The structure of a class of finite ramified coverings and canonical forms of analytic matrix-functions in a neighborhood of a ramified turning point

Abstract

Let X be the germ of a complex- or real-analytic manifold M at a point x_o ∈ M, or the henselian germ of an algebraic manifold M over a field k of characteristic zero at a point x_o ∈ M(k), D ⊂ X a divisor. Under some assumptions on D and its singularities we give a description of the structure, the singularities, and the divisor class group of all finite normal coverings of X ramified over D. Let g : X → gl(n) be an analytic or a k-algebraic family, respectively, of semisimple matrices, the eigenvalues of which are ramified on D as functions of x ∈ X. Put U = X − D. Using the above results under some quite general assumptions on g and D we construct an irreducible nonsingular variety U_c, a finite etale morphism a_c : U_c → U, and a morphism u_c : U_c → GL(n) (all in the same category as X and g), such that t_c(x) = u_c(x)g(x)u_c(x)⁻¹ is a diagonal matrix, for all x ∈ U_c. This construction gives, among other things, an extension in a refined form (on the level of U_c-sections) of the classical one-parameter Perturbation Theory of matrices to the case of many parameters, ramified eigenvalues, not necessarily hermitian matrices, etc. We also prove the stable triviality of the eigenbundles of g on U and vanishing of their Chern classes.

1. Introduction and Notation

Let k be a field of characteristic zero.

1.1. In this paper, unless it is explicitly stated otherwise, A will be a local noetherian k-algebra of one of the following two types:

1.1.1. A = B^h is the henselization of a smooth finitely generated k-algebra B of finite type over k with respect to a maximal ideal m of B.

1.1.2. A = k{T₁, T₂, … , T_n} is the algebra of convergent power series over a field k, where k = C the field of complex or k = R the field of real numbers.

In the algebraic case of 1.1.1, denote X = Spec A, and in the analytic case of 1.1.2, denote X = Specan A, the analytic spectrum of A (see ref. 1, Ch. 2, and sections 3.3 and 2.1 below). Denote by x_o the closed point of X in the k-algebraic case and the center of the germ X in the k-analytic case. We shall call such an X k-henselian or k-analytic germ, respectively, and shall refer to these cases as the local k-algebraic henselian and the local k-analytic cases, respectively.

1.1.3. Let g : X → gl(n) be a k-analytic or k-polynomial morphism in the same category. We shall call such a g a k-analytic or k-algebraic matrix valued function.

We say that the matrix valued function g is pointwise diagonalizable over k if g(x) is a matrix diagonalizable over the residue field k(x) of x, for all x in X(k̄), where k̄ is the algebraic closure of k.

1.1.4. We say that x_o is a turning or a transition point for a matrix function g, if some eigenvalues e_i1, e_i2, … , e_im of g change their multiplicities in X. We say that a turning point x_o is unramified if all the eigenvalues e_i(x) of g(x) are regular functions in X in the same category, as functions of the parameter x, i.e. they are analytic or polynomial functions of the parameter x in X in the analytic or algebraic cases, respectively. Otherwise, we shall say that the turning point x_o is ramified. In this case, some of the eigenvalues e_ij changing their multiplicities in X are the branches of one or several ramified analytic functions.

1.2. We say that a pointwise diagonalizable matrix function g is locally diagonalizable or admits a local diagonalization in a neighborhood of x_o, if there exists an invertible matrix valued function u : X → GL(n) in the same category as X and g, such that the following equality holds:

1

where t(x) is a diagonal matrix function.

The existence of a local diagonalization of a matrix-function g when dimX ≥ 1 is a classical problem that frequently arises and plays an important role in many areas and contexts in mathematics and physics, including nearly all the main branches of differential equations, perturbation theory, gauge theories, etc. See refs. 2–7 and sections 4 and 5 of this paper and literature quoted there for further discussions of these links.

It is well known (see for example, ref. 4, section 2) that the main difficulties presents a diagonalization in a neighborhood of a turning point x_o. In ref. 4 this problem was studied in the case when dim X ≥ 1 and the turning point x_o is unramified. Under this assumption and some additional assumptions on g(x), it was shown in ref. 4 that a diagonalization exists on each stratum of a certain stratification of the germ X. However, the ramification of the eigenvalues e_j is a very common feature when dim X ≥ 2. For this reason, the assumption of the regularity of the eigenvalues is too restrictive for many important questions and potential applications. The situation here is the opposite to that encountered in the classical one-parameter real-analytic Perturbation theory of Hermitian matrices and operators, where the problem of ramification did not arise, because of a theorem of F. Rellich that established its absence on the real line (2, 3, 5).

