
Theorem 1.2 to obtain, for each point $y\in Y$, a small neighborhood $N_{y}$
of $y$ and small enough values $\varepsilon_{y},\delta_{y}>0$ such that the
maps

$f_{y^{\prime}}:B_{\varepsilon_{y}}\cap{f_{y^{\prime}}}^{-1}(D_{\delta_{y}}^{*})\to
D_{\delta_{y}}^{*}$

as $y^{\prime}$ varies in $N_{y}$ fit together into a smooth locally trivial
fibration over $D_{\delta_{y}}^{*}\times N_{y}$. In particular, these maps
define smooth locally trivial fibrations over the punctured disk which are all
fiber diffeomorphic.

However, the equivariance of homogeneous polynomials under scaling implies per
Remark 2.13 that for a homogeneous polynomial every radius is a Milnor radius
— that is, every choice of $\varepsilon>0$, no matter how large, can be used
to define the Milnor fibration. In particular, the fibrations we have defined
for the $f_{y^{\prime}}$ in the ball $B_{\varepsilon_{y}}$ are, in fact, their
Milnor fibrations at the origin, and hence fiber diffeomorphic to their affine
Milnor fibrations. As such, the fiber diffeomorphism type of the affine Milnor
fibration of $f_{y}$ is locally constant on $Y$, hence constant over all of
$Y$ if we assume connectedness. As discussed, the result follows.

######  Remark 6.2.

Note that here the homogeneity is crucial in guaranteeing that the smooth
fibers we get from Theorem 1.2 as we vary the parameters are actually the
Milnor fibers of the functions in the family. In general, this need not be
true — indeed, the fact that it is not is essential if we are to use the
theorem to obtain splittings into simpler singularities as in the case of a
Morsification of an isolated singularity or of our later Example 9.2.

We can apply this result to obtain a clearer picture of how the Milnor
fibration varies over the space of all homogeneous polynomials:

######  Corollary 6.3.

Fix integers $n\geq 0$ and $d\geq 1$ and let
$H_{n,d}\cong\mathbb{P}^{\binom{n+d}{n}-1}$ be the space of degree-$d$
hypersurfaces in $\mathbb{P}^{n}$, so that the $\binom{n+d}{n}$ projective
coordinates of each point give the coefficients up to scaling of the monomial
terms in a homogeneous polynomial defining the corresponding hypersurface.
Then iteratively applying Corollary 6.1 gives us a partition of $H_{n,d}$ into
finitely many disjoint Zariski-locally-closed subsets so that the fiber
diffeomorphism type of the affine Milnor fibrations of the corresponding
defining polynomials is constant along each subset.

Proof

Let $\Sigma_{n,d}$ be the closed subscheme of $\mathbb{P}^{n}\times H_{n,d}$
such that the fiber of $\Sigma_{n,d}$ over each point of $H_{n,d}$ is the
scheme-theoretic singular locus of the corresponding projective hypersurface,
and denote by $U_{n,d}\cong\mathbb{C}^{\binom{n+d}{n}}\setminus 0$ the space
of nonzero degree-$d$ homogeneous polynomials in $n+1$ variables. We will
apply Corollary 6.1 with the universal polynomial

$F:\mathbb{C}^{n+1}\times U_{n,d}\to\mathbb{C}$

given by $F(x_{0},\ldots,x_{n},h)=h(x_{0},\ldots,x_{n})$. The map
$\Sigma_{F\times\pi}\to U_{n,d}$ is then the pullback along the natural
$\mathbb{C}^{*}$-bundle map $U_{n,d}\to H_{n,d}$ of the projection
$\Sigma_{n,d}\to H_{n,d}$ by dint of the fact that a homogeneous polynomial is
contained in its own Jacobian ideal and so $\Sigma_{n,d}$ is defined by
partial derivatives alone. Hence, since bundle maps are flat, Lemma 3.2 tells
us for any map $\psi:S\to H_{n,d}$ that pulling back the normal cone to
$\psi^{*}\Sigma_{n,d}$ in $\mathbb{P}^{n}\times S$ along the bundle map
$\psi^{*}U_{n,d}\to S$ yields the normal cone to $\psi^{*}\Sigma_{F\times\pi}$
in $\mathbb{P}^{n}\times\psi^{*}U_{n,d}$. Thus, by Corollary 6.1, a partition
of $H_{n,d}$ into finitely many disjoint Zariski-locally-closed subsets $S$
such that the normal cone to $\Sigma_{n,d}|_{S}$ in $\mathbb{P}^{n}\times S$
is flat over $S$ will satisfy the desired constancy of the Milnor fibrations.

To construct such a partition, we proceed as follows. Suppose $S$ is a closed
subset of $H_{n,d}$. Then, by an appropriate modification (to account for the
algebraicity of all objects involved) of the argument used in the proof of
Theorem 1.1, the non-flat locus of the normal cone to $\Sigma_{n,d}|_{S}$ in
$\mathbb{P}^{n}\times S$ over $S$ is a closed algebraic subset of
$\mathbb{P}^{n}\times S$. Since the projection $\mathbb{P}^{n}\times S\to S$
is projective and hence proper, the image of this locus in $S$ is also Zariski
closed. Moreover, by generic flatness (e.g., Theorem 5.12 of [Fan+05] or
Theorem 24.1 of [Mat89]) applied to any irreducible component of $S$, this
image cannot be all of $S$.

As such, starting with $S=H_{n,d}$, we obtain the desired partition
iteratively by letting $S^{\prime}$ be this image of the non-flat locus,
adding $S\setminus S^{\prime}$ as one of the subsets in our partition, and
then replacing $S$ with $S^{\prime}$ and repeating the procedure. Since
$H_{n,d}$ is Noetherian and the successive closed subsets $S$ form a
descending chain, this procedure must terminate in finitely many steps. The
result follows.

