[PPT] - Perverse Sheaves, Finite Maps, and Numerical Invariants AMS Special PowerPoint Presentation

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Perverse Sheaves, Finite Maps, and Numerical Invariants

AMS Special Session on Singularities Northeastern University Brian Hepler April 21st, 2018

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

The Program

We will define a new perverse sheaf, the multiple-point

complex N•

X, naturally associated to any “parameterized”

LCI [H., Massey 2017]

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

The Program

We will define a new perverse sheaf, the multiple-point

complex N•

X, naturally associated to any “parameterized”

LCI [H., Massey 2017]

In the hypersurface case, N•

X and the vanishing cycles of the

constant sheaf should be considered “fundamental invariants”. The characteristic polar multiplicities of these sheaves allow us to extract important numerical data for these spaces.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

The Program

We will define a new perverse sheaf, the multiple-point

complex N•

X, naturally associated to any “parameterized”

LCI [H., Massey 2017]

In the hypersurface case, N•

X and the vanishing cycles of the

constant sheaf should be considered “fundamental invariants”. The characteristic polar multiplicities of these sheaves allow us to extract important numerical data for these spaces.

We examine how these invariants “deform” in a
ne-parameter family (via one-parameter unfoldings, or

IPA-deformations). We compare these deformation formulas with Milnor’s classical formula for the Milnor number in terms

f double-points.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

The Set-Up

Let (X, 0) be the germ of an n-dimensional LCI in some

(CN, 0), and (after picking a suitable representative of X) let π : Y → X be a surjective, finite, and generically one-to-one morphism (e.g. π is a parameterization, or the normalization

f X).

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

The Set-Up

Let (X, 0) be the germ of an n-dimensional LCI in some

(CN, 0), and (after picking a suitable representative of X) let π : Y → X be a surjective, finite, and generically one-to-one morphism (e.g. π is a parameterization, or the normalization

f X).
Then, there is a natural surjection of perverse sheaves

Z•

X[n] → I• X → 0 on X, where I• X is the complex of

intersection cohomology on X with constant Z-local system. Since Perv(X) is an Abelian category, this morphism has a kernel, N•

X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

The Set-Up

Let (X, 0) be the germ of an n-dimensional LCI in some

(CN, 0), and (after picking a suitable representative of X) let π : Y → X be a surjective, finite, and generically one-to-one morphism (e.g. π is a parameterization, or the normalization

f X).
Then, there is a natural surjection of perverse sheaves

Z•

X[n] → I• X → 0 on X, where I• X is the complex of

intersection cohomology on X with constant Z-local system. Since Perv(X) is an Abelian category, this morphism has a kernel, N•

X.

Consequently, there is a short exact sequence of perverse

sheaves on X: 0 → N•

X → Z• X[n] → I• X → 0.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Fundamental Short Exact Sequence of π

Since π is a finite map (really, we just need a small map the

sense of Goresky and Macpherson), π pushes forward intersection cohomology on Y to intersection cohomology on X, i.e., I•

X ∼

= π∗I•

Y ;

0 → N•

X → Z• X[n] → π∗I• Y → 0.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Fundamental Short Exact Sequence of π

Since π is a finite map (really, we just need a small map the

sense of Goresky and Macpherson), π pushes forward intersection cohomology on Y to intersection cohomology on X, i.e., I•

X ∼

= π∗I•

Y ;

0 → N•

X → Z• X[n] → π∗I• Y → 0.

This is the fundamental short exact sequence of the map π.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

We will investigate this short exact sequence in two cases:

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

We will investigate this short exact sequence in two cases:
Y is smooth and the morphism π parameterizes the total

space V (f ) of a one-parameter family of hypersurfaces V (ft0), and is of the form π(z, t) = (πt(z), t), where π0(z) is a generically one-to-one parameterization of V (f0) := V (f , t).