Thus, assume now that the eigenvalues of g do ramify on a divisor D ⊂ X, as functions of the (multi-)parameter x ∈ X and x_o ∈ D is a ramified turning point. In this case it is known that a regular diagonalizing family u for g satisfying Eq. 1 above may not exist on X, in general. Furthermore, in section 4.6 below we give an example of g for which a diagonalizing morphism u does not exist on U = X − D and even on any finite etale covering of U.

One of the main purposes of this paper is to prove the existence of a diagonalizing morphism u_c : U_c → GL(n) for g over a suitable finite etale cover a_c : U_c → U under some restrictions on g (Theorem 3.5). This is the first positive result on the local diagonalization in a neighborhood of a ramified turning point when dim X ≥ 2, to our knowledge. In the analytic case, this gives, among other things, an extension in a refined form of the classical Perturbation Theory onto the multi-parameter ramified case, not-necessarily-Hermitian matrices, and in some other directions. Indeed, these results give the triviality of the eigenbundles of g over U_c, rather than just the existence of analytic projections onto them, as the classical analytic constructions of the Perturbation Theory do, and the first property is by far stronger and more delicate than the second in the multi-parameter case, as Example 4.6 shows.

1.3. The results outlined in section 1.2 are based on several more general results that have an independent interest in broader algebra-geometric and analytic contexts. The main of them are an explicit construction and a description of the singularities of all finite normal coverings Y → X, ramified along a divisor D, which satisfies the following condition: there exists a closed subgerm N ⊂ D of codimension ≥ 2 in D such that D is a divisor with normal crossings outside N (and D may have arbitrary singularities inside N) (Theorem 2.9). Theorem 2.9 extends the classical Abhyankar Lemma (ref. 8, XIII, section 5), and it provides a much greater flexibility in applications than this Lemma. In particular, using this extension, we construct under some quite general assumptions on g and its eigenvalues an irreducible reduced factorial germ X_c and a finite ramified cover a_c : X_c → X, such that all the (ramified) eigenvalues e_j of g give rise to functions e_c,j on X_c that are well defined everywhere and are regular in the corresponding category (section 3). These nice properties of X_c allow us to define on X_c the eigensheaves E_c,j of g corresponding to the “regularized” eigenvalues e_c,j, for all j, and to show that the restrictions of E_c,j onto U_c = X_c − D_c are trivial bundles, for all j, where D_c = a(D).

The last facts imply the results mentioned in section 1.2 (Theorem 3.5) and they have many other applications. Applications to the stable triviality of the eigenbundles (including the kernel bundles) of g on U and vanishing of their Chern classes are given in sections 4 and 5. It seems these are the first general results on properties of the eigenbundles in a neighborhood of a ramified turning point. Many explicit calculations of the local Chern classes and holonomies of the eigenbundles in some special low-dimensional (dim X ≤ 3) cases can be found in the literature on the geometric (Berry) phases, quantum Hall effect, and anomalies (see refs. 6 and 7, and literature quoted there). The algebra-geometric methods of this paper are significantly different from those used in the previous works on these questions and they give more precise and complete results under much broader assumptions (e.g., for dim X ≥ 4), where the differential-geometric methods of the previous works (the Chern-Weil theory, etc.) are inapplicable in principle (see section 4.7).

The results of sections 4 and 5 can be extended onto ramified eigenvalues and the corresponding eigenbundles of analytic families of differential operators. The eigenbundles, their Chern classes, and other properties for families of differential operators arising in various branches of geometry and physics (Laplac, Dirac, and Schroedinger operators, etc.) have geometrical or physical interpretations and carry information important for these fields.

2. Construction and Properties of a Class of Finite Ramified Coverings

2.1. Complex-Analytic Varieties Defined over R.

For our purposes it will be convenient to use systematically in the analytic category analogues of the concepts of complex algebraic varieties defined over the real field R and their morphisms defined over R, which have become standard in algebraic geometry since A. Weil and A. Grothendieck. They are defined as follows.

We say that an analytic function f : U → C on an open subset U ⊂ Cⁿ is defined over R, or simply is R-analytic if in suitable systems of coordinates in U and C its Taylor series expansion has only real coefficients. We say that a complex-analytic variety Y (or a complex-analytic germ Y) is defined over R if there exists a collection of local charts (U_i, i ∈ I) covering Y, where each U_i is analytically isomorphic to an open subset of Cⁿ, such that the set of equations (f_i,j, j ∈ J_i) defining Y ∩ U_i in U_i and the functions u_i,j gluing the charts U_i can be chosen R-analytic. For a simplicity we shall call such a Y an R-analytic variety or R-analytic germ, respectively.