######  Example 6.4.

For any $n\geq 0$, the space $\Sigma_{n,2}$ of Corollary 6.3 with $d=2$ will
be defined by equations which are homogeneous linear in the variables
$x_{0},\ldots,x_{n}$. In particular, the partition of $H_{n,2}$ we obtain from
our algorithm is simply the partition by the dimensions of the singular loci
of the corresponding hypersurfaces, since $\Sigma_{n,2}$’s restriction over
each constant-dimensional locus is a projectivized vector bundle and so we can
see by Proposition 2.3 that its normal cone will be a vector bundle over a
projectivized vector bundle over the base, hence flat.

In principle, Corollary 6.3 can likewise be applied to obtain explicit
partitions of $H_{n,d}$ for higher degrees $d$ — however, in practice the
number of variables involved makes direct computations expensive even for low
$n\geq 2$ and $d\geq 3$.

##  7 Complete Intersections

One of the main restrictions which appears frequently in the works of Sierma’s
school is the hypothesis that the function to be deformed have a critical
locus which is set-theoretically a complete intersection ([Sie83, Sie87,
Sie88, Pel85, Pel88, Pel89, Pel90, Zah94, Ném99, Gaf07]). Here we prove
general results about the well-behavedness of deformations of such functions
which are in an appropriate sense compatible with the complete intersection
structure; compare these in particular with the results of Section 3 of
[Zah94] and Section 4 of [Gaf07].

To apply our techniques, of course, it is necessary to require some additional
restriction on the scheme structure of the critical locus as well; we will
discuss the case where the critical locus is scheme-theoretically a complete
intersection, except possibly at the origin. Then the normal cone to the
critical locus will behave well under deformations which respect the complete
intersection structure:

######  Proposition 7.1.

Let $I=(g_{1},\ldots,g_{c})\subset\mathcal{O}_{\mathbb{C}^{n+1},0}$ be an
ideal defining a germ at the origin of a complete intersection of codimension
$c$ in $\mathbb{C}^{n+1}$. Let $f:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ be a
non-constant holomorphic function germ such that the Jacobian ideal $J_{f}$ is
contained in $I$ and $j(f):=\dim_{\mathbb{C}}I/J_{f}$ is finite.

Suppose we have germs
$G_{1},\ldots,G_{c},F:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C},0)$ of deformations
of $g_{1},\ldots,g_{c},f$ respectively such that the ideal $J_{F\times\pi}$ of
maximal minors of the Jacobian matrix of $F\times\pi$ is contained in
$\tilde{I}:=(G_{1},\ldots,G_{c})$, where
$\pi:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C}^{u},0)$ is the projection. Then the
normal cone in $(\mathbb{C}^{n+1+u},0)$ to the vanishing $(C_{F\times\pi},0)$
of $J_{F\times\pi}$ is flat over $(\mathbb{C}^{u},0)$ everywhere on
$(\mathbb{C}^{n+1}\times 0,0)$, except possibly at the origin if $j(f)>0$.

Proof

Let $(V(\tilde{I}),0)\subset(\mathbb{C}^{n+1+u},0)$ denote the complex-
analytic space germ cut out by $\tilde{I}$. Then, by the Principal Ideal
Theorem (e.g., Theorem 10.2 of [Eis04]), the dimension of $(V(\tilde{I}),0)$
is at least $n+1+u-c$; on the other hand, since its fiber over
$0\in\mathbb{C}^{u}$ is the vanishing of the ideal $I$ in
$(\mathbb{C}^{n+1},0)$ and hence has pure dimension $n+1-c$, we can see by,
e.g., Theorem 10.10 of [Eis04] that its dimension is at most $n+1+u-c$ as well
and so it is a complete intersection of codimension $c$ in
$(\mathbb{C}^{n+1+u},0)$. In particular, it is Cohen-Macaulay, and hence its
flatness over $(\mathbb{C}^{u},0)$ follows from the fiber dimension by well-
known properties of maps from Cohen-Macaulay schemes to smooth bases — see,
e.g., Theorem 18.16 of [Eis04].

We now compare $(V(\tilde{I}),0)$ and $(C_{F\times\pi},0)$. Consider the
following short exact sequence of
$\mathcal{O}_{\mathbb{C}^{n+1+u},0}$-modules:

$0\to\tilde{I}/J_{F\times\pi}\to\mathcal{O}_{C_{F\times\pi},0}\to\mathcal{O}_{V(\tilde{I}),0}\to
0$

By the flatness of $(V(\tilde{I}),0)$ over $(\mathbb{C}^{u},0)$ and the long
exact sequence in Tor, we can see that this sequence remains exact on
restriction over $0\in\mathbb{C}^{u}$. As such, the fiber over
$0\in\mathbb{C}^{u}$ of $\tilde{I}/J_{F\times\pi}$ is precisely $I/J_{f}$. Our
requirement that $j(f)<\infty$ tells us exactly that $I/J_{f}$ has finite
length — in particular, its support is zero-dimensional, hence concentrated at
the origin in $\mathbb{C}^{n+1}\times 0$, and indeed it is empty if we have
$j(f)=0$. As such, Nakayama’s Lemma (e.g., Corollary 4.8 of [Eis04]) tells us
for any sufficiently small representative that the stalks of
$\tilde{I}/J_{F\times\pi}$ will be zero at all points of
$\mathbb{C}^{n+1}\times 0$, except possibly at the origin if $j(f)>0$. Thus
$C_{F\times\pi}$ agrees with $V(\tilde{I})$ at such points and so it will be
enough to show that the normal cone to $(V(\tilde{I}),0)$ in
$(\mathbb{C}^{n+1+u},0)$ is flat over $(\mathbb{C}^{u},0)$.

Since $(V(\tilde{I}),0)$ is a complete intersection in
$(\mathbb{C}^{n+1+u},0)$, however, its normal cone in $(\mathbb{C}^{n+1+u},0)$
is a vector bundle by Proposition 2.3 and hence flat over it. As such, the
result follows from the flatness of $(V(\tilde{I}),0)$ over
$(\mathbb{C}^{u},0)$, which we have already shown.

