Computation of Topological Numbers Via Linear Algebra:

Hypersurfaces, Vector Fields and Vector Fields on Hypersurfaces

by

Luis Giraldo, Xavier Gómez-Mont and Pavao Mardešić

The pattern we develop is that for problems over the complex numbers the dimension of a finite dimensional vector space computes a topologically defined number, while over the real numbers one needs to introduce a natural symmetric non-degenerate bilinear form in the real part of this finite dimensional vector space and then its signature is the topological number searched for. This is carried out for hypersurfaces in section 1, for vector fields in section 2 and for vector fields tangent to a hypersurface in section 3. A significant role is played by Algebra structures in finite dimensional vector spaces and non-degenerate dualities defined in them due to the existence of a ‘minimal ideal’ which is generated by a natural element: The Hessian determinant of a function, the Jacobian determinant of a vector field or the relative Jacobian determinant, for a vector field restricted to a hypersurface.

Let f be a germ of a holomorphic function at 0 in C^{n+1}, having 0 as an isolated critical point. We can associate to f the finite dimensional algebras obtained by considering the ring of germs of holomorphic functions O_{C^{n+1},0} modulo the ideal generated by the partial derivatives (f_{0}, …, f_{n}) of f

A := \frac{O_{C^{n+1},0}}{(f_{0}, …, f_{n})}  ,  f_{j} := \frac{∂f}{∂z_{j}}

The dimension of this algebra corresponds to the number of independent ‘vanishing cycles’ ([8]), i.e.

dimH_{n}(V_{ε}, C)  ,  V_{ε} = f^{−1}(ε) ∩ {|z| < ε′}.

The algebra structure in A is remarkable: The class of the Hessian determinant of f

Hess(f) := det ( \begin{matrix}\frac{∂^{2}f}{∂z_{1}^{2}} & … \frac{∂^{2}f}{∂z_{1}∂z_{n}} \\ & … \\ \frac{∂^{2}f}{∂z_{n}z_{1}} & … \frac{∂^{2}f}{∂z_{n}z_{2}} \end{matrix}  )  (1)

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Supported by CONACYT grant 0329P-E, Mexico

in A (that is, the class of the Hessian determinant of f modulo the partials of f) generates an ideal (Hess(f)) ⊂ A which has dimension 1 as a C-vector space, and further it is contained in every non-zero ideal of A ([2],[7]). It is a ‘minimal ideal’ and plays a dual role to the maximal ideal in the local algebra A. We can make this duality precise as follows: Choose any linear map L : A → C with L(Hess(f)) ≠ 0, then the symmetric bilinear form obtained by the composition of multiplication in A with the linear map L

< , >_{L}: A × A→^{.}A→^{L}C

is a non-degenerate bilinear form ([2],[7]). Under this duality, the orthogonal of an ideal corresponds to the annihilator ideal, and in particular, the orthogonal to the maximal ideal of A is the ideal (Hess(f)). The dual property to the maximality of the maximal ideal is that every non-zero ideal in A contains the 1-dimensional ideal (Hess(f)).

Doing the same thing over the real numbers, and asking the linear map A_{R}→^{L}R to satisfy L(Hess(f)) > 0, one obtains that the symmetric bilinear form

< , >_{L}: A_{R} × A_{R}→^{.}A_{R}→^{L}R  ,  A_{R} := \frac{A_{C^{n+1},0}}{(f_{0}, …, f_{n})}

is non-degenerate, where A_{C^{n+1},0} is the ring of germs of real analytic functions. In the real case, one has in addition the invariant provided by the signature σ of the bilinear form < . > L , number which is independent of the choice of L. The geometric meaning of σ is that 1 ± σ is the Euler characteristic of V_{±ε} ([1],Theorem 1.2).

