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Matrix Singularities Vanishing Topology of Isolated Matrix Singularities Vanishing Topology of Isolated Matrix Singularities Overview

Towers of Solvable Groups, Free Divisors andthe Topology of Nonisolated Matrix

Singularities

James Damon (in joint work with Brian Pike)1

1University of North Carolina, Chapel Hill, N C, U S A

Singularities in Aarhus, August 2009In Honor of Andy Du Plessis’s 60th Birthday


Happy Birthday Andy !

From those of us who are versions 60.0 or later


Outline

Matrix Singularities

Examples of Matrix SingularitiesSingularity Theory for Matrix Singularities

Vanishing Topology of Isolated Matrix Singularities

Overview of the Strategy for Nonisolated Matrix Singularities

Solvable Group Representations and Free DivisorsLie Group Representations and Linear Free DivisorsBlock Representation of a Solvable Linear Algebraic GroupTowers of Solvable Lie Groups and RepresentationsExamples of Towers of Solvable Groups and Representations

Singular Milnor Numbers for Nonisolated Matrix Singularities


M : the space of m ! m complex matrices which may besymmetric, skew-symmetric, or m ! p general matricesV : the subvariety of singular matrices in M.

Definition

A matrix singularity is defined by a holomorphic germ

f0 : Cn, 0 "# M, 0 (1)

(or more generally f0 : X , 0 # M, 0 for analytic germ X , 0).The pull-back variety V0 = f !1

0 (V) is the “matrix singularity”defined by f0.

Cn, 0f0""""# M, 0

!

"

"

!

"

"

f !10 (V) V0, 0 """"# V, 0

(2)


Example (Isolated Cohen-Macaulay Surface Singularity in C4)

f0(x , y , z ,w) =

#

x y zy z w

$

(3)

V = matrices of rank $ 1. V0 is defined by the ideal of 2 ! 2minors and is Cohen-Macaulay by Hilbert-Burch Theorem

(xz " y2, xw " yz , yw " z2)

Example (Obstructed Isolated Surface Singularity in C5)

f0(x , y , z , u, v) =

#

x y z uy z u v

$

(4)

V = matrices of rank $ 1. V0 is defined by the ideal of 2 ! 2minors

(xz " y2, xu " yz , xv " yu, yu " z2, yv " zu, zv " u2)


Example (Isolated Gorenstein Surface Singularity in C5)

f0(x , y , z , u, v) =

%

&

&

&

&

'

0 x u v z"x 0 y z v"u "y 0 x 0"v "z "x 0 y"z "v 0 "y 0

(

)

)

)

)

*

(5)

V consists of the “singular matrices”having rank $ 2; and V0 isdefined by the ideal of 4 ! 4 Pfa!ans

(y2 + xv , uy + xz , xy + z2 " v2, yz " uv , x2 + yv " uz).

By Buchsbaum-Eisenbud Theorem, V0 is an isolated Gorensteinsurface singularity (in fact, a unimodal singularity).


Example (Simple Symmetric Matrix Singularity)

f0(x , y) =

#

x yk

yk xy

$

(6)

The matrix is symmetric, V consists of symmetric matrices of rank$ 1, and V0 is an isolated singularity of type D2k+1 defined byx2y " y2k .This is one of simple symmetric matrix singularities classified byBruce.There is a body of further work on simple m ! m matrixsingularities both symmetric and skew-symmetric (m even) -Bruce, Tari, Haslinger, Goryunov, Zakalyukin


Singularity Theory for Matrix Singularities

Questions: classification, stability, deformation theory, geometricproperties, and topology.

Contrasting Approaches:

Kodaira-Spencer viewpoint: (algebraic geometry) requiresflat deformations - obstructed deformation theories, e.g.Example 2, while Examples 1 and 3 are unobstructed.