This result can, of course, be applied with Theorem 1.1 to obtain
stratifications satisfying the Thom condition with respect to particular pairs
of strata, in the spirit of Section 4 of [Gaf07]. It is then immediate that
such deformations preserve the smooth fiber — we state this fact separately
for easy reference:

######  Corollary 7.2.

Let $I=(g_{1},\ldots,g_{c})\subset\mathcal{O}_{\mathbb{C}^{n+1},0}$ be an
ideal defining a germ at the origin of a complete intersection of codimension
$c$ in $\mathbb{C}^{n+1}$. Let $f:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ be a
non-constant holomorphic function germ such that the Jacobian ideal $J_{f}$ is
contained in $I$ and $j(f):=\dim_{\mathbb{C}}I/J_{f}$ is finite.

Suppose we have germs
$G_{1},\ldots,G_{c},F:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C},0)$ of deformations
of $g_{1},\ldots,g_{c},f$ respectively such that the ideal $J_{F\times\pi}$ of
maximal minors of the Jacobian matrix of $F\times\pi$ is contained in
$\tilde{I}:=(G_{1},\ldots,G_{c})$, where
$\pi:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C}^{u},0)$ is the projection. Also let
$(C_{F\times\pi},0)$ be the vanishing of $J_{F\times\pi}$. Then the conclusion
of Theorem 1.2 holds for $F\times\pi$ — that is, if we fix a small enough
$\varepsilon>0$, we can choose $\delta>0$ and $\gamma>0$ small enough relative
to $\varepsilon$ so that the following are true: The representatives
$F:B_{\varepsilon}\times B_{\gamma}\to\mathbb{C}$ and $C_{F\times\pi}\subset
B_{\varepsilon}\times B_{\gamma}$ exist and, if we let
$\Delta:=(F\times\pi)(C_{F\times\pi})\subset\mathbb{C}\times B_{\gamma}$ be
the discriminant, the restriction

$F\times\pi:(B_{\varepsilon}\times
B_{\gamma})\cap(F\times\pi)^{-1}((D_{\delta}\times
B_{\gamma})\setminus\bar{\Delta})\to(D_{\delta}\times
B_{\gamma})\setminus\bar{\Delta}$

defines a smooth locally trivial fibration, where
$B_{\varepsilon}\subset\mathbb{C}^{n+1}$, $D_{\delta}\subset\mathbb{C}$, and
$B_{\gamma}\subset\mathbb{C}^{u}$ are the open balls of the specified radii.

Proof

This follows from Proposition 7.1 and Condition (ii) of Theorem 1.2.

Hence a function with scheme-theoretic critical locus a complete intersection
can be studied in terms of deformations of that complete intersection,
provided compatible deformations of the function can be found. This is easiest
in the case where the function, $f$, lies in the square of the ideal $I$
defining the complete intersection, since we can then write
$f=\vec{g}^{T}H\vec{g}$ for $\vec{g}$ a vector with entries the minimal
generators of $I$ and simply deform $H$ and $\vec{g}$ separately to produce
such a deformation of $f$. However, the following proposition shows that this
in fact imposes fairly stringent restrictions on the complete intersection in
question — compare Theorem 1.13 of [Pel90], Theorem 2.2 of [Zah94], and Lemma
4.4 of [Gaf07].

######  Proposition 7.3.

Let $I=(g_{1},\ldots,g_{c})\subset\mathcal{O}_{\mathbb{C}^{n+1},0}$ be an
ideal defining a germ at the origin of a complete intersection of codimension
$c\leq n$ in $\mathbb{C}^{n+1}$. Let $f:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$
be a holomorphic function germ such that the Jacobian ideal $J_{f}$ is
contained in $I$ and $j(f):=\dim_{\mathbb{C}}I/J_{f}$ is finite. Then the
following are equivalent:

  1. (i)

$f\in I^{2}$.

  2. (ii)

$I$ is radical.

  3. (iii)

$I$ defines a reduced isolated complete intersection singularity.

Proof

(iii) implies (ii) by definition. To see that (ii) implies (i), we use the
fact that $f\in\sqrt{J_{f}}$ (because $0\in\mathbb{C}$ is an isolated critical
value of $f$ and we work locally) — since $J_{f}\subseteq I$, we then have
$f\in\sqrt{I}=I$. Thus, by Proposition 1.8 of [Pel88], $J_{f}\subseteq I$
implies that $f\in I^{2}$.

To show that (i) implies (iii), we suppose that $f\in I^{2}$ and write
$f=\vec{g}^{T}H\vec{g}$ for

$\vec{g}=\begin{bmatrix}g_{1}\\\ \vdots\\\ g_{c}\end{bmatrix}$

and $H$ a $c\times c$ matrix with entries in
$\mathcal{O}_{\mathbb{C}^{n+1},0}$. Then, for any of the differential
operators $\partial_{i}:=\tfrac{\partial}{\partial x_{i}}$, we have

$\displaystyle\partial_{i}f$ $\displaystyle=\partial_{i}(\vec{g}^{T}H\vec{g})$
$\displaystyle=(\partial_{i}\vec{g}^{T})H\vec{g}+\vec{g}^{T}(\partial_{i}H)\vec{g}+\vec{g}^{T}H(\partial_{i}\vec{g})$
$\displaystyle=((\partial_{i}\vec{g}^{T})H\vec{g})^{T}+\vec{g}^{T}(\partial_{i}H)\vec{g}+\vec{g}^{T}H(\partial_{i}\vec{g})$
$\displaystyle=\vec{g}^{T}(H^{T}(\partial_{i}\vec{g})+(\partial_{i}H)\vec{g}+H(\partial_{i}\vec{g}))$
$\displaystyle=\vec{g}^{T}((H^{T}+H)(\partial_{i}\vec{g})+(\partial_{i}H)\vec{g}).$

Now, if we let $R:=\mathcal{O}_{\mathbb{C}^{n+1},0}$, we can see that the
quotient map $R\to R/J_{f}$ is the cokernel of the map $R^{\oplus(n+1)}\to R$
given by the Jacobian matrix of $f$, the $i$th column of which is
$\partial_{i}f$. The prior computation then shows that we can decompose this
map as $R^{\oplus(n+1)}\xrightarrow{M}R^{\oplus c}\xrightarrow{\vec{g}^{T}}R$,
where $M$ is the matrix with $i$th column
$(H^{T}+H)(\partial_{i}\vec{g})+(\partial_{i}H)\vec{g}$.