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

We will investigate this short exact sequence in two cases:
Y is smooth and the morphism π parameterizes the total

space V (f ) of a one-parameter family of hypersurfaces V (ft0), and is of the form π(z, t) = (πt(z), t), where π0(z) is a generically one-to-one parameterization of V (f0) := V (f , t). This means that π is a one-parameter unfolding of π0.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

We will investigate this short exact sequence in two cases:
Y is smooth and the morphism π parameterizes the total

space V (f ) of a one-parameter family of hypersurfaces V (ft0), and is of the form π(z, t) = (πt(z), t), where π0(z) is a generically one-to-one parameterization of V (f0) := V (f , t). This means that π is a one-parameter unfolding of π0.

Y is the normalization of a LCI X. When Y is additionally a

rational homology manifold, we call π a Q-parameterization of X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

N•

X in General

In general, from the short exact sequence

0 → N•

X → Z• X[n] → π∗I• Y → 0,

we immediately conclude that the perverse sheaf N•

X has

support contained in the singular locus of X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

N•

X in General

In general, from the short exact sequence

0 → N•

X → Z• X[n] → π∗I• Y → 0,

we immediately conclude that the perverse sheaf N•

X has

support contained in the singular locus of X.

When Y is smooth, we additionally have:

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

N•

X in General

In general, from the short exact sequence

0 → N•

X → Z• X[n] → π∗I• Y → 0,

we immediately conclude that the perverse sheaf N•

X has

support contained in the singular locus of X.

When Y is smooth, we additionally have:
N•

X is supported on the image multiple-point set

D := {x ∈ X | |π−1(x)| > 1}.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

N•

X in General

In general, from the short exact sequence

0 → N•

X → Z• X[n] → π∗I• Y → 0,

we immediately conclude that the perverse sheaf N•

X has

support contained in the singular locus of X.

When Y is smooth, we additionally have:
N•

X is supported on the image multiple-point set

D := {x ∈ X | |π−1(x)| > 1}.

N•

X has nonzero stalk cohomology only in degree −(n − 1),

where n = dim0 X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

N•

X in General

In general, from the short exact sequence

0 → N•

X → Z• X[n] → π∗I• Y → 0,

we immediately conclude that the perverse sheaf N•

X has

support contained in the singular locus of X.

When Y is smooth, we additionally have:
N•

X is supported on the image multiple-point set

D := {x ∈ X | |π−1(x)| > 1}.

N•

X has nonzero stalk cohomology only in degree −(n − 1),

where n = dim0 X.

In degree −(n − 1), the stalk cohomology is very easy to

describe: for p ∈ X, H−(n−1)(N•

X)p ∼

= Zm(p). where m(p) := |π−1(p)| − 1, as before.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

N•

X in General

In general, from the short exact sequence

0 → N•

X → Z• X[n] → π∗I• Y → 0,

we immediately conclude that the perverse sheaf N•

X has

support contained in the singular locus of X.

When Y is smooth, we additionally have:
N•

X is supported on the image multiple-point set

D := {x ∈ X | |π−1(x)| > 1}.

N•

X has nonzero stalk cohomology only in degree −(n − 1),

where n = dim0 X.

In degree −(n − 1), the stalk cohomology is very easy to

describe: for p ∈ X, H−(n−1)(N•

X)p ∼

= Zm(p). where m(p) := |π−1(p)| − 1, as before.

In this case, we call N•

X the multiple-point complex of X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Characteristic Polar Multiplicities and the Lˆ e Numbers

For any perverse sheaf P• on an open subset U of some CN,

the characteristic polar multiplicities of P• with respect to a “nice” choice of linear forms z = (z0, · · · , zs), denoted λi

P•,z(p) (defined in [Massey ’94]) are non-negative

integer-valued functions that mimic the purpose of the Lˆ e numbers associated to non-isolated hypersurface singularities.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Characteristic Polar Multiplicities and the Lˆ e Numbers

For any perverse sheaf P• on an open subset U of some CN,

the characteristic polar multiplicities of P• with respect to a “nice” choice of linear forms z = (z0, · · · , zs), denoted λi

P•,z(p) (defined in [Massey ’94]) are non-negative

integer-valued functions that mimic the purpose of the Lˆ e numbers associated to non-isolated hypersurface singularities.