Thom-Mather viewpoint: Equivalences defined by groupsG of di"eomorphisms acting on both the spaces of germs andtheir unfoldings. For “geometric subgroups of A or K ”, basictheorems of singularity theory are valid :finite determinacy theorem, versal unfolding theorem,infinitesimal stability implies stability under deformations,classification theorems, and topological versions of thesetheorems


Equivalence Groups for Matrix Singularities: KM , KV , and KH

i) KM–equivalence: used for classification of simple matixsingularities.

f0(x) %# f1(x) = B(x) · f0 & !(x)

where B(x) acts by an appropriate matrix action and! : Cn, 0 # Cn, 0 is a germ of a di"eomorphism.

ii) KV–equivalence: f0 viewed as a “nonlinear section ofV”(gives equivalence of V0, 0 ' Cn, 0). Defined by the actionsof pairs of di"eomorphisms (#,!), preserving Cn ! V.

Cn ! CN , 0!

""""# Cn ! CN , 0i

("""" Cn ! V, 0

!

"

"

+

!

"

"

+

Cn, 0"

""""# Cn, 0

(7)

iii) KH– equivalence: , where # preserves all of the level sets ofH, a good defining equation for V.


Relevance of Thom-Mather Approach for Matrix Singularities:

a) Three groups are “geometric subgroups of A or K”so basictheorems of singularity theory apply :

b) These results from the examples require “transversality of f0to V”o" 0 in an appropriate algebraic sense which isequivalent to finite determinacy.

c) Lie group classification methods of Bruce, Du Plessis, Wall,Kirk, etc apply.

d KM and KV have the same tangent spaces; hence they givethe same equivalence.

e) If f0 is weighted homogeneous for the same set of weights asV, then the extended tangent spaces of f0 for KV and KH arethe same. Hence,

KV ,e " codim(f0) = KH,e " codim(f0)


Vanishing Topology of Isolated Matrix Singularities I

If n $ codim(sing(V)) then V0 has an isolated singularity.

For m ! m matrices which are symmetric, skew-symmeric (with meven), or general matrices, V is a hypersurface defined by thedeterminant function (or Pfa!an in the skew-symmetric case),which we denote by H.Then H & f0 defines V0 and has an isolated singularity.

Bruce observed a “µ = "”-type result:For simple symmetric matrix singularities f0 withn = 2 (= codim(sing(V)) " 1)

µ(H & f0) = KM,e " codim(f0) (8)

Because all of these simple matrix singularites f0 are weightedhomogeneous, d) and e) above imply

µ(H & f0) = KH,e " codim(f0) (9)


Theorem (Goryunov-Mond)

For any of the three types of m ! m matrices (with m even inskew-symmetric case) provided n $ codim(sing(V)) (= 3symmetric case, 6 skew-symmetric case, or 4 for general case).Then, the Milnor number of the isolated singularity H & f0 is givenby

µ(H & f0) = " + (#0 " #1) (10)

where " = KH,e " codim(f0) and #0 " #1 is a two term Euler

characteristic with #i = rank(TorONi (ON/J(H),On)), where

N = dim M.In addition, in the case n = codim(sing(V)) " 1, the correctionterm vanishes (implying the observed results of Bruce in thesymmetric case).


Three Basic Topological Questions:

1) If n > codim(sing(V)), then the matrix singularities are nolonger isolated. What can we say about the vanishingtopology in this case?

2) Can we understand topologically the correction terms to the“µ = "”-type result in the more general case?

3) Do these results extend beyond matrix singularities?


Overview of the Strategy for Nonisolated Matrix Singularities

1) If n ) codim(sing(V)), then perturbing f0 to a genericmapping ft will give a mapping which still meets sing(V).Hence, f !1

t (V) will not be smooth and hence not a Milnorfiber of H & f0.

2) We use instead the “singular Milnor fiber”obtained as thefiber Vt = f !1

t (V) for a stabilization ft : B# # M of f0. Byresults of D- and Mond (using a result of Le), Vt is homotopyequivalent to a bouquet of spheres of real dimension n " 1,whose number we denote by µV(f0).