Since $j(f)<\infty$, the origin is an isolated point of the support of
$I/J_{f}$. In particular, the map $J_{f}/IJ_{f}\to I/I^{2}$ from the
restricted conormal sheaf of the critical locus to the conormal sheaf of the
complete intersection germ $(V(I),0)$ defined by $I$ fails to be an
isomorphism only possibly at the origin, since these subspace germs are
everywhere else the same. If we regard our map $R^{\oplus
c}\xrightarrow{\vec{g}^{T}}R$ instead as a map $R^{\oplus c}\to I$, we can see
that tensor by $R/I$ gives an isomorphism $(R/I)^{\oplus c}\cong I/I^{2}$ per
Proposition 2.3 since $g_{1},\ldots,g_{c}$ is a regular sequence. It can then
be seen that the map $(R/I)^{\oplus(n+1)}\to(R/I)^{\oplus c}$ induced by $M$
factors as

$(R/I)^{\oplus(n+1)}\twoheadrightarrow J_{f}/IJ_{f}\to
I/I^{2}\cong(R/I)^{\oplus c};$

in particular, its cokernel is supported only possibly at the origin and hence
so too the locus in $V(I)$ where the rank of $M$ drops below $c$.

However, if we let $G:(\mathbb{C}^{n+1},0)\to(\mathbb{C}^{c},0)$ be the map
whose components are the $g_{i}$, we can see from the definition of $M$ that
$M$ is equivalent modulo $I$ to the matrix $(H^{T}+H)\operatorname{Jac}G$,
where $\operatorname{Jac}G$ is the Jacobian matrix of $G$. Hence we find that
both $H^{T}+H$ and $\operatorname{Jac}G$ must have full rank everywhere on
$V(I)$ except possibly at the origin; by the Jacobian criterion (e.g., Theorem
16.19 of [Eis04] or Corollary 2 of Section 2.19 of [Fis76]), the vanishing of
the sum of $I$ and the ideal of maximal minors of $\operatorname{Jac}G$ gives
us exactly the locus where $V(I)$ fails to be scheme-theoretically smooth, and
so this tells us that $I$ defines an isolated complete intersection
singularity which is moreover reduced except possibly at the origin. Since
$c\leq n$ and so $V(I)$ has dimension at least $1$, the fact that complete
intersections lack embedded components tells us that it is reduced at the
origin as well. This concludes the proof.

######  Remark 7.4.

This result suggests also that, for a complete intersection of positive
dimension, appearing scheme-theoretically as the critical locus of a function
should actually be a fairly strong condition — there are certainly, for
example, many reduced complete intersections which have more than simply
isolated singularities, but the proposition shows that they cannot arise in
this context.

That the requirement $f\in I^{2}$ forces $V(I)$ to be an isolated complete
intersection singularity should actually in some sense not be too surprising
given our general philosophy and results so far — in this context we can
produce deformations of $f$ compatible with any given deformation of $V(I)$,
and Corollary 7.2 shows that the local smooth fibers should remain consistent
over the parameter space in such deformations. However, if we believe that the
structure of the local smooth fibers should reflect that of the critical loci,
we should expect the local type of the deformed $V(I)$ to be constant over the
parameter space as well in every such deformation — in particular, there is
some heuristic reason to believe that $V(I)$ itself should admit a Milnor
fibration, something which is true of isolated complete intersection
singularities (see, e.g., [Loo84]) but not of complete intersections in
general.

Since we have a programmatic way to produce good deformations of our function
in the situation where the conditions of Proposition 7.3 hold, it should be
possible to compute the homology of the Milnor fiber in general in these cases
— and, indeed, Zaharia has already done so for $c<n$ in Proposition 5.4 of
[Zah94], using a more direct approach to verify the consistency of the smooth
fibers (see also Theorem 3.12 of [Gaf07] for Gaffney’s computation of the
invariant needed to use Zaharia’s formula in practice). Thus the main use of
Corollary 7.2 will be in moving beyond the case of reduced complete
intersections, which has been the focus of much of the existing literature —
as mentioned, however, this will require a more involved approach to obtaining
admissible deformations, which we will not pursue further here.

We conclude with a brief discussion of the case of isolated singularities. As
mentioned in Section 1, every perturbation of a function defining an isolated
hypersurface singularity preserves the smooth fiber, and it is not difficult
to see that such deformations trivially satisfy the hypotheses of Theorem 1.2.
Less obviously, though, it turns out that in this case things will behave
nicely even at the origin:

######  Remark 7.5.

If $F:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C},0)$ is a deformation of a function
germ $f$ defining an isolated singularity, then applying Proposition 7.1 with
$I=J_{f}$ and $\tilde{I}=J_{F\times\pi}$ tells us that the normal cone to the
critical locus is flat over the parameter space, even at the origin.

This fact is not at all necessary in showing that these deformations are well-
behaved, but we include it here as a curiosity.

##  8 Comparison with Existing Techniques

As outlined in Section 1, Theorem 1.2 is the latest in a long line of results
establishing conditions under which some kind of consistency can be obtained
for the smooth fibers of germs of families of holomorphic functions. Here we
clarify its relationships with these existing concepts and past treatments of
the flatness of the critical locus by producing proofs and examples as
appropriate.

We begin with finite determinacy. Specifically, we show that any unfolding
through an ideal with respect to which our function has finite extended
codimension as defined in [Bob04] satisfies the hypotheses of our theorem:

######  Proposition 8.1.

Let $n\geq 0$ be an integer, $f:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ a
nonzero holomorphic map-germ, and $I\subseteq\mathcal{O}_{\mathbb{C}^{n+1},0}$
an ideal with respect to which $f$ has finite extended codimension. Then, if
$F:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C},0)$ is an $I$-unfolding of $f$ and
$\pi:(\mathbb{C}^{n+1+u},0)\to(\mathbb{C}^{u},0)$ is the projection to the
parameter space, with $(C_{F\times\pi},0)$ the vanishing locus of the ideal
$J_{F\times\pi}$ of maximal minors of the Jacobian matrix of $F\times\pi$, the
origin will be at worst an isolated point of the failure of flatness of the
normal cone $C_{C_{F\times\pi}}\mathbb{C}^{n+1+u}$ on $\mathbb{C}^{n+1}\times
0$.