Indeed, for 0 ≤ i ≤ dim0 Σf , and all p in U, one has the

equalities λi

f ,z(p) = λi φf [−1]Z•

U[N](p).

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Characteristic Polar Multiplicities and the Lˆ e Numbers

For any perverse sheaf P• on an open subset U of some CN,

the characteristic polar multiplicities of P• with respect to a “nice” choice of linear forms z = (z0, · · · , zs), denoted λi

P•,z(p) (defined in [Massey ’94]) are non-negative

integer-valued functions that mimic the purpose of the Lˆ e numbers associated to non-isolated hypersurface singularities.

Indeed, for 0 ≤ i ≤ dim0 Σf , and all p in U, one has the

equalities λi

f ,z(p) = λi φf [−1]Z•

U[N](p).

Example: If dim0 Σf = 0, λ0

f ,z(0) = µ0(f ) is the Milnor

number of f .

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Characteristic Polar Multiplicities and the Lˆ e Numbers

For any perverse sheaf P• on an open subset U of some CN,

the characteristic polar multiplicities of P• with respect to a “nice” choice of linear forms z = (z0, · · · , zs), denoted λi

P•,z(p) (defined in [Massey ’94]) are non-negative

integer-valued functions that mimic the purpose of the Lˆ e numbers associated to non-isolated hypersurface singularities.

Indeed, for 0 ≤ i ≤ dim0 Σf , and all p in U, one has the

equalities λi

f ,z(p) = λi φf [−1]Z•

U[N](p).

Example: If dim0 Σf = 0, λ0

f ,z(0) = µ0(f ) is the Milnor

number of f .

Example: If dim0 Σf = 1 and dim0 Σ
f|V (z)
= 0,

λ1

f ,z(0) =

C⊆Σf irred.
µC (C · V (z))0 ,

where

µC denotes the generic transverse Milnor number of f

along C.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Deforming a Parameterized Hypersurface

We recall the well-known result of [Milnor, 1968], relating the Milnor number µ0(f0) of a plane curve singularity to the number of double points δ which occur in a generic (stable) deformation of f0 by µ0(f0) = 2δ − r + 1, where r is the number of irreducible components of the curve V (f0) at 0.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Deforming a Parameterized Hypersurface

We recall the well-known result of [Milnor, 1968], relating the Milnor number µ0(f0) of a plane curve singularity to the number of double points δ which occur in a generic (stable) deformation of f0 by µ0(f0) = 2δ − r + 1, where r is the number of irreducible components of the curve V (f0) at 0. We wish to generalize this formula, in light of recent work in [H., Massey 2017], in which we obtain a quick proof of the above formula, using the multiple-point complex N•

X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Deforming a Parameterized Hypersurface

The first question we ask is: what if we didn’t have such a “stable deformation” of the plane curve V (f0)? That is, what if we didn’t know that the origin 0 ∈ V (f0) splits into δ nodes? We can still use the techniques of [H.,Massey 2017] in this situation.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Deforming a Parameterized Hypersurface

The first question we ask is: what if we didn’t have such a “stable deformation” of the plane curve V (f0)? That is, what if we didn’t know that the origin 0 ∈ V (f0) splits into δ nodes? We can still use the techniques of [H.,Massey 2017] in this situation. In this case, if π is a finite, generically one-to-one morphism which parameterizes the deformation of V (f0), we have µ0(f0) = −r + 1 +

p∈Bǫ∩V (t−t0)

(µp(ft0) + m(p)) where m(p) := |π−1(p)| − 1.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Deforming a Parameterized Hypersurface

The first question we ask is: what if we didn’t have such a “stable deformation” of the plane curve V (f0)? That is, what if we didn’t know that the origin 0 ∈ V (f0) splits into δ nodes? We can still use the techniques of [H.,Massey 2017] in this situation. In this case, if π is a finite, generically one-to-one morphism which parameterizes the deformation of V (f0), we have µ0(f0) = −r + 1 +

p∈Bǫ∩V (t−t0)

(µp(ft0) + m(p)) where m(p) := |π−1(p)| − 1. But m(p) = rank H0(N•

X)p; we find

then that, if we let N•

Xt0 = N• X |V (t−t0)[−1], then

λ0

N•

X0,z(0) = r − 1, and λ0

N•

Xt0 ,z(p) = m(p).