3) If V were a free divisor, then the results in D- and Mond andD- (’96) give a formula for the singular Milnor number

µV(f0) = KH,e " codim(f0)

However, V for matrices is essentially never a free divisor.


Overview of the Strategy (cont)

4) Replace the algebraic resolutions of Goryunov–Mond by ageometric configuration of free divisors and free completeintersections.

5) To obtain the free divisors we use “Cholesky-typeFactorization”and obtain “linear free divisors”fromrepresentations of solvable linear algeberaic groups.

6) In fact, we use a “tower of solvable groups andrepresentations”to inductively obtain the geometricconfiguration.

7) Then, we use a “Free Completion Lemma”to inductivelycompute

µV =,

i

ai µVi(11)

for the configuration {Vi} of free divisors on smoothsubspaces.


8) This is analogue of the Le-Greuel theorem which computes foran ICIS f = (f1, f2) : Cn, 0 # Cp+1, 0, with f2 : Cn, 0 # Cp, 0 alsoan ICIS, the sum

µ(f ) + µ(f2) = dim C(On/(f"2 mp + Jac(f ))

It doesn’t compute µ(f ) directly, but rather as an alternating sumof lengths of algebras, determined by a configuration of subspacesin Cp

There is a module version of the Le-Greuel formula from D- (’96)which replaces the algebra by a determinantal (Cohen-Macaulay)module, and also applies to sections of free divisors on an ICIS.


Singular Milnor Number via Free Completions

We will inductively arrive at a formula by finding free completionsfor both V and certain associated varieties.

Definition A hypersurface singularity V, 0 ' Cn, 0 has a freecompletion if there is a free divisor W, 0 ' Cn, 0 such thatV *W, 0 is again a free divisor.

Lemma

If V, 0 ' Cn, 0 has a free completion then

µV = µV#W " µW + ("1)n!1$V$W (12)

where $V$W(f0) denotes the vanishing Euler characteristic of f0.

The first two terms of RHS are computed as lengths ofdeterminantal modules (by results of D- and Mond and D- (’96)).


Free Divisors

Definition: A hypersurface germ V, 0 ' Cp is a free divisor if

Derlog(V) = {% + &p : %(I (V)) , I (V)}

is a free Op-module (necessarily) of rank p.

Saito’s Criterion: If there are p vector fields %i =-

i bi ,j'

'zjin

Derlog(V) so that the “coe!cient determinant”det(bi ,j) defines Vwith reduced structure then V is a free divisor and Derlog(V) is afree Op-module generated by the {%i}.

Basic Examples : Discriminants (Saito, Looijenga), Bifurcationsets (Bruce), Hyperplane arrangements (Terao)General Theorem (D- ’98 - ’06): For a geometric subgroup Gof A or K:If G is Cohen-Macaulay and generically has “Morse-Typesingularities”, then G-discriminants for G-versal unfoldings are freedivisors.


Lie Group Representations and Lie Algebra of Representation VectorFields

Let ( : G # GL(V ) be a representation of complex Lie group G ,with Lie algebra g. There is a natural commutative diagram(functorial under equivariant inclusions (G ,V ) )# (H,W )).

g$

""""# gl(V )i

""""# &(V )

exp

"

"

+

exp

"

"

+

exp

"

"

+

G$

""""# GL(V )i

""""# Di! (V , 0)

(13)

where Di! (V , 0) is the group of germs of di"eomorphisms on V ,&(V ) is the module of germs of vector fields on V , 0. Also,i is a Liealgebra homomorphism where we use the negative of the usual Liebracket on &(V ).For any u + g we denote the image by *u and refer to it as the“associated representation vector field”.


Linear Free Divisors for Reductive Groups

Suppose V and G have the same dimension, and therepresentation is faithful, we shall refer to this as anequidimensional representation.