Proof

Taking a representative of $f$ with $\tilde{I}$ a sheaf of ideals on its
domain satisfying $\tilde{I}_{0}=I$, we define as in Section 1 of [Bob04] a
coherent sheaf

$\mathcal{F}:=\frac{\tilde{I}}{\Theta_{\tilde{I},e}(f)}$

whose concentration at the origin witnesses the finite extended codimension
condition. Indeed, the dimension over $\mathbb{C}$ of the stalk of
$\mathcal{F}$ at any point $p$ of our domain gives the extended
$\tilde{I}_{p}$-codimension of $f$ as a germ at $p$ — as such, we find that
the extended $\tilde{I}_{p}$-codimension of $f$ is zero except possibly when
$p=0$.

Theorem 1.12 of [Bob04] then tells us that, at every $p\neq 0$ in our
representative’s domain, the 0-parameter unfolding of $f$ is a versal
$\tilde{I}_{p}$-unfolding. So long as we choose a sufficiently small
representative of $F$, we can see that it will give an
$\tilde{I}_{p}$-unfolding of the germ of $f$ at each such $p\times 0$.
Therefore, by the definition of versality, at each point of
$\mathbb{C}^{n+1}\times 0$ in the domain away from the origin, we can in fact
write $F\times\pi$ locally up to isomorphism as $f\times\pi$. In particular,
we find in a neighborhood of each such point $p\times 0$ that, up to an
automorphism of the ambient space, the inclusion
$C_{F\times\pi}\hookrightarrow\mathbb{C}^{n+1+u}$ is simply the inclusion
$C_{f}\times\mathbb{C}^{u}\hookrightarrow\mathbb{C}^{n+1+u}$, where $C_{f}$ is
the critical locus of $f$ with the scheme structure given by regarding it as
the vanishing of the Jacobian ideal of $f$. Its normal cone is then a product
over $\mathbb{C}^{u}$ as well, hence flat.

Thus Theorem 1.2 generalizes Theorem 2.2 of [Bob04]. On the other hand, there
are many cases where the hypotheses of Theorem 1.2 are satisfied which do not
fall under the rubric of finite extended codimension — see, e.g., Example 9.2.

The relationship with the Lê numbers is much more tenuous:

######  Remark 8.2.

The constancy of the Lê numbers in a family is independent of the hypotheses
of Theorem 1.2. It is not difficult to see that there should be many examples
where these hypotheses hold but the Lê numbers at the origin do not remain
constant — even an appropriate deformation by translations of an arbitrarily-
chosen nontrivial hypersurface singularity will do the trick, but we can also
consider any family of isolated singularities with non-constant Milnor number
at the origin (see Remark 7.5) or, again, Example 9.2.

In the other direction, the one-parameter family of Example 9 in [Bob05] has
constant Lê numbers at the origin, as proven in that paper, but one can easily
verify that the part of the non-flat locus of its critical scheme lying over
$t=0$ has positive dimension.

Before moving on to the promised examples, we note that the idea of focusing
on deformations which make the critical locus consistent over the parameter
space in some flatness-related sense seems to have been considered
independently at least twice already — with the important distinction that the
prior works both impose such conditions on the reduced critical locus. In
[JS90], de Jong and van Straten develop a theory of admissible deformations —
in our language, the combination of $F$ and some flat deformation over the
same parameter space of a subspace $\Sigma$ of $V(f)$ will define such a
deformation precisely when the deformed $\Sigma$ includes into
$C_{F\times\pi}\cap V(F)$, the idea being that $\Sigma$ should be chosen to be
the reduced critical locus of $f$. However, Example 9.3 will show that such
deformations need not preserve the nearby smooth fiber, and hence may be of
limited interest in the study of the Milnor fibration. Additionally, even if
one does try to take $\Sigma$ to be the non-reduced critical locus, the
framework as stated and in particular the requirement that the deformation of
$\Sigma$ include into $V(F)$ impose limitations on the types of functions and
deformations we can consider.

Bobadilla’s later paper [Bob13] proves that a related set of conditions, some
of them specific to the one-dimensional case, guarantees not just the
topological consistency of the Milnor fiber but even the topological
$R$-equisingularity of the functions in the family. The precise relationship
between these conditions — in particular, the controls imposed on the generic
transversal Milnor numbers and Milnor number at the origin — and the flatness
of the normal cone to the non-reduced critical locus seems worthy of further
study, but we will not pursue the matter here.

##  9 Examples

Here we collect various concrete examples pertaining to topics discussed
elsewhere in the paper.

Consider Conditions (i) and (ii) of Theorem 1.2. The condition of flatness of
the critical locus in a family, as in Condition (i), is easier to verify than
the condition of the flatness of its normal cone as in Condition (ii) — as
discussed in Section 1, the difference between these cases is essentially
whether we require consistency of the critical locus merely in an intrinsic
sense or also in an embedded one. The idea that this intrinsic consistency
might be sufficient is theoretically appealing, and moreover would greatly
reduce the computational cost of these methods in practice — in the procedure
of Corollary 6.3, for example, the computation of the normal cone can be quite
expensive. However, the following example shows that not every function
satisfies the hypothesis on the existence of vector field germs necessary for
Condition (i) to hold:

######  Example 9.1.

Let $f:(\mathbb{C}^{3},0)\to(\mathbb{C},0)$ be the holomorphic function germ
given by $f(x,y,z)=x^{6}+y^{6}+x^{2}y^{2}z^{2}$. Then we claim that no
deformation of $f$ can satisfy Condition (i) of Theorem 1.1.