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

We generalize this construction to deformations of parameterized hypersurfaces with codimension-one singularities.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

We generalize this construction to deformations of parameterized hypersurfaces with codimension-one singularities. We don’t necessarily have a deformation into something as nice as double-points. We choose the notion of an IPA-deformation–these are deformations which, intuitively, are those where the only “interesting” behavior happens at the origin.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

We generalize this construction to deformations of parameterized hypersurfaces with codimension-one singularities. We don’t necessarily have a deformation into something as nice as double-points. We choose the notion of an IPA-deformation–these are deformations which, intuitively, are those where the only “interesting” behavior happens at the origin.

Theorem (H., 2017)

Suppose that π : (W, {0} × S) → (U, 0) is a one-parameter unfolding with parameter t, with im π = V (f ) for some f ∈ Oanal

U,0 .

Suppose further that z = (z1, · · · , zn) is chosen such that z is an IPA-tuple for f0 = f|V (t) at 0. Then, the following relationship holds for the Lˆ e numbers of f0 and the characteristic polar multiplicities

f N•

Xt0 := N• X |V (t−t0)[−1] with respect to z at 0: for

0 < |t0| ≪ ǫ ≪ 1 and 0 ≤ i ≤ n − 2: λi

f0,z(0) + λi N•

X0,z(0) =

p∈Bǫ∩V (t−t0,z1,z2,··· ,zi)
λi

ft0,z(p) + λi N•

Xt0 ,z(p)

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Rational Homology Manifolds and Q-Parameterizations

Let’s return to the fundamental short exact sequence of the normalization of an LCI X, but now with Q-coefficients.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Rational Homology Manifolds and Q-Parameterizations

Let’s return to the fundamental short exact sequence of the normalization of an LCI X, but now with Q-coefficients.

One immediately notices that the natural surjection

Q•

X[n] → I• X is an isomorphism precisely when N• X = 0; that

is, the LCI X is a Q-homology manifold precisely when the complex N•

X vanishes ([Borho, MacPherson]).

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Rational Homology Manifolds and Q-Parameterizations

Let’s return to the fundamental short exact sequence of the normalization of an LCI X, but now with Q-coefficients.

One immediately notices that the natural surjection

Q•

X[n] → I• X is an isomorphism precisely when N• X = 0; that

is, the LCI X is a Q-homology manifold precisely when the complex N•

X vanishes ([Borho, MacPherson]).

It is then natural to ask that, given the normalization Y of X

and the resulting fundamental short exact sequence, is there a similar result relating N•

X to whether or not Y is a

Q-homology manifold?

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Rational Homology Manifolds and Q-Parameterizations

Let’s return to the fundamental short exact sequence of the normalization of an LCI X, but now with Q-coefficients.

One immediately notices that the natural surjection

Q•

X[n] → I• X is an isomorphism precisely when N• X = 0; that

is, the LCI X is a Q-homology manifold precisely when the complex N•

X vanishes ([Borho, MacPherson]).

It is then natural to ask that, given the normalization Y of X

and the resulting fundamental short exact sequence, is there a similar result relating N•

X to whether or not Y is a

Q-homology manifold?