David Mond’s Basic Observation: For an equidimensionalrepresentation with an open orbit U , Saito’s criterion can beapplied to V = V \U using the associated representation vectorfields *ui

for a basis {ui} of g. If the associated coe!cientdeterminant defines V with reduced structure, then V is a freedivisor. He calls these linear free divisors.

Theorem (Mond and Buchweitz)

The equidimensional representations V for reductive groups Garising from quiver representations of finite type define linear freedivisors.


Cholesky-type Factorizations and Solvable Lie Group Representations

A = (ai j) : m ! m matrix of any of the three types.A[k] denotes the upper left hand k ! k submatrix. of A.

Cholesky-Type Factorization: i) If A is an m ! m matrix suchthat det(A[k]) -= 0 for 1 $ k $ m, then A = L · Ufor a unique lower triangular matrix L with 1’s on the diagonal anda unique upper triangular matrix U.

ii) If A is an m ! m symmetric matrix such that det(A[k]) -= 0 for

1 ) k ) m, then A = B · BT

for a unique lower triangular matrix B .

The complement of the variety where det(A[k]) = 0 for some1 $ k $ m is a single open orbit of the corresponding solvablegroup L!B T or B, where B is the Borel subgroup of lowertriangular matrices, and L is the unipotent subgroup of B ofmatrices with 1’s on the diagonal.


Solvable Linear Algebraic Groups

A linear algebraic group G is solvable if there is a series ofsubgroups G = G0 . G1 . G2 . · · ·Gk!1 . Gk = {e} with Gj+1

normal in Gj such that Gj/Gj+1 is abelian for all j .

Theorem ( Lie–Kolchin theorem)

Let V be a finite dimensional representation of a solvable linearalgebraic group G, then there exists a flag of G-invariant subspaces

V = VN . VN!1 . · · · . V1 . V0 = {0}

where dim Vj = j for all j .

Hence, there is a basis for V such that that matrix representationsof the action of the elements of G lie in the upper triangularmatrices, which form a Borel subgroup B . Converse also holds.


Block Representation of a Solvable Linear Algebraic Group

Definition

A equidimensional representation V of a solvable linear algebraicgroup G will be called a block representation if:

i) there exists a sequence of G -invariant subspaces

V = Wk . Wk!1 . · · · . W1 . W0 = (0)

ii) for the induced representation (j : G # GL(V /Wj), we letKj = ker((j), then dim Kj = dim Wj for all j and theequidimensional action of Kj/Kj!1 on Wj/Wj!1 has an openorbit for all j .

iii) the “coe!cient determinants”pj for the representation ofKj/Kj!1 on Wj/Wj!1 are all reduced and relatively prime inOV ,0.


Basic Observation:

A Block representation ( : G # GL(V ) has a coe!cient matrix forthe representation vector fields with respect to an appropriate basisfor V which has the block upper triangular form.The pj = det(Dj) are coe"cient determinants. We refer to thevariety defined by p1 as the generalized determinant variety.

%

&

&

&

&

&

&

'

Dk / / / /0 Dk!1 / / /

0 0. . . / /

0 0 0. . . /

0 0 0 0 D1

(

)

)

)

)

)

)

*

(14)

Proposition(D- and Pike) For a Block representation of a solvablelinear algebraic group G , with determinants pj that are reducedand relatively prime, the “exceptional orbit variety”(thecomplement of the open orbit) is a free divisor defined by

.

pj = 0


Towers of Solvable Lie Groups and Representations

Rather than consider individual Block representations, we considersimultaneously towers of solvable groups and representations.A tower of solvable lie groups G is a sequence

{e} = G0 ' G1 ' G2 ' · · · ' Gk ' · · ·

Such a tower has a tower of representations V = {Vj} if

(0) = V0 ' V1 ' V2 ' · · · ' Vk ' · · ·

if each Vj is a representation of Gj , and (Gj!1,Vj!1) # (Gj ,Vj) isa homomorphism of representations, i.e.with Vj a Gj!1 representation via restriction, Vj!1 )# Vj isGj!1-equivariant.