One may proceed further: For every h ∈ A_{R} one may repeat the above method applied to the algebra \frac{A_{R}}{Ann_{A_{R}} (h)} with the relative Hessian determinant \frac{Hess(f)}{h} ∈

\frac{A_{R}}{Ann_{A_{R}} (h)} as generator of the ‘minimal’ ideal of \frac{A_{R}}{Ann_{A_{R}} (h)} (see [5])

< , >_{L}: \frac{A_{R}}{Ann_{A_{R}}(h)} × \frac{A_{R}}{Ann_{A_{R}}(h)} →^{.} \frac{A_{R}}{Ann_{A_{R}}(h)} →^{L}R  ,  L( \frac{Hess(f)}{h}) > 0

The signature of the associated non-degenerate bilinear forms defines an integer valued function

Sign_{A_{R}} : A_{R} ⟶ Z  (2)

Since all the forms obtained are non-degenerate, then Sign_{A_{R}} is constant as long as the ideal (h) ⊂ A_{R} has constant dimension. That is, we may stratify A_{R} into a finite number of real semi-algebraic subsets, defined by the rank of the multiplication map h : A_{R} → A_{R}, and (2) is the association to each strata an integer. The integer corresponding to the 0 strata is 0. The exact geometric meaning of this signature function has still elluded us. Nevertheless, we will see in section 3 that it appears in a formula for the index of vector fields on V_{0}^{R} and n an odd positive integer.

Similarly, let X̃ := ∑_{j=0}^{n} X^{j} \frac{∂}{∂z_{j}} be a germ of a holomorphic vector field at 0 in C^{n+1} with an isolated zero at 0. We can associate to X the finite dimensional algebras obtained by considering the ring of germs of holomorphic functions O_{C^{n+1},0} modulo the ideal generated by the entries of the vector field (X^{0}, …, X^{n}):

B := \frac{O_{B,0}}{(X^{0}, …, X^{n})}

This algebra is as remarkable as A, with the the role of the Hessian determinant of f taken by the Jacobian determinant of X:

J(X) := det ( \begin{matrix}\frac{∂X^{0}}{∂z_{1}} & … \frac{∂X^{0}}{∂z_{n}} \\ & … \\ \frac{∂X^{n}}{∂z_{1}} & … \frac{∂X^{n}}{∂z_{n}} \end{matrix}  ) .  (3)

In this case, the dimension of B and the signature of its natural bilinear form in B_{R} correspond to the Poincaré-Hopf index of the holomorphic or real analytic vector field, respectively (see section 2).

Now assume that X is tangent to the hypersurface V := f^{−1}(0) with an isolated singularity at 0. The tangency condition is equivalent to the existence of a germ of a holomorphic function h satisfying

df(X̃) = ∑_{j=0}^{n} f_{j}X^{j} = f · h  h := \frac{df}{f} (X̃) ∈ O_{C^{n+1},0}  (4)

Observe that the relation (4) imply that fh = 0 in both rings A and B, so that in particular we have

(f) ⊂ Ann_{A}(h) ⊂ A  ,  (f) ⊂ Ann_{B}(h) ⊂ B

and dually

(h) ⊂ Ann_{A}(f) ⊂ A  ,  (h) ⊂ Ann_{B}(f) ⊂ B

The index of the vector field X := X̃|_{V} on the hypersurface V at 0 may be computed algebraically by ([3]):

Ind_{V,0}(X) = \{ \begin{matrix}dim_{C} \frac{B}{Ann_{B}(h)} + dim_{C} \frac{A}{(f)}  for even n \\ dim_{C} \frac{B}{(f)} − dim_{C} \frac{A}{(f)}  for odd n \end{matrix}

This then gives a procedure using Linear Algebra to compute a topologically defined number. We describe in section 3 how one introduces bilinear forms in certain algebras to solve the computation of the index of a real analytic vector field restricted to an algebraically isolated singularity. We use the for this the fact that the following modules are equidimensional as vector spaces ([3]):

\frac{Ann_{A}(f)}{(h)} , \frac{Ann_{A}(h)}{(f)} , \frac{Ann_{B}(f)}{(h)} , \frac{Ann_{B}(h)}{(f)}.