Theorem (H., 2018)

Y is a Q-homology manifold if and only if N•

X has stalk

cohomology concentrated in degree −n + 1; i.e., for all p ∈ X, Hk(N•

X)p is non-zero only possibly when k = −n + 1.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Sketch of Proof (= ⇒)

Suppose first that the normalization Y is a Q-homology

manifold, and let p ∈ X be arbitrary. Taking stalk cohomology at p yields the short exact sequence 0 → Q → H−n(π∗I•

Y )p → H−n+1(N• X)p → 0,

together with isomorphisms Hk(N•

X)p ∼

= Hk−1(π∗I•

Y )p ∼

=

q∈π−1(p)

Hk−1(I•

Y )q

for all −n + 2 ≤ k ≤ 0.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Sketch of Proof (= ⇒)

Suppose first that the normalization Y is a Q-homology

manifold, and let p ∈ X be arbitrary. Taking stalk cohomology at p yields the short exact sequence 0 → Q → H−n(π∗I•

Y )p → H−n+1(N• X)p → 0,

together with isomorphisms Hk(N•

X)p ∼

= Hk−1(π∗I•

Y )p ∼

=

q∈π−1(p)

Hk−1(I•

Y )q

for all −n + 2 ≤ k ≤ 0.

Since Y is a Q-homology manifold, Q•

Y [n] ∼

= I•

Y , implying in

particular that the stalk cohomology of I•

Y is concentrated in

degree −n. Hence, the stalk cohomology of N•

X is

concentrated in degree −n + 1.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Sketch of Proof (⇐ =)

Suppose now that, for all p ∈ X, Hk(N•

X)p is non-zero only

possibly when k = −n + 1. We wish to show that the natural morphism Q•

Y [n] → I• Y is an isomorphism in Db c (Y ).

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Sketch of Proof (⇐ =)

Suppose now that, for all p ∈ X, Hk(N•

X)p is non-zero only

possibly when k = −n + 1. We wish to show that the natural morphism Q•

Y [n] → I• Y is an isomorphism in Db c (Y ).

It is immediately clear, from the isomorphisms

Hk(π∗I•

Y )p ∼

= Hk+1(N•

X)p for −n + 1 ≤ k ≤ −1, that the

stalk cohomology of I•

Y is concentrated in degree −n.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Sketch of Proof (⇐ =)

Suppose now that, for all p ∈ X, Hk(N•

X)p is non-zero only

possibly when k = −n + 1. We wish to show that the natural morphism Q•

Y [n] → I• Y is an isomorphism in Db c (Y ).

It is immediately clear, from the isomorphisms

Hk(π∗I•

Y )p ∼

= Hk+1(N•

X)p for −n + 1 ≤ k ≤ −1, that the

stalk cohomology of I•

Y is concentrated in degree −n.

By the Lemma, H−n(I•

Y )q ∼

= Q for all q ∈ Y , since Y is a normal space, and thus locally irreducible.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Sketch of Proof (⇐ =)

Suppose now that, for all p ∈ X, Hk(N•

X)p is non-zero only

possibly when k = −n + 1. We wish to show that the natural morphism Q•

Y [n] → I• Y is an isomorphism in Db c (Y ).

It is immediately clear, from the isomorphisms

Hk(π∗I•

Y )p ∼

= Hk+1(N•

X)p for −n + 1 ≤ k ≤ −1, that the

stalk cohomology of I•

Y is concentrated in degree −n.

By the Lemma, H−n(I•

Y )q ∼

= Q for all q ∈ Y , since Y is a normal space, and thus locally irreducible.

Consequently, the morphism Q•

Y [n] → I• Y is an isomorphism in

Db

c (Y ) if and only if the map

H−n(Q•

Y [n])q ∼

= Q → Q ∼ = H−n(I•

Y )q

is not the zero map.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Sketch of Proof (⇐ =)

Suppose now that, for all p ∈ X, Hk(N•

X)p is non-zero only

possibly when k = −n + 1. We wish to show that the natural morphism Q•

Y [n] → I• Y is an isomorphism in Db c (Y ).

It is immediately clear, from the isomorphisms

Hk(π∗I•

Y )p ∼

= Hk+1(N•

X)p for −n + 1 ≤ k ≤ −1, that the

stalk cohomology of I•

Y is concentrated in degree −n.