To show this, observe that the critical locus of $f$ is set-theoretically the
$z$-axis in $\mathbb{C}^{3}$ and consider a point $p=(0,0,\zeta)$ for
$\zeta\neq 0$. Suppose toward a contradiction that there exists a vector field
germ $V$ at $p$ satisfying $Vf=0$ such that $V(p)$ is not contained in the
tangent space to $S_{|p|}$ at $p$. Then we see that $(dz\cdot V)(p)\neq 0$, so
we have $V=a\frac{\partial}{\partial x}+b\frac{\partial}{\partial
y}+u\frac{\partial}{\partial z}$ for holomorphic function germs
$a,b,u\in\mathcal{O}_{\mathbb{C}^{3},p}$ with $u$ a unit. In particular, since
$Vf=0$, we can see that $\frac{\partial f}{\partial z}$ is contained in the
ideal of this convergent power series ring generated by $\frac{\partial
f}{\partial x}$ and $\frac{\partial f}{\partial y}$.

Letting $J_{f}$ be the Jacobian ideal of $f$ in $\mathbb{C}[x,y,z]$, $I$ the
ideal generated by $\frac{\partial f}{\partial x}=6x^{5}+2xy^{2}z^{2}$ and
$\frac{\partial f}{\partial y}=6y^{5}+2x^{2}yz^{2}$, and
$\mathfrak{m}=(x,y,z-\zeta)$, we can then see by the faithful flatness of the
map $\mathbb{C}[x,y,z]_{\mathfrak{m}}\to\mathcal{O}_{\mathbb{C}^{3},p}$ (as in
Remark 2.7) that $(J_{f}/I)_{\mathfrak{m}}$ must be zero. However, a
straightforward computation in a computer algebra system such as Macaulay2,
Singular, or Oscar reveals that this is not the case, yielding the desired
contradiction.

Therefore our current proof methods are not enough to establish the
sufficiency of the intrinsic condition in general, and it seems ambiguous at
this point whether we should expect it to be true. One natural approach to
attempting to prove or disprove it would of course be to search for
deformations of functions such as the one in the previous example which have
flat critical locus but not flat normal cone — however, in practice even
finding interesting deformations of a given function simply satisfying the
intrinsic condition can be quite difficult, so we defer such investigations to
a later work.

######  Example 9.2.

Consider the holomorphic function $f:\mathbb{C}^{3}\to\mathbb{C}$ given by the
polynomial equation $f(x,y,z)=x^{3}+xy^{2}z$ and the two-parameter deformation
given by $F((x,y,z),s,t)=(x^{2}+y^{2}z-s)(x-t)$. We will use this deformation
to compute the Milnor fiber of $f$ at the origin. To verify that the local
smooth fiber at the origin varies consistently in this family, we can apply
Theorem 1.2 with either Condition (i) or Condition (ii).

For the former, we note that the critical locus of $f$ is set-theoretically
the union of the $y$-axis and the $z$-axis. At all points of this set away
from the origin, the germ of the vector field $y\frac{\partial}{\partial
y}-2z\frac{\partial}{\partial z}$ will have the desired properties. Moreover,
noting by Remark 2.7 that we can work algebraically, we compute

$\operatorname{Tor}_{1}^{\mathbb{C}[s,t]}\left(\frac{\mathbb{C}[x,y,z,s,t]}{(\tfrac{\partial
F}{\partial x},\tfrac{\partial F}{\partial y},\tfrac{\partial F}{\partial
z})},\frac{\mathbb{C}[s,t]}{(s,t)}\right)=0,$

either manually or with the aid of a computer algebra system. By the local
criterion (e.g., Theorem 6.8 of [Eis04]) and the well-behavedness of Tor under
the various localizations involved, this is sufficient to guarantee that in
particular we have flatness at the origin.

For the latter, we note by Remarks 2.7 and 3.3 that we can again work entirely
algebraically, and directly compute

$\operatorname{Tor}_{1}^{\mathbb{C}[s,t]}\left(\operatorname{gr}_{\left(\tfrac{\partial
F}{\partial x},\tfrac{\partial F}{\partial y},\tfrac{\partial F}{\partial
z}\right)}\mathbb{C}[x,y,z,s,t],\frac{\mathbb{C}[s,t]}{(s,t)}\right)=0.$

As before, this is enough to give us the flatness condition we seek.

Figure 1: The critical loci of Example 9.2 over $t=0$ and $t\neq 0$
respectively ($s=5t^{2}$)

Hence, whichever approach we choose to take, we can conclude by Theorem 1.2
that this deformation preserves the local smooth fiber at the origin. To
compute the homology of this fiber, we restrict over the curve $s=5t^{2}$ in
the parameter space, obtaining a one-parameter deformation of $f$ which we
will henceforth call $F$ in a slight abuse of notation. For each $t$, denote
by $f_{t}$ the restriction of $F$ to $\mathbb{C}^{3}\times\\{t\\}$.

Before we proceed to the computation, since we are interested in having a
picture of the scheme-theoretic critical locus and how the smooth fiber can be
gleaned from it, we will say a little about how to visualize the critical loci
of the functions in this family, as depicted in Figure 1. In the $t=0$ case,
we can observe $J_{f}=(3x^{2}+y^{2}z,2xyz,xy^{2})$ and find by computation
that this ideal has associated primes $(x,y)$, $(x,z)$, and $(x,y,z)$ — it is
then straightforward to show that the multiplicities (in the sense of Section
6 of Chapter 3 of [Eis04]) of these primes in $\mathbb{C}[x,y,z]/J_{f}$ are 4,
1, and 2 respectively.

To visualize the $t\neq 0$ case, on the other hand, we find that the ideal
$J_{F\times\pi}=(3x^{2}+y^{2}z-5t^{2}-2tx,2(x-t)yz,(x-t)y^{2})$ cutting out
the family of critical loci has associated primes $(x-t,y^{2}z-4x^{2})$,
$(x+t,y)$, $(x+t,y,z)$, $(3x-5t,y)$, and $(3x-5t,y,z)$, each corresponding to
a component of multiplicity 1. Now, since the isomorphism
$\mathbb{C}^{3}\times\mathbb{C}^{*}\cong\mathbb{C}^{3}\times\mathbb{C}^{*}$
which sends $(x,y,z,t)$ to $(tx,ty,z,t)$ gives a trivialization of the
vanishing of $J_{F\times\pi}$ in $\mathbb{C}^{3}\times\mathbb{C}^{*}$ over
$\mathbb{C}^{*}$, we find for any fixed $t\neq 0$ that the components and
multiplicities of $C_{f_{t}}$ are given simply by restricting the
aforementioned components over this value. In particular, each of the minimal
primes in this list corresponds to a connected component of $J_{f_{t}}$, and
these components are sent by $f_{t}$ to $0$, $8t^{3}$, and
$-\tfrac{40}{27}t^{3}$ respectively.