For R-analytic variety Y we can consider the set Y(R) of its real points—i.e., real solutions of the local equations f_i,j, as well as the set Y(C) of its complex points. R-analytic morphisms f : Y → Z between R-analytic varieties Y and Z are defined using R-analytic functions in the obvious way.

For each real-analytic local R-algebra B we can consider R-analytic germ Y, the set of complex points of which is Y(C) = Specan(B ⊗_R C) ⊂ Cⁿ (see ref. 1, Ch. 2, section 3). We shall denote this R-analytic germ Y by the symbol Specan B and shall refer to it as an R-analytic germ corresponding to B.

As in algebraic geometry, all the standard geometric notions and properties concerning R-analytic varieties Y, Z … and their morphisms f : Y → Z (for example, the dimension, the codimension, the finiteness, or the surjectivity of morphisms, etc.) will be defined as they usually are defined for the complex varieties of their C-points Y(C), Z(C) … and their analytic morphisms, rather than for their sets of real points Y(R), Z(R) only.

2.2. Let Y be a normal irreducible k-germ in the same category as X and a : Y → X a finite Galois (ramified) k-covering of X. Then Y = Spec A_Y for a suitable local henselian k-algebra A_Y in the henselian algebraic case, or Y = Specan A_Y for a local analytic k-algebra A_Y in the k-analytic case.

Denote by K = Fract(A) the field of fractions of A—i.e. K is the field of rational functions on X in the algebraic case and the field of meromorphic functions on X in the analytic case—and by K_Y = Fract(A_Y).

Denote by D = Ram (a) ⊂ X the ramification divisor in X of the morphism a. Let D = ∪ D_i be the decomposition of D into a union of its irreducible components D_i, r_i the ramification index of Y with respect to D_i (defined, for example, in ref. 9, section 2) and r_Y := ∏r_i the total ramification index of a. Because X is a nonsingular germ, the divisor D (respectively each D_i) is defined by a single equation f = 0 (resp. f_i = 0) in X.

2.3. We say that a divisor D = ∪_i D_i is a k-divisor if all its irreducible components D_i are defined over k and are geometrically irreducible. In this case, the equations f_i of D_i also can be chosen with the coefficients in k.

2.4. For a divisor D, denote by N(D) the closed subgerm of D with the reduced structure consisting of all points x ∈ D, at which D is not a divisor with normal crossings. (Notice that the irreducible components D_i of D may have self-intersections.) Denote by ir(D) the number of the irreducible components D_i of D and let c(D) = codim_DN(D).

2.5. We say that a divisor D satisfies condition (SD_m) if it is a k-divisor, ir(D) ≤ dimX and c(D) ≥ m.

2.6. Denote by ℋ_o the homotopy category of pointed simplicial sets, and by Pro − ℋ_o the category of pro-objects from ℋ_o, which was defined and studied by Artin and Mazur in ref. 10. For a noetherian scheme X or an analytic space X or an analytic germ X over C denote by ht(X) its homotopy type—i.e., the corresponding class in Pro − ℋ_o; see ref. 10 for its definition and properties. For a scheme X, its homotopy groups π_m(X) are defined as π_m(ht(X)) and π₁(ht(X)) coincides with the etale (profinite) fundamental group π(X) defined in ref. 8. For an analytic space or germ X over C, π_m(ht(X)) coincides with the topological homotopy groups π(X) of this space or germ, for all m. For a scheme X over C, denote by X(C)^an the associated analytic space, and by X_cl the “classical” topos of X defined in refs. 10 and 11.

Proposition 2.7. Let D = ∪ D_ibe a divisor in X with the irreducible components D_i, U = X − D, U_i = X − D_i. Assume that D satisfies SD₂.

(1) If X = Specan A is a nonsingular complex-analytic germ as in 1.1.2, then

(2) If X = Spec A is local henselian algebraic over C as in 1.1.1, then

Proof: The second statement of (1) was proved in ref. 12, Corollary 2.4.1. Let D_i be an irreducible component of D given by an equation f_i = 0 for some analytic function f_i : X → C. It is proved in ref. 12 that the Milnor fiber of f_i is simply connected. The first statement of (1) follows then from the exact sequence of homotopy groups of the Milnor fibration of f_i, which was constructed in ref. 13. Both statements of (2) follows from those of (1) and Proposition 2.8 below applied to Y = U and Y = U_i.