By the Lemma, H−n(I•

Y )q ∼

= Q for all q ∈ Y , since Y is a normal space, and thus locally irreducible.

Consequently, the morphism Q•

Y [n] → I• Y is an isomorphism in

Db

c (Y ) if and only if the map

H−n(Q•

Y [n])q ∼

= Q → Q ∼ = H−n(I•

Y )q

is not the zero map. But this is just the “diagonal” morphism from a single copy of Q to the number of connected components of Y \{p}, which is clearly non-zero. Thus, Y is a Q-homology manifold.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Q-Parameterizations

Let π0 : Y0 → X0 be the normalization of a hypersurface X0 = V (f0), and suppose Y0 is a Q-homology manifold. Let π : Y0 × C → X be a one-parameter unfolding of π0. Then, we have the following theorem:

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Q-Parameterizations

Let π0 : Y0 → X0 be the normalization of a hypersurface X0 = V (f0), and suppose Y0 is a Q-homology manifold. Let π : Y0 × C → X be a one-parameter unfolding of π0. Then, we have the following theorem:

Theorem (H.,2018)

Everything works exactly the same as in the case of a parameterization with smooth domain, and the same relationship holds between the characteristic polar multiplicities of N•

X and

φf [−1]Q•

U[n + 1].

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

Let f (x, y, z) = xz2 − y2(y + x3), so that X = V (f ) ⊆ C3 has Σf = V (y, z). Then, if Y = V (u2 − x(y + x3), uy − xz, uz − y(y + x3)) ⊆ C4, the projection map π : Y → X is the normalization of X. It is easy to check that ΣY = V (x, y, z, u), and π−1(Σf ) = V (u2 − x4, y, z).

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

Let f (x, y, z) = xz2 − y2(y + x3), so that X = V (f ) ⊆ C3 has Σf = V (y, z). Then, if Y = V (u2 − x(y + x3), uy − xz, uz − y(y + x3)) ⊆ C4, the projection map π : Y → X is the normalization of X. It is easy to check that ΣY = V (x, y, z, u), and π−1(Σf ) = V (u2 − x4, y, z). Let Xk := {p ∈ X ||π−1(p)| = k}.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

Let f (x, y, z) = xz2 − y2(y + x3), so that X = V (f ) ⊆ C3 has Σf = V (y, z). Then, if Y = V (u2 − x(y + x3), uy − xz, uz − y(y + x3)) ⊆ C4, the projection map π : Y → X is the normalization of X. It is easy to check that ΣY = V (x, y, z, u), and π−1(Σf ) = V (u2 − x4, y, z). Let Xk := {p ∈ X ||π−1(p)| = k}. It then follows that Xk = ∅ if k > 2, and X2 = V (y, z)\{0}, so that supp N•

X = V (y, z) = Σf .

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

We have the short exact sequence

0 → Q → H−2(π∗I•

Y )p → H−1(N• Y )p → 0

and isomorphism H−1(π∗I•

Y )p ∼

= H0(N•

Y )p.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

We have the short exact sequence

0 → Q → H−2(π∗I•

Y )p → H−1(N• Y )p → 0

and isomorphism H−1(π∗I•

Y )p ∼

= H0(N•

Y )p.

We can then see that verifying the stalk cohomology of N•

X is

concentrated in degree −1 is equivalent to verifying H−1(π∗I•

Y )p = 0 for all p ∈ X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

We have the short exact sequence

0 → Q → H−2(π∗I•

Y )p → H−1(N• Y )p → 0

and isomorphism H−1(π∗I•

Y )p ∼

= H0(N•

Y )p.

We can then see that verifying the stalk cohomology of N•

X is

concentrated in degree −1 is equivalent to verifying H−1(π∗I•

Y )p = 0 for all p ∈ X.