Now let $\varepsilon>0$ be arbitrary and consider the restriction of $F$ to
$B_{\varepsilon}\times\mathbb{C}$, for $B_{\varepsilon}$ the open ball of
radius $\varepsilon$ centered at the origin in $\mathbb{C}^{3}$. Then,
provided that $t$ is small enough relative to $\varepsilon$, the critical
locus of $f_{t}$ in $B_{\varepsilon}$ will be as depicted in Figure 1, since
the formation of the critical locus commutes with restriction to open sets.

Although the deformation we have used is not exactly of the type considered in
[ST17], we can still use a slight modification of the methods of that paper,
which were originally introduced in [Sie87], to compute the homology of the
smooth fiber. Specifically, we observe that, if we take the inverse image
under $f$ of a small enough disk $D_{\delta}$ around the origin and intersect
it with $B_{\varepsilon}$, the result will be homotopy-equivalent to the fiber
$f^{-1}(0)\cap B_{\varepsilon}$, hence contractible. By including the boundary
and applying Lemma 2.10, we find that, for small enough $t\neq 0$, the space
$\mathbb{E}:={f_{t}}^{-1}(D_{\delta})\cap B_{\varepsilon}$ is contractible as
well. Letting $\mathbb{F}$ be a smooth fiber of $f_{t}$ in $B_{\varepsilon}$,
we then find that
$H_{*+1}(\mathbb{E},\mathbb{F})\cong\tilde{H}_{*}(\mathbb{F})$ by the long
exact sequence in homology, and Theorem 1.2 tells us that
$\tilde{H}_{*}(\mathbb{F})$ is precisely the reduced homology of the Milnor
fiber of $f$.

As in [ST17], we can now decompose this relative homology group into a direct
sum of the analogous groups around each critical value individually using
homotopy retraction and excision. Specifically, we proceed as follows. Let
$D_{0},D_{+},D_{-}$ be sufficiently small disjoint open disks inside
$D_{\delta}$ around the critical values $0$, $8t^{3}$, and
$-\tfrac{40}{27}t^{3}$ respectively. Observing that the transversal Milnor
fibration given by fixing $z$ is a local product at all points of the critical
locus where $z\neq 0$, we find that the special points (in the sense of
[ST17]) are exactly $q_{+}:=(-t,0,0)$ and $q_{-}:=(\tfrac{5}{3}t,0,0)$; let
$B_{+}$ and $B_{-}$ be sufficiently small Milnor balls in $B_{\varepsilon}$
around these points respectively. Also let
$\mathcal{U}_{0},\mathcal{U}_{+},\mathcal{U}_{-}$ be sufficiently small
tubular neighborhoods in $B_{\varepsilon}$ of the connected components
$\Sigma_{0}:=V(x-t,y^{2}z-4x^{2})\cap B_{\varepsilon},\Sigma_{+}:=V(x+t,y)\cap
B_{\varepsilon},\Sigma_{-}:=V(3x-5t,y)\cap B_{\varepsilon}$ of the critical
locus. Then, if we let
$\mathbb{E}_{0}:={f_{t}}^{-1}(D_{0})\cap\mathcal{U}_{0}$,
$\mathbb{E}_{+}:={f_{t}}^{-1}(D_{+})\cap(\mathcal{U}_{+}\cup B_{+})$, and
$\mathbb{E}_{-}:={f_{t}}^{-1}(D_{-})\cap(\mathcal{U}_{-}\cup B_{-})$ and
$\mathbb{F}_{0},\mathbb{F}_{+},\mathbb{F}_{-}$ be the intersections of the
smooth fibers of $f_{t}$ over small non-critical values nearby the appropriate
critical values with $\mathbb{E}_{0},\mathbb{E}_{+},\mathbb{E}_{-}$
respectively, we have

$H_{*+1}(\mathbb{E},\mathbb{F})\cong
H_{*+1}(\mathbb{E}_{0},\mathbb{F}_{0})\oplus
H_{*+1}(\mathbb{E}_{+},\mathbb{F}_{+})\oplus
H_{*+1}(\mathbb{E}_{-},\mathbb{F}_{-})$

by homotopy retraction and excision.

We can now consider each of these relative homology groups independently. For
each of the groups $H_{*+1}(\mathbb{E}_{\pm},\mathbb{F}_{\pm})$, we can take a
further retraction of pairs to get the relative homology of
${f_{t}}^{-1}(D_{\pm})\cap B_{\pm}$ and its intersection with a nearby smooth
fiber; by our reasoning above, this is simply the reduced homology of the
Milnor fiber of $f_{t}$ at the point $q_{\pm}$. Since $x-t$ is a unit in the
local power series ring at this point and multiplication by units does not
affect the Milnor fiber, the point is a $D_{\infty}$ singularity and hence the
reduced homology of the Milnor fiber consists of a single $\mathbb{Z}$-summand
in degree 2.

Consider the remaining component $\Sigma_{0}$ of the critical locus and note
that $\mathbb{E}_{0}\simeq\Sigma_{0}\simeq S^{1}$. Thus, from the long exact
sequence of a pair in homology and the fact that the bounds of [KM75] imply
$H_{i+1}(\mathbb{E},\mathbb{F})\cong\tilde{H}_{*}(\mathbb{F})\cong 0$ and
hence $H_{i+1}(\mathbb{E}_{0},\mathbb{F}_{0})\cong 0$ for $i\not\in\\{1,2\\}$,
we obtain a short exact sequence

$0\to H_{2}(\mathbb{E}_{0},\mathbb{F}_{0})\to H_{1}(\mathbb{F}_{0})\to
H_{1}(\mathbb{E}_{0})\to 0$

with $H_{1}(\mathbb{E}_{0})\cong\mathbb{Z}$ and an isomorphism
$H_{3}(\mathbb{E}_{0},\mathbb{F}_{0})\cong H_{2}(\mathbb{F}_{0})$.