The following proposition is a local henselian version of the “generalized Riemann existence theorem,” which was proved by Artin and Mazur for algebraic varieties of finite type over C (ref. 10, Theorem 12.1):

Proposition 2.8. Let V be a normal complex algebraic variety, x_o ∈ V(C) a C-point on a V, and Z a closed subvariety of V containing x_o, Y = V − Z. Let V^h and Z^h be the henselian germs of V and Z, respectively, at x_o, V^anand Z^an the analytic germs of V and Z, respectively, at x_o. Put Y^h = V^h − Z^hand Y^an = V^an − Z^an. Then there exists a map of the homotopy types ɛ : h(Y(C)^an) → h(Y_et) in the category Pro − ℋo, and the natural homomorphism of the homotopy groups: π(Y(C)^an) → π(Y^h), induced by ɛ, becomes an isomorphism after profinite completion of π, for all m ≥ 0.

Proof: The construction of the map ɛ is a local version of that given in ref. 10, Theorem 12.1. As in 10, it is sufficient to prove that it induces an isomorphism between the analytic and etale cohomology with the twisted finite coefficients. For this we use the strong comparison (or base change) theorem between the analytic and etale cohomology (ref. 11, XVI, 4.1), applied to the natural embedding i : U → X.

The following theorem is an extension of the classical (absolute) Abhyankar Lemma (ref. 8, XIII, 5.1–5.3), which assumes that D is a divisor with normal crossings.

Theorem 2.9. Let X = Spec A or X = Specan A be a nonsingular k-germ in the k-henselian or k-analytic category as in section 1.1.1 or 1.1.2, respectively. Let Y be a normal irreducible k-germ, a : Y → X a finite Galois ramified k-covering of X in the same category as X, and let the objects A_Y, D, D_i, etc. be defined for this covering as in section 2.2. Assume, in addition, that condition (SD₂) for the ramification divisor D is satisfied and that the extension Y/X induces an isomorphism k →∼ k_Yof the residue fields.

(1) If d_Y = deg(Y/X) = r_Y, then

(1i) Y coincides with Spec A′_Yin the henselian k-algebraic case and with Specan A′_Yin the k-analytic case, where

and it is a complete intersection, germ, finite flat over X in the both k-algebraic and k-analytic cases;

(1ii) the singular locus Sin(Y) of Y is contained in the inverse image N(D)_Y = a⁻¹(N(D)) in Y under a of the non-normal crossing subgerm N(D) of D.

(2) If d_Y = deg(Y/X) < r_Y, then there exists a normal irreducible reduced k-germ Y_cand a finite Galois abelian purely ramified k-covering b : Y_c → Y of degree r_Y/d_Y(both in the corresponding category), such that the following properties (2i) and (2ii) are satisfied:

(2i) The ramification divisor Ram(a_c) of the composite projection a_c = a ○ b : Y_c → X in X coincides with D as a set.

(2ii) The germ Y_cis a complete intersection and its singular locus Sin(Y_c) is contained in the inverse image germ N(D)_c := a(N(D)) ⊂ X_c.

Remark 2.9.1: If the field k is algebraically closed, then automatically k = k_Y, and by Proposition 2.7, any finite covering Y → X, etale over U = X − D, is Galois.

An outline of the Proof: Assume first that k = C. In the both cases (algebraic henselian over C case and the complex-analytic case) it follows from Proposition 2.7 that

where the ith factor R_i = Z/r_i of G is the ramification subgroup of G with respect to the component D_i of D and r_i is the ramification index of K_Y/K with respect to D_i. This implies that K_Y = K(f, … , f). This K_Y is, clearly, the field of fractions of the ring A′_Y, which was defined in claim (1i) above.

Using the results of ref. 14 and an induction on s one can prove that the ring A′_Y is a complete intersection ring. Denote Y′ = Spec A′_Y in the k-algebraic case and Y′ = Specan A′_Y in the k-analytic case. Let a′ : Y′ → X be the natural projection and put N(D)′_Y := (a′)⁻¹(N(D)).

The classical Abhyankar Lemma (ref. 8, expose XIII, 5.1) implies that all the points x ∈ Y′ − N(D)′_Y are smooth. This establishes claim (1ii). Furthermore, (1ii) and the condition c(D) ≥ 2 implies that codim_Y′Sin(Y′) ≥ 3 and by the criterion of normality of Krull–Serre (15) Y′ is a normal germ. Therefore, it coincides with the normalization Y of X in K_Y.