Conversely, since I•

Y |Y \ΣY ∼

= Q•

Y \ΣY [2], Y is a Q-homology

manifold if the stalk cohomology of I•

Y at 0 ∈ Y is non-zero

nly in degree −2, where it is of dimension one.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

At 0 ∈ Y , we find

H−2(I•

Y )0 ∼

= H−2(KY ,0; I•

Y ) ∼

= H1(KY ,0; Q), where KY ,0 = Y ∩ Sǫ (for 0 < ǫ ≪ 1) is the real link of Y at 0.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

At 0 ∈ Y , we find

H−2(I•

Y )0 ∼

= H−2(KY ,0; I•

Y ) ∼

= H1(KY ,0; Q), where KY ,0 = Y ∩ Sǫ (for 0 < ǫ ≪ 1) is the real link of Y at 0.

Since Y has an isolated singularity at the origin, the link KY ,0

is a compact, orientable, smooth manifold of real dimension 3.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

At 0 ∈ Y , we find

H−2(I•

Y )0 ∼

= H−2(KY ,0; I•

Y ) ∼

= H1(KY ,0; Q), where KY ,0 = Y ∩ Sǫ (for 0 < ǫ ≪ 1) is the real link of Y at 0.

Since Y has an isolated singularity at the origin, the link KY ,0

is a compact, orientable, smooth manifold of real dimension 3. Hence, H0(KY ,0; Q) ∼ = H3(KY ,0; Q) ∼ = Q.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

The standard parameterization of the twisted cubic in P3 lifts

to a parameterization of Y , which is isomorphic to the affine cone over the twisted cubic. This parameterization is 3-to-1, from which it follows (with some work) that we have a 3-to-1 cover of KY ,0 by the 3-sphere in C2. Hence, H1(KY ,0; Z) ∼ = H2(KY ,0; Z) ∼ = Z/3Z.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Example

The standard parameterization of the twisted cubic in P3 lifts

to a parameterization of Y , which is isomorphic to the affine cone over the twisted cubic. This parameterization is 3-to-1, from which it follows (with some work) that we have a 3-to-1 cover of KY ,0 by the 3-sphere in C2. Hence, H1(KY ,0; Z) ∼ = H2(KY ,0; Z) ∼ = Z/3Z. By the Universal Coefficient Theorem and Poincar´ e Duality, H2(KY ,0; Q) ∼ = H1(KY ,0; Q) = 0, so that Y is a Q-homology manifold.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

N•

X in the Literature

When X = V (f ) ⊆ U ⊆ Cn+1 is a hypersurface, [D. Massey,

2018] has recently shown that N•

X ∼

= Ker{id −

∼

Tf }, where

∼

Tf is the monodromy automorphism on the vanishing cycles φf [−1]Z•

U[n + 1], and the kernel is taken in the

category Perv(V (f )).

When X is a reduced complex algebraic variety of pure

dimension n, [M. Saito, 2018] has recently shown that W0H1(X; Q) ∼ = Coker{H0(Y ; Q) → H−n+1(X; N•

X)}

where W0H1(X; Q) denotes the weight zero part of the cohomology H1(X; Q), considered as a mixed Hodge structure, and Y is the normalization of X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Future Directions

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Future Directions

Hypersurface Arrangements and Combinatorial Formulas.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Future Directions

Hypersurface Arrangements and Combinatorial Formulas.
The LCI case, if it is possible.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Future Directions

Hypersurface Arrangements and Combinatorial Formulas.
The LCI case, if it is possible.
Small Maps and N•

X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Future Directions

Hypersurface Arrangements and Combinatorial Formulas.
The LCI case, if it is possible.
Small Maps and N•

X.

Saito’s MHM viewpoint for N•

X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Future Directions

Hypersurface Arrangements and Combinatorial Formulas.

Problem: It seems like IPA-Deformations aren’t quite what we need.

The LCI case, if it is possible.
Small Maps and N•

X.

Saito’s MHM viewpoint for N•

X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Future Directions

Hypersurface Arrangements and Combinatorial Formulas.

Problem: It seems like IPA-Deformations aren’t quite what we need.

The LCI case, if it is possible.

Problem: What replaces the Milnor fiber/ Vanishing Cycles?