1. Real and Complex Hypersurfaces

Let n ≥ 1 be an integer, and we will denote throughout by B the ball of radius r > 0 in C^{n+1}, and by B_{R} the ball of radius r in R^{n+1}.

1.1. The Holomorphic Case

Let f : B̄ ⟶ C be a holomorphic function with 0 as its only critical point and denote f^{−1}(δ) by V_{δ}. Assume further that V_{0} is a contractible topological space. Denote by

A := \frac{O_{B,0}}{(f_{0}, …, f_{n})}

the finite dimensional algebra obtained by considering the ring of germs of holomorphic functions O_{B,0} modulo the ideal generated by the partial derivatives (f_{0}, …, f_{n}) of f. It is a local commutative algebra. If we call the Hessian determinant of f the determinant of the Hessian matrix (1) and we consider its class in A, then it generates a 1 dimensional ideal, called the socle of A. It plays a dual role to the maximal ideal: It is contained in every non-zero ideal of A ([2],[7]). With the help of this fundamental property, one associates a family of non-degenerate complexvalued bilinear forms defined on A as follows: Choose any C-linear map L : A → C which sends Hess(f) ∈ A to a non-zero complex number, and define the symmetric non-degenerate bilinear form as

< , >_{L}: A × A→^{.}A→^{L}C

These bilinear forms have the property that for every ideal J ⊂ A its L-orthogonal is the annihilator Ann_{A}(J) (and hence independent of the choice of L).

The geometric meaning of the algebra A appears as:

Theorem 1.1 ([8]). : The dimension of A is equal to the dimension of the homology group H_{n−1}(V_{ϵ}, C), with |ϵ| small and non-zero.

They are the only non-vanishing reduced homology groups of V_{ϵ} and are called ‘vanishing cycles’. The dimension of A is then related to the ‘hidden topology’ inside the singularity V_{0}.

1.2. The Real Analytic Case

Let f_{R} : B̄_{R} ⟶ R be a real analytic function that accepts an analytic extension f to a holomorphic function on the closed unit ball B̄ in C^{n}. Assume that 0 is the only critical point of f on B̄ (i.e. 0 is an algebraically isolated singularity of f_{R}) and denote the real hypersurfaces f_{R}^{−1}(δ) by V_{δ}^{R} , δ ∈ R. Assume further that V_{0}^{R} is a contractible topological space and denote by V_{+}^{R} and V_{−}^{R} the differentiable type of the fibers of f_{R} for small positive and negative ϵ. Denote by

A_{R} := \frac{A_{B_{R},0}}{(f_{0}, …, f_{n})}

the finite dimensional algebra obtained by considering the ring of germs of real analytic functions A_{B_{R},0} modulo the ideal generated by the partial derivatives of f. It is a local commutative algebra. The class of the Hessian determinant (1) of f in A_{R}, generates the socle: A 1 dimensional ideal that is contained in every nonzero ideal of A_{R} ([2],[7]). One associates a family of non-degenerate real-valued bilinear forms defined in A_{R} as before:

L : A → R , L(Hess(f)) > 0 , < , >_{L}: A_{R} × A_{R}→^{.}A_{R}→^{L}R

Call σ the signature of any of the associated non-degenerate bilinear forms. The geometric meaning of the signature σ appears since it determines the Euler characteristic χ(V_{±}^{R}) of V_{±}^{R}:

Theorem 1.2 ([1]). Under the above hypothesis, we have

χ(V_{±}^{R}) := \{ \begin{matrix}1 − σ  n is odd \\ 1 ∓ σ  n is even \end{matrix}

2. The Index of Real and Complex Vector Fields

2.1. The Holomorphic Case

Let X = ∑_{j=0}^{n} X^{j} \frac{∂}{∂z_{j}} be a holomorphic vector field on B̄ ⊂ C^{n+1} vanishing only at 0. Denote by