(2) Assume now that d_Y = deg(a) < r_Y. Define Y′ = Spec A′_Y in the algebraic case and Y′ = Specan A′_Y in the analytic case, as in 1(i) above, and put Y_c = Y′. Let K_c be the field of rational functions on Y_c in the k-algebraic case or meromorphic functions in the k-analytic case. Then by (1i) Y_c is the normalization of X in K_c and Y_c satisfies property (2i) by its construction and property (2ii) by (1ii).

The case of an arbitrary field k of characteristic zero can be reduced to that of k = C.

Corollary 2.10. Under the assumptions of Theorem 2.9, the germ Y is the factor Y_c/H of a complete intersection germ Y_cby a finite abelian group H := Ker(G_c → G), where G = Gal(Y/X) and G_c = Gal(Y_c/X).

2.11. For a normal Noetherian scheme Z or a normal k-analytic space Z denote by Cl (Z) and Pic (Z) the (Weil) divisor class group of Z and the Picard group of Z respectively.

2.12. Denote U = X − D and let D_c = a(D) and U_c = a(U) be the inverse images of D and U in Y_c, respectively.

Corollary 2.13. Under the notation and assumptions of Theorem 2.9 assume in addition that condition (SD₃) is satisfied. Then Cl(Y_c) = 0—i.e. this scheme is factorial, and

(i) in the k-algebraic henselian case, Cl(U_c) = Pic(U_c) = 0;

(ii) in the k-analytic case, let V be a line bundle on U_cthat has an extension V∧ onto the whole Y_cas a coherent sheaf. Then its class in Pic(U_c) vanishes.

Proof: By Theorems 2.9(1i) and (2ii) Y_c is a complete intersection germ and condition (SD₃) and 2.9(1ii) imply that codim_Yc(Sin(Y_c)) ≥ 4. The vanishing of Cl(Y_c) follows now from the factoriality theorem of Grothendieck, which proved the Samuel Conjecture (ref. 16, XI, 3.14). In the algebraic case, the natural homomorphism Cl(Y_c) → Cl(U_c) is surjective. Therefore, the triviality of Cl(Y_c) implies (i).

In the analytic case, denote by W the double dual V∧** of 𝒪_Yc-module V∧. It is a coherent reflexive 𝒪_Yc-module of rank 1 on Y_c that extends V∧. The description of the (Weil) divisor class groups given in ref. 17 shows that V is in the image of the natural homomorphism Cl(Y_c) → Cl(U_c). This implies (ii).

3. A Construction of a Local Diagonal Form and of a Perturbation Theory in the Case of Ramified Eigenvalues

Let A and X be as in section 1.1.1 or section 1.1.2, K the field of fractions of A, g : X → gl(n) a k-morphism in the k-algebraic henselian or the k-analytic category, respectively.

3.1. Consider the characteristic equation

of the matrix g(x) as a polynomial equation with respect to λ over the field K. Here I_n is the unit matrix of order n. Assume that p(λ) is irreducible over K. Let L/K be a decomposition field of p(λ) over K and e₁, … , e_n the collection of all the roots of p in L.

Denote by K_p the (minimal) Galois field extension of K generated by all the roots e_j of p over K and by A′ = A[e₁, … , e_n] the A-subalgebra of K_p generated by all the e_j. Let A_p be the integral closure of A in K. It is a henselian local k-algebra in the algebraic case and an analytic k-algebra in the analytic case. For each root e_j, the characteristic equation is an equation of an integral dependence of e_j over A. Hence, A′ ⊂ A_p and A_p is the integral closure of A′ in K.

3.2. Denote X_p = Spec A_p in the k-algebraic henselian case and X_p = Specan A_p in the k-analytic case, and let a_p : X_p → X be the natural projection. Let the germs D = Ram(a), D_j, 1 ≤ j ≤ s, N(D) and U = X − D be defined as in sections 2.2–2.4 for Y = X_p and a = a_p.

Assume that condition (SD₂) and the following condition (RF) are satisfied: (RF) The extension A_p/A induces an isomorphism k →∼ k_p of the residue fields of these rings.

Let X_c be the complete intersection germ and b : X_c → X_p the ramified covering constructed for Y = X_p in Theorem 2.9. Put a_c = a_p ○ b : X_c → X. By their constructions, the maps a_p, b, and a_c are defined over k. Denote by g_c = a*_c(g) the pullback of g onto X_c and by e_c,j(x), j = 1, … , n, the eigenvalues of the matrix g_c(x), for x ∈ X_c. Notice that the element e_j ∈ A_p gives rise to a regular function a*_c(e_j) on X_c in the corresponding category, for all j. (However, these functions may not be defined over k, in general.)