Small Maps and N•

X.

Saito’s MHM viewpoint for N•

X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Future Directions

Hypersurface Arrangements and Combinatorial Formulas.

Problem: It seems like IPA-Deformations aren’t quite what we need.

The LCI case, if it is possible.

Problem: What replaces the Milnor fiber/ Vanishing Cycles?

Small Maps and N•

X.

Problem: Fibers no longer need to be 0-dimensional, so N•

X

will be more complicated.

Saito’s MHM viewpoint for N•

X.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Future Directions

Hypersurface Arrangements and Combinatorial Formulas.

Problem: It seems like IPA-Deformations aren’t quite what we need.

The LCI case, if it is possible.

Problem: What replaces the Milnor fiber/ Vanishing Cycles?

Small Maps and N•

X.

Problem: Fibers no longer need to be 0-dimensional, so N•

X

will be more complicated.

Saito’s MHM viewpoint for N•

X.

Problem: Need to understand.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Thank You!

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Appendix: IPA-Deformations

In order to compute the characteristic polar multiplicities and Lˆ e numbers, we need to choose linear forms that “cut down” the support in a certain way. We now give several equivalent conditions for this “cutting” procedure.

Proposition

f is a deformation of f|V (L) with isolated polar activity at 0 if any of the equivalent hold:

1. dim0 Γ1

f ,L ∩ V (L) ≤ 0.

2. dim(0,d0L) im dL ∩ (f ◦ π)−1(0) ∩ T ∗

f U, where π : T ∗U → U is

the canonical projection map.

3. dim(0,d0L) SS(ψf [−1]Z•

U[n + 1]) ∩ im dL ≤ 0.

4. dim(0,d0L) SS(Z•

V (f )[n]) ∩ im dL ≤ 0.

5. dim0 supp φL[−1]Z•

V (f )[n] ≤ 0.

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Appendix: Characteristic Polar Multiplicities [Massey, 1994]

Given a perverse sheaf P• on a complex analytic subset X of

CN, and choice of “nice” tuple of linear forms z = (z0, . . . , zs)

n CN (where dim supp P• = s), the characteristic polar

multiplicities of P• with respect to z at a point p ∈ X are the non-negative integers λi

P•,z(p) = rankZ H0(φzi−zi(p)[−1]ψzi−1−zi−1(p)[−1] · · · ψz0−z0(p)[−1]P

for 0 ≤ i ≤ s.

Such numbers exist more generally for objects of Db

C−c(X),

but they are slightly more cumbersome to define (and no longer need to be non-negative!)

Why are these useful? For all p ∈ X, one has

χ(P•)p =

s

i=0

(−1)iλi

P•,z(p).

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Appendix: Proof of Relationship

A large part of the proof is verifying that the IPA-condition is sufficient to simultaneously guarantee the numbers λi

f ,(t,z)(0) and

λi

N•

X ,(t,z)(0) are defined.

After applying the dynamic intersection property to rewrite λi

f0,z(0)

as a sum in the t = 0 slice, it suffices to prove λ0

N•

X ,(t,z) = −λ0

N•

X0,z(0) +

p∈Bǫ∩V (t−t0)

λ0

N•

Xt0 ,z(p).

Since (t, z) is an IPA-tuple for f at 0, we have λ0

N•

X0,z(0) = λ1

N•

X ,(t,z)(0) − λ0

N•

X ,(t,z)(0).

The claim follows from again applying the dynamic intersection property for proper intersections with Λ1

N•

X ,(t,z).

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Introduction Deforming Hypersurfaces Rational Homology Manifolds Trivial, Non-Trivial Example Future Directions

Appendix: N•

X and Small Maps Example

f (x, y, z, w) = xw − yz, X = V (f ) ⊆ C4, and set Y = V (xt0 + yt1, zt0 + wt1) ⊆ C4 × P1. Then, the projection π : Y → X is a small map, where Y is smooth, and one-to-one away from 0 in X. We then find: Hk(N•