B := \frac{O_{B,0}}{(X^{0}, …, X^{n})}

the finite dimensional algebra obtained by considering germs of holomorphic functions modulo the ideal generated by the components of the vector field X. It is a local commutative algebra. Define the Jacobian determinant of X as (3) and we consider its class in B. It is shown in [2],[7], that it generates the socle of B: A 1 dimensional ideal, which is contained in every non-zero ideal of B. One associates a family of non-degenerate complex-valued bilinear forms defined in B as before:

L : B → C , L(Hess(f)) ≠ 0 , < , >_{L}: B × B→^{.}B→^{L}C

The geometric meaning of the algebra B is:

Theorem 2.1([6]). The dimension of B (over C) is the Poincaré-Hopf index of the vector field X at the isolated singular point 0 ∈ C^{n+1}.

2.2. The Real Analytic Case

Let B_{R} be the ball of radius 1 in R^{n+1}, n ≥ 1, and let X = ∑_{j=0}^{n} X^{j} \frac{∂}{∂x_{j}} be a real analytic vector field on B̄_{R} vanishing only at 0 which extends analytically to a holomorphic vector field X on B̄ vanishing only at 0. Such points are called algebraically isolated singular points of the vector field X. Denote by

B_{R} := \frac{A_{B_{R},0}}{(X^{0}, …, X^{n})}  (2.1)

the finite dimensional algebra obtained by considering the ring A_{B_{R},0} of germs of real analytic functions modulo the ideal generated by the components of the vector field X. It is a local commutative algebra. The class in B_{R} of the Jacobian determinant of X as in (3) generates the socle of B_{R}. One associate a family of non-degenerate real-valued bilinear forms defined in B_{R} as before:

L : B → R , L(Hess(f)) > 0 , < , >_{L}: B_{R} × B_{R}→^{.}B_{R}→^{L}R

Let σ(X) be the signature of < , >_{L}. The geometric meaning of this signature is:

Theorem 2.2 ([2],[7]). Under the above hypothesis, the signature σ(X) of the above family of non-degenerate bilinear forms in B_{R} is the Poincaré-Hopf index of the vector field X at the algebraically isolated critical point 0 ∈ R^{n+1}.

3. The Index of Vector Fields Tangent to a Hypersurface

3.1. The Holomorphic Case

Let B be the ball of radius 1 in C^{n+1}, n ≥ 1, and f : B̄ ⟶ C a holomorphic function with 0 as its only critical point and denote f^{−1}(δ) by V_{δ}. Let X̃ = ∑_{j=0}^{n} X^{j} \frac{∂}{∂z_{j}} be a holomorphic vector field on B̄ vanishing only at 0 and tangent

to V := V_{0}. Denote by X the vector field induced on V by X̃. The condition that X̃ is tangent to V is equivalent to having df(X̃) a multiple of f, so that \frac{df}{f} (X̃) is a holomorphic function on B, called the cofactor:

df(X̃) = ∑_{j=0}^{n} f_{j}X^{j} = f · h  h := \frac{df}{f} (X̃) ∈ O_{B,0}  (3.1)

Topologically, one has the Poincaré-Hopf index, Ind_{V_{ϵ},0}(X), defined by transporting the vector field X to vector fields X_{ϵ} on V_{ϵ} ∩ ∂B_{R}, extending then continuously to all V_{ϵ} with isolated critical points, and adding up the usual Poincaré-Hopf indices of X_{ϵ} at the critical points:

Ind_{V,0}(X) = ∑ ; \begin{matrix}q ∈ V_{ϵ} \\ X_{ϵ}(q) = 0 \end{matrix} Ind_{V_{ϵ},q}(X_{ϵ})

The index measures the number of equilibrium points with sign of the vector field X_{ϵ} in the smoothening (V_{ϵ}, X_{ϵ}) before it degenerated to (V, X).