Lemma 3.3. Assume that the matrix function g satisfies the following condition (PWD/U):

(PWD/U) the matrix function g is pointwise diagonalizable over k on U = X − D (cf. 1.1.3).

Then: (i) If k = R, the restriction of the eigenvalue e_jon U(R) is a single-valued real-analytic function on U(R), for all j.

(ii) If in addition conditions (RF) and (SD₂) are satisfied, the eigenvalue e_c,jof g_ccoincides with some of the elements a*_c(e₁), … , a*_c(e_n), say a*_c(e_j), for each j. Furthermore, e_c,jis a k-analytic function on X_cin the k-analytic case and it is a k-polynomial function on X_cin the k-algebraic henselian case.

3.4. Preserve the assumptions of Lemma 3.3. For each j and for each x ∈ X_c denote by E_c,j(x) the space of all the eigenvectors of g_c(x) in the vector space k(x)ⁿ belonging to the eigenvalue e_c,j(x). The collection of k(x)-vector spaces E_c,j(x)) for all x in X_c forms a coherent 𝒪_Xc-submodule of the free 𝒪_Xc-module 𝒪 of rank n on X_c (ref. 4, section 2). It follows from the constructions of X_c and D given above that the eigenvalues e_c,j, j = 1, … , n may coincide only inside D_c. Hence, they have constant multiplicities m_c,j(x) = dim_k(x)(E_c,j(x)) over U_c. By ref. 4, Proposition 2.4, this implies that the restriction E_c,j|U_c of E_c,j onto U_c is a vector bundle, for all j.

Theorem 3.5. In the notation above, assume that the characteristic polynomial p(λ) is irreducible over K and that conditions (SD₃) of section 2.5, (RF) of section 3.2, and (PWD/U) of section 3.3, as well as condition (Mu1/U) below, are satisfied:

(Mu1/U) The multiplicities m_j(x) of all the eigenvalues e_j(x) are equal to one, for all x ∈ U.

Then the k-germ X_cis factorial and the following properties hold:

(i) The restriction E_c,j|U_cof the eigensheaf E_c,jon U_cis a trivial vector bundle in the corresponding category, for all j.

(ii) There exists a k-morphism u_c : U_c → GL(n) in the same category as g, which diagonalizes the matrix-function g_c = g ○ a_con U_c—i.e., the equality

is valid, where t_c(x) = diag(e_c,1(x), … , e_c,n(x)) is the diagonal matrix with the eigenvalues e_c,j(x) of g_c(x) on the main diagonal.

Proof: Under the assumptions of Theorem 3.5, the covering a_c : X_c → X constructed in sections 3.1 and 3.2 satisfies the conditions of Theorem 2.9 with Y = X_c and a = a_c. Then all the eigenbundles E_c,j are trivial on U_c by Corollary 2.13(i) in the k-algebraic henselian case and by Corollary 2.13(ii) in the k-analytic case. This proves (i). The existence of a diagonalizing k-morphism u_c : U_c → GL(n) for g_c on U_c follows from the triviality of E_c,j and Proposition 2.4 of ref. 4.

4. Rational Stable Triviality of the Eigenbundles of g(x) on U(R)

In this section k = C or R, the fields or complex or real numbers. We shall preserve in this section the general notation of sections 1–3 and assume that g satisfies condition (PWD/U) of section 3.3.

4.1. By Lemma 3.3(i), the eigenvalues e_j are single-valued R-analytic functions on U(R). Therefore, we can consider the eigenbundle E_j over U(R), corresponding to the eigenvalue e_j, for each j. It is an R-analytic bundle. Denote by E_j,C = E_j ⊗_R C the complexification of E_j.

4.2. For a differentiable (C^∞) (respectively real-analytic) manifold M, denote by Bun^τ(M, C) the category of topological (τ = t) (respectively real-analytic (τ = ran)) vector bundles on M with the complex coefficients, and by K(M, C) the Grothendieck group of this category.

Theorem 4.3. Under the notation above, assume that conditions (SD₃), (RF), (PWD/U) and (Mu1/U) of sections 2.5, 3.2, 3.3, and 3.5, respectively, are satisfied. Let M be a closed differentiable (C^∞) submanifold of U(R), m = dim_RM, E_j,C|M the restriction of the complexified eigenbundle E_j,C onto M. Then

(i) The complex vector bundle d_c(E_j,C|M) is topologically stably trivial on M, where d_cis the degree of the extension X_c/X. Furthermore, if d_c > m/2, then this bundle is topologically trivial.