Now, by considering the transversal slices given by the level sets of the
linear function $y$, we see that the transversal type of the singularity is
everywhere $A_{1}$ on $\Sigma_{0}$ and the transversal Milnor fibration is a
local product as before. Indeed, by noting $y\neq 0$ around $\Sigma_{0}$ and
writing $f_{t}=(\tilde{x}^{2}+\tilde{z}-5t^{2})(\tilde{x}-t)$ in the
coordinates $(\tilde{x},\tilde{y},\tilde{z})=(x,y,y^{2}z)$, we obtain a global
trivialization of the transversal Milnor fibration, so $\mathbb{F}_{0}$ is
homotopy equivalent to a torus and the map $S^{1}\times
S^{1}\simeq\mathbb{F}_{0}\to\mathbb{E}_{0}\simeq S^{1}$ is simply the
projection onto one factor — here we are able to ignore any issues near the
boundary of $B_{\varepsilon}$ by an appropriate homotopy retraction.

As such, $H_{3}(\mathbb{E}_{0},\mathbb{F}_{0})\cong
H_{2}(\mathbb{F}_{0})\cong\mathbb{Z}$, and the map
$\mathbb{Z}\oplus\mathbb{Z}\cong H_{1}(\mathbb{F}_{0})\to
H_{1}(\mathbb{E}_{0})\cong\mathbb{Z}$ is the projection onto a single factor
and hence has kernel $H_{2}(\mathbb{E}_{0},\mathbb{F}_{0})\cong\mathbb{Z}$.
Combining this with our previous computation, we find that the reduced
homology of the Milnor fiber of $f$ is given by

$\tilde{H}_{i}(\mathbb{F})\cong\begin{cases}\mathbb{Z}&i=1\\\
\mathbb{Z}^{\oplus 3}&i=2\\\ 0&\text{else.}\end{cases}$

Thus we can apply Theorem 1.2 to compute Milnor fiber homology by splitting
off not only Morse points but also higher-dimensional features, such as the
two $D_{\infty}$ singularities seen above.

As promised in Section 8, we now give a family where the reduced critical
locus is deformed flatly, as in [JS90], as is its normal cone, but the smooth
fibers are not consistent even in very coarse senses:

######  Example 9.3.

Let $f:(\mathbb{C}^{3},0)\to(\mathbb{C},0)$ be the holomorphic function germ
given by $f=y^{2}-x^{3}$ and consider the one-parameter deformation of $f$
given by $F=y^{2}+x^{2}(tz-x)$. Then, in the language of Theorem 1.2,
$C_{F\times\pi}$ has a non-isolated failure of flatness over the origin in the
parameter space. However, we can see that the trivial deformation of the
reduced critical locus $V(x,y)$ of $f$ does include into $C_{F\times\pi}$,
since $F\in(x,y)^{2}$, so $(V(x,y)\times\mathbb{C},F)$ is an admissible
deformation over the parameter space $\mathbb{C}$ in the sense of [JS90].
Indeed, the normal cone to the subspace cut out by $\sqrt{J_{F\times\pi}}$ is
a product over the parameter space, hence flat.

Now, the Milnor fiber of $f$ is given up to homotopy by the Milnor fiber for
the germ $(\mathbb{C}^{2},0)\to(\mathbb{C},0)$ given by the same expression
$y^{2}-x^{3}$, since if we are looking at the restriction of $f$ to
$B_{\varepsilon}\cap f^{-1}(D_{\delta}^{*})$ for $\varepsilon,\delta>0$ small
enough to define the Milnor fibration we can simply collapse the vertical
coordinate to get the 2-dimensional picture. Thus, since the Milnor number of
this cusp is 2, we find that the Milnor fiber of $f$ has the homotopy type of
the bouquet $S^{1}\vee S^{1}$.

On the other hand, keeping the same $\varepsilon$, we can see for any $t\neq
0$ that the restriction $f_{t}$ of $F$ to $B_{\varepsilon}\times\\{t\\}$ gives
a $D_{\infty}$ singularity at the origin — the isomorphism to the Whitney
umbrella can be obtained by a straightforward change of coordinates. By well-
known properties of the Whitney umbrella, we find immediately that the
stratification given by the filtration $\mathbb{C}^{3}\supset V(f_{t})\supset
V(x,y)\supset V(x,y,z)$ satisfies the Thom ($a_{f_{t}}$) condition; thus, as
mentioned in Remark 2.13, we can compute the Milnor fibration of $f_{t}$ using
$B_{\varepsilon}$ so long as $S_{\varepsilon}$ and all smaller spheres at the
origin are transverse to its strata. This is immediate for all strata except
the regular part of $V(f_{t})$.

On this stratum, failure of transversality will occur precisely when the
$3\times 6$ real matrix whose rows are the gradients of the real and imaginary
parts of $f_{t}$ and the squared distance from the origin fails to be full
rank. By reinterpreting every second column as an imaginary coordinate and
using the Cauchy-Riemann equations, we can see that, away from the origin,
this condition is equivalent to the requirement that the vector of complex
conjugates of partial derivatives of $f_{t}$ — the gradient of $f_{t}$, in the
language of [Mil68] — is a complex multiple of the vector $(x,y,z)$ of
coordinates of the ambient space. Some calculations, which we omit for the
sake of brevity, then tell us that this failure of transversality can be
excluded by keeping $t$ small enough relative to $\varepsilon$.

Hence the smooth fiber of $f_{t}$ in $B_{\varepsilon}\times\\{t\\}$ over any
value near enough to the origin in $D_{\delta}$ will be the Milnor fiber of a
$D_{\infty}$ singularity and so have the homotopy type of $S^{2}$. This is not
consistent with the Milnor fiber at $t=0$ even at the level of Euler
characteristic, much less homology, homotopy type, or diffeomorphism type.

Thus deformations of this sort are of limited utility in obtaining Milnor
fiber information — for such purposes, it really is necessary to keep track of
the non-reduced structure of the critical locus, as in Theorems 1.1 and 1.2.

##  10 Bibliography

## References

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