Theorem 3.1 ([3]). Under the above hypothesis, the index of the vector field X on the hypersurface V at 0 is:

Ind_{V,0}(X) = \{ \begin{matrix}dim_{C} \frac{B}{Ann_{B}(h)} + dim_{C} \frac{A}{(f)}  for even n \\ dim_{C} \frac{B}{(f)} − dim_{C} \frac{A}{(f)}  for odd n \end{matrix}  (3.2)

Formula (3.2) may be interpreted geometrically as follows. The number of singular points in the ambient B of X̃ at 0 can be measured by the C-dimension of B and it may be pictured geometrically by the point SpecB, consisting of the point {0} and the algebra of functions B. There are 2 subpoints of SpecB associated to the tangency with the hypersurface V: The relation (3.1) in the ring B says that f and h annihilate each other, and so we have the inclusions

0 ⊂ (f) ⊂ Ann_{B}(h) ⊂ B

which give rise to the inclusions

Spec\frac{B}{Ann_{B}(h)} ⊂ Spec\frac{B}{(f)} ⊂ SpecB.  (3.3)

That is, the middle point in (3.3) is obtained by imposing on the zero set SpecB of X̃ the condition f = 0. One obtains the smaller point by imposing the further conditions contained in Ann_{B}(h). This smaller point is the dual (complementary or orthogonal) point to Spec\frac{B}{(h)}. Formula (3.2) then is saying thay fon n even we should measure the index with the smaller point, while for n odd we should measure with the bigger point.

3.2. The Real Analytic Case

Let f_{R} : B̄_{R} ⟶ R be a real analytic function which has an analytic extension f to B̄ with 0 as its only critical point. Denote f_{R}^{−1}(δ) by V_{δ}^{R}. Let X̃_{R} = ∑_{j=0}^{n} X^{j} \frac{∂}{∂x_{j}} be a real analytic vector field on B̄_{R} which extends analytically to a holomorphic vector field X̃_{R} on B̄ vanishing only at 0. Let B_{R} be the finite dimensional algebra (2.1). As we saw in section 2.2. the class of the Jacobian determinant JX̃_{R} of X̃_{R} in B_{R} is a generator of the 1 dimensional socle.

Assume further that X̃_{R} is tangent to V^{R} := V_{0}^{R}. Denote by X the vector field induced on V^{R} by X̃_{R}. The condition that X̃_{R} is tangent to V^{R} is equivalent to (3.1).

Topologically, one has the Poincaré-Hopf index, Ind_{±,0}(X), defined by transporting the vector field X|_{V ∩∂B_{R}} to vector fields X_{±} on V_{±} ∩ ∂B_{R}, extending them continuously to all V_{±} with isolated critical points, and adding up the usual Poincaré-Hopf indices of X_{±} at the critical points:

Ind_{V_{±},0}(X) = ∑ ; \begin{matrix}q ∈ V_{±} \\ X_{±}(q) = 0 \end{matrix} Ind_{V_{±},q}(X_{±})

The index measures the number of equilibrium points with sign of the vector field X_{±} in the smoothening (V_{±}, X_{±}) before it degenerated to (V, X).

Algebraically, we can now consider the non-degenerate bilinear form in \frac{B_{R}}{Ann_{B_{R}} (h)}

induced by the choice of \frac{J(X̃_{R})}{h} as generator of its socle:

< , >_{L}: \frac{B_{R}}{Ann_{B_{R}}(h)} × \frac{B_{R}}{Ann_{B_{R}}(h)}→^{.} \frac{B_{R}}{Ann_{B_{R}}(h)}→^{L}R

L : \frac{B_{R}}{Ann_{B_{R}}(h)} → R , L(\frac{J(X̃_{R})}{h}) > 0

Denote by Sign_{f,0}(X̃_{R}) the common signature of these bilinear forms.

Theorem 3.2 ([4]). Let n ∈ N be an even positive number, and let f be a real analytic function with an algebraically isolated singularity as above. There are integers K_{±} that depend only on f such that

Ind_{±,0}(X) = Sgn_{f,0}(X̃_{R}) + K_{±}  (3.4)

for every real analytic vector field X̃_{R} with an algebraically isolated singularity at 0 and tangent to V, X := X̃_{R}|_{V^{R}}.