(ii) If M is a closed reduced coherent (in the sense of ref. 18, Ch. 2) real-analytic subvariety of U(R), then the vector bundle d_c(E_j,C|M) is real-analytically stably trivial on M. Furthermore, if d_c > m/2, then it is real-analytically trivial.

An outline of the Proof: Denote by M_c := a(M) ⊂ X_c(C) the inverse image of M under the projection a_c : X_c → X. Then the restriction of a_c onto M_c gives a continuous etale surjective map μ : M_c → M of topological spaces of degree d_c. This map induces the direct image (or transfer) homomorphism of the K_o-groups μ_*^t : K(M_c, C) → K(M, C), and the inverse image homomorphism μ*_t : K_o(M, C) → K_o(M_c, C), such that μ_*^t ○ μ*_t = d_c.

The topological stable triviality of d_cE_j,C|M follows from Theorem 3.1(i) and these properties of the transfer. The analytic stable trivality in (ii) follows from that in (i) and a real analytic version of the Grauert–Oka principle (ref. 18, Ch. 8, Theorem 2.2). The last triviality statement in (i) follows from the stabilization theorems for topological bundles (ref. 19, Ch. 8, Theorem 1.5). The triviality statement in (ii) follows from that in (i) and the same version of the Grauert–Oka principle.

4.4. Let H*(M, R) be a cohomology theory with the coefficients in a commutative ring R that admits a theory of Chern classes c : K(M, C) → H^2r(M, R) with the standard properties on the category Bun^τ(M, C), for τ = t or τ = ran, (see ref. 19, Ch. 16). If R is a Q-algebra, then for this cohomology theory the Chern character ch^τ : K(M) → H*(M, R) is also defined and it is a ring homomorphism.

As examples of such cohomology theories we can take the following: (i) the singular cohomology H*_sin(M, R) of the underlying topological space M with R = Z or Q; (ii) the de Rham cohomology theory H*_dR(M, R) of smooth (C^∞) differential forms on the C^∞-manifold M with R = R.

Corollary 4.5. Under the notation and the assumptions of Theorem 4.3(i) (respectively 4.3(ii)) we have:

(i) The first (integral) Chern class c(d_cE_j,C|M) of the restriction of the eigenbundle d_cE_j,C onto M is zero, for all j and τ = t (respectively τ = ran).

(ii) If R is a Q-algebra, then ch^τ(E_j,C|M) = 0, for τ = t (respectively for τ = ran).

Example 4.6: Let X = (C³, 0) the germ of C³ at zero, g(x) = ∑x_iσ_i ∈ gl(2), where σ_i are the Pauli matrices, x = (x₁, x₂, x₃) ∈ C³ [see Example 3.7 of Avron's paper (6)]. Then D = (∑_ix = 0) is a cone in the germ X. It has an isolated singularity at its vertex 0 = D(R) of codimension 2 in D. The matrix g(x) has two eigenvalues e₊ = |x| and e₋ = −|x|, where |x| = (∑_ix)^1/2. According the Chern–Weil theory, the Chern classes of the eigenbundles E₊ and E₋ of g can be calculated by integrating over a small 2-sphere S² ⊂ U(R) with the center at 0 the curvature forms of the Berry connections on E₊ and E₋—i.e., the connections induced by the exterior differential d on C³ (refs. 6 and 7). This gives c₁(E₊) = −1 ∈ Z, c₁(E₋) = 1 ∈ Z (ref. 6, p. 50). Corollary 4.5 above and its proof imply then that this g(x) cannot be analytically (and even continuously) diagonalized on any finite cover of U. This shows that the condition codim_DN(D) ≥ 3 of (SD₃) in Theorems 3.5 and 4.3, and all other statements depending on them, is sharp.

Remark 4.7: In the general cituation of the beginning of section 4, the method of the Chern–Weil theory indicated in section 4.6 is applicable only when U(R) is homotopy equivalent to a compact closed (without a boundary) orientable submanifold M ⊂ U(R). For a germ X it is possible only when U(R) = X(R) − x_o, or equivalently D(R) = x_o. The last equality is a very strong restriction on the pair (X, g). The methods and results here, including those of sections 4 and 5, do not require this restriction.

5. The Local Structure and Characteristic Classes of the Kernel Bundle

We return in this section to the general notation and assumptions of section 1.1.

5.1. Let Ker g be the kernel sheaf of g considered as a homomorphism of free sheaves of 𝒪_X-modules g : 𝒪 → 𝒪. We say that x_o is a turning or a transition point for Ker g at x_o if the dimension of the fibers of Ker g jumps up at x_o. It means that some eigenvalues of g, which are not zero identically on X, vanish at x_o. We say that x_o is a ramified turning or transition point for Ker g if some of the nonzero eigenvalues vanishing at x_o ramify at x_o as the functions of x.