Towers of Solvable Lie Groups and Representations (cont)

Definition

A tower of solvable groups and representations (G,V) is a Blockrepresentation if: for all + ) 0 the following hold:

i) Each V% is a Block representation of G% via

G% = K %k . K %

k!1 . · · · . K %1 . K %

0 = {e}

andV% = W %

k . W %k!1 . · · · . W %

1 . W %0 = (0)

ii) For each + the composition of the natural homomorphisms ofrepresentations

(G%!1,V%!1) # (G%,V%) # (G%/K%1 ,V%/W

%1 )

is an isomorphism of representations.


Towers of Solvable Groups and Free Divisors

Theorem (D- and Pike)

Suppose (G,V) is a tower of solvable groups and representationswhich is a Block representation. Let Ej be the exceptional orbitvariety for the action of Gj on Vj . Then

i) For each j, Ej is a free divisor

ii) The generalized determinant variety Dj has a free completionEj = Dj * Ej!1

iii)µDj

= µEj" µEj!1 + ("1)n!1$Dj$Ej!1 (15)


Examples of Towers of Solvable Groups and Representations

Theorem (D- and Pike)

Towers of solvable groups and representations that are Blockrepresentations:1) Cholesky Factorization: Gm = Bm acting onSymm(C) = m ! m symmetric matrices.

2) Modified Cholesky Factorization:

Gm = Bm !

#

1 00 BT

m!1

$

(16)

acting on m ! m matrices.

3) Gm = Bm!1 !

#

1 00 BT

m!1

$

(17)

acting on (m " 1) ! m matrices.


Examples continued

Theorem (continued)

4) Restrictions of representations:

Gm =

/#

D2 02,m!2

/m!2,2 Bm!2

$0

(18)

acting onVm = {m!m symmetric matrices with upper left entry = 0}

5) Nonlinear Solvable Groups:

Gm = nonlinear solvable extension of

/#

D2 02,m!2

0m!2,2 Bm!2

$0

(19)acting on Skm(C) = m ! m skew-symmetric matrices (meven).


Examples of Formulas for Matrix Singular Milnor Numbers

2 ! 2 Symmetric Matrices: B2 acting on

/#

a bb c

$0

Esy2 defined by a (ac " b2) = 0

µDsy2

= µEsy2" (µa + µa,b) where µE

sy2

= KB2,e" codim

2 ! 2 Matrices: E2 = B2 !

/#

1 00 /

$0

acting on

/#

a bc d

$0

E2 defined by a b (ad " bc) = 0

µD2 = µE2"((µa+µa,cb)+ (µb+µb,ad)) where µE2 = KE2,e"codim


Examples continued

3 ! 3 Symmetric Matrices: B3 acting on

1

2

3

%

'

a b cb d ec e f

(

*

4

5

6

Esy3 = Esy

2 *Dsy3 defined by a (ad " b2) · det(A) = 0

µDsy3

= µEsy3"

7

(µa+µa,b)+µa,Q"

3+µD

sy2 #Q"

3"µQ3+µa,b,c,d

8

(20)

where µEsy3

= KB3,e" codim.

Also, Q3 = {det(A1) = 0} and Q%3 = {det(A2) = 0}

where A1 and A2 are obtained from A by setting f = 0,respectively, a = 0. Both Q3 and Q%

3 have free completions.

µQ3 = µP3 "7

(µc + µc,de) + (µe + µe,ac)8


Ongoing Work and Questions

Block Representations and Free Divisors

Block representations from representations of semisimple Liegroups?

Restriction of a Block representation is again a Blockrepresentation?

Extending a representation of a solvable Lie group to anonlinear solvable group which has a Block Representation?

Computing Singular Milnor Numbers

Understanding geometric configurations via blockrepresentations .

Computing $Dj$Ej!1 without the geometric configuration.

Relating #i with the geometric configurations.

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