Hence, for n even in formula (3.2), we have the same pattern as the previous results for the smooth case in sections 1 and 2, providing an algebraic method to compute a topological number. For n odd, formula (3.2) does not give at first hand a procedure to define a non-degenerate bilinear form so that its signature is related to the index. We must first show that formula (3.2) may be written for n odd as:

Ind_{f,0}(X̃) = dim_{C} \frac{B}{Ann_{B}(h)} − dim_{C} \frac{A}{Ann_{A}(h)}  (3.5)

¿From this, again under the same pattern that in the smooth case, we have:

Theorem 3.3([5]). Let n ≥ 1 be an odd positive integer and f_{R} and X̃_{R} as above, then the index of the vector field X = X̃|_{V^{R}} on V^{R} at 0 is:

Ind_{±,0}(X) = Sgn_{f,0}(X̃_{R}) − Sgn_{A}(\frac{df}{f} (X̃_{R})),  (3.6)

A geometric interpretation compatible with formula (3.6) is the following: There are 2 kinds of singular points of X at 0:

a) The singular points captured in \frac{B}{Ann_{B}(h)} and measured by the relative Jacobian.

b) The singular points of SpecA which are not in {X = 0}, and which are measured by the relative Hessian determinant in \frac{A}{Ann_{A}(h)}.

One then counts the first type of points minus the second kind of points. This way of counting is particular to the case when n is odd. Fon n even, one only counts the first kind of points, and adds a constant K_{±}, as in (3.4).

The validity of formula (3.5) follows directly from the isomorphism of modules

\frac{Ann_{A}(h)}{(f)} ≃ \frac{Ann_{B}(h)}{(f)}  (3.7)

since

dim\frac{B}{(f)} −dim\frac{A}{(f)} = dim\frac{B}{Ann_{B}(h)} +dim\frac{Ann_{B}(h)}{(f)} −dim\frac{A}{Ann_{A}(h)} −dim\frac{Ann_{A}(h)}{(f)}

= dim\frac{B}{Ann_{B}(h)} − dim\frac{A}{Ann_{A}(h)}

The isomorphism (3.7) is somewhat surprising, since the modules are formed with very different rings A and B. Nevertheless, the relation (3.1) supplies a certain duality.

References

1. V. Arnold, The index of a singular point of a vector field, the Petrovski-Oleinik inequality and mixed Hodge Structures, Funct. Anal. Appl. 12 (1978), 1 - 14.

2. D. Eisenbud, H. Levine, An algebraic formula for the degree of a C^{∞} map germ., Ann. Math. 106 (1977), 19 - 38.

3. X. Gómez–Mont, An Algebraic formula for the index of a vector field on a hypersurface with an isolated singularity, J. Alg. Geom., to appear.

4. X. Gómez–Mont, P. Mardešić, The index of a vector field tangent to an even dimensional hypersurface and the signature of the relative Jacobian, Ann. Inst. Fourier 47, 1523-1539.

5. X. Gómez–Mont, P. Mardešić, The index of a vector field tangent to an odd dimensional hypersurface and the signature of the relative Hessian, preprint.

6. Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, J. Wiley (1978).

7. Khimishishiashvili, G.N., On the local degree of a smooth map, Trudi Tbilisi math. Inst (1980), 105-124.

8. J. Milnor, Singular points of Complex Hypersurfaces, Princeton U. P. (1968).

Luis Giraldo, Departamento de Algebra, Universidad Complutense, Madrid, España

giraldo@eucmax.sim.ucm.es

Xavier Gómez-Mont, CIMAT, A.P. 402, Guanajuato, 36000, Mexico. gmont@ fractal.cimat.mx

Pavao Mardešić, U. Dijon, France. B.P 400, Dijon, 21011, France mardesic @ satie.u-bourgogne.fr