In this paper we consider only the case when x_o is a ramified transition point for Ker g. In the unramifed case a description of the local structure of Ker g under some natural assumptions on g follows easily from the results of ref. 4.

5.2. Assume from now on that x_o is a ramified turning point for Ker g. Let e_o = 0 be the eigenvalue of g that is identically zero on X, and m_o its multiplicity in g. Let e_j, m_o + 1 ≤ j ≤ n be the collection of all nonzero eigenvalues of g that vanish at x_o. Then the characteristic polynomial of g over K has a form

Assume also that the polynomial q(λ) is irreducible over K. Denote by K_q, A_q, X_q, a_q : X_q → X, D, U, etc. all the objects defined or constructed for the irreducible polynomal q in the same way as in sections 3.1 and 3.2 above for p. Assume that conditions (SD₂), (RF), and (PWD/U) are satisfied. Let X_c and b : X_c → X_q be the k-germ and the finite covering constructed for Y = X_q in Theorem 2.9, a_c = a_q ○ b : X_c → X, D_c, and U_c the inverse images of D and U in X_c. Denote d_c = deg(X_c/X).

Let g_c be a pullback of g onto X_c, e_c,j the k-regular eigenvalue of g_c on X_c corresponding to e_j, m_o + 1 ≤ j ≤ n, by sections 3.2 and 3.3. Put e_c = Πe_c,j. Denote by Z_c the divisor of zeros of e_c in X_c, Z = a_c(Z_c) ⊂ X, and V = X − (D ∪ Z). Because a_c is a finite k-morphism, Z is a closed subgerm of X of codimension one, defined over k. Therefore, V is open in X and it is defined over k. Notice that by the constructions the restriction Ker g|V of the coherent sheaf Ker g onto V is a locally free 𝒪_V-module.

5.3. For a noetherian k-scheme (respectively for a k-analytic space) Z denote by K_o(Z) the Grothendieck group of the category of coherent, locally free 𝒪_Z-modules, or equivalently k-algebraic (respectively k-analytic) vector bundles on Z.

Let Z → H*(Z) be any cohomology theory on the category of k-schemes or k-analytic spaces for which there exists the theory of Chern classes c_r : K_o(Z) → H^2r(Z) with the standard properties (ref. 19, Ch. 16). If H*(Z) is a vector space over a field F of characteristic zero, then the Chern character ch(?) is defined with the values in H*(Z).

Theorem 5.4. Assume that q is an irreducible polynomial over K and that conditions (SD₃), (RF), (PWD/U), and (Mu1/U) for the ramified eigenvalues e_jof g are satisfied. Then

(1) In the k-algebraic category, the restriction Ker g|V is a stably trivial 𝒪_V-module on V. In particular, if rank_𝒪V(Ker g|V) > dim X, then Ker g|V is a free 𝒪_V-module on V.

(2) in the k-analytic category, the d_c-multiple d_c(Ker g|V) of the restriction Ker g|V is a stably trivial 𝒪_V-module on V. In particular, if rank d_c(Ker g|V) > dim X, then d_c(Ker g|V) is a free 𝒪_V-module on V.

An outline of Proof: Let E_c,j be the eigenbundle of the eigenvalue e_c,j on X_c, m_o + 1 ≤ j ≤ n. Denote V_c = a(V). Then we have the following relation on V_c:

Because all the eigenbundles E_c,j, are trivial on U_c, the equality above shows the stable triviality of Ker g_c|V_c. Using the transfer arguments as in the proof of Theorem 4.3, we can derive from this the stable triviality of d_c(Ker g|V) on V in the both k-algebraic and k-analytic cases. But in the k-algebraic case the group K_o(V) = Z has no torsion, so Ker g|V is stably trivial on V itself. The stable rank of 𝒪_V is ≤ dim V (20). Combining these facts we see that if rank_𝒪V(Ker g|V) > dim V, it is free on V.

Corollary 5.5. Under the conditions of 5.4 we have:

(1) In the k-algebraic category the (integral) Chern classes c_r(Ker g|V) of the bundle Ker g|V vanish, for all r > 0.

(2) In the k-analytic category all the Chern classes c_r(d_c(Ker g|V)) of the bundle d_c(Ker g|V) in H^2r(V) vanish, for all r > 0.

(3) If H*(V) is a vector space over a field F of characteristic zero, then the (rational) Chern character ch(Ker g|V) of Ker g|V in H*(V) also vanishes.