# Is there a theory that describes eigenspaces of matrix functions near degeneracies?

Let $$X$$ be a smooth manifold and let $$h$$ be a smooth map from $$X$$ to hermitian $$N \times N$$ matrices. Under favorable circumstances (one sufficient condition: all eigenvalues of $$h(x)$$ are simple for every $$x \in X$$) this datum determines a number of vector bundles over $$X$$ whose fibers are eigenspaces of $$h(x)$$. These bundles are all subbundles of a fixed trivial bundle with fiber $$\mathbb C^N$$, whose direct sum is the whole trivial bundle.

More generally one could consider the situation in which the above is true generically but fails on some locus $$Y \subset X$$ (e.g. the subset of $$X$$ on which $$h(x)$$ does not have simple spectrum). One nice example is $$X= \mathbb R^3$$, $$h(x) = \begin{bmatrix} x_3 & x_1 - \mathrm{i} x_2 \\ x_1 + \mathrm{i} x_2 & -x_3 \end{bmatrix}$$ for which the "bad locus" is $$\{ 0 \}$$.

I would like to know if there is some general theory which describes the behaviour of eigenspaces at "bad points". I suspect that this problem should have interesting local and global aspects; I am interested in both.

• What about considering maximal flags at each point? This is equivalent to make a triangulation of your matrix. I think this should result in a filtration in subbundles defined at each point, and this should degenerate smoothly. Commented Mar 29, 2021 at 11:36

Question: "I would like to know if there is some general theory which describes the behaviour of eigenspaces at "bad points". I suspect that this problem should have interesting local and global aspects; I am interested in both."

Remark: I post this remark since one of the tags is an "algebraic geometry" tag: Let $$k$$ be the real numbers and let $$K$$ be the complex numbers.

Let $$A:=k[x_1,x_2,x_3]/(f)$$ where $$f:=x_1^2+x_2^2+x_3^2-1$$. We may view the "zero set" $$Z:=V(f)$$ as a "sub set" of real affine 3-space $$\mathbb{A}^3_{k}:=Spec(k[x_1,x_2,x_3])$$. If $$h(x)$$ is your matrix, you get the following:

$$h(x) \circ h(x) = (x_1^2+x_2^2+x_3^2)I(2)$$

where $$I(2)$$ is the $$2 \times 2$$ identity matrix and $$\circ$$ denotes matrix multiplication. Here we view $$h(x)$$ as an element of $$Mat(K[x_1,x_2,x_3],2)$$ - the ring of $$2\times 2$$-matrices with coefficients in $$K[x_1,x_2,x_3]$$.

Hence if $$Z:=V(x_1^2+x_2^2+x_3^1-1) \subseteq X:=\mathbb{A}^3_{k}$$ is the real 2-sphere and if you consider the complexification $$Z_K:=K\times_k Z$$ and restrict the matrix $$h(x)$$ to $$Z_K$$, you get a $$2\times 2$$-matrix

$$h(x) \in Mat(K\otimes_k A,2)$$

with the property that $$h(x) \circ h(x)= I(2) \in Mat(K\otimes_k A,2)$$. If $$\phi:=\frac{1}{2}(h(x) +I(2)) \in Mat(K\otimes_k A,2)$$ you get an idempotent element:

$$\phi \circ \phi = \phi$$

and to $$\phi$$ you get a finite rank projective $$K\otimes_k A$$-module $$E(\phi)$$. To this you get a finite rank algebraic vector bundle on the complex 2-sphere $$Z_K$$. If $$B:=K[x_1,x_2,x_3]$$ it follows $$\phi \in Mat(B,2)$$ is a $$2\times 2$$-matrix with the property that when you restrict it to the complex 2-sphere, it is an idempotent and defines a vector bundle. Let $$E:=\tilde{B^2}$$ be the trivial vector bundle of rank 2 on $$B$$ and let $$F:=\tilde{(K\otimes_k A)^2}$$ be the trivial rank $$2$$ vector bundle on $$K\otimes_k A$$. The map $$\phi$$ gives a morphism of vector bundles (or $$B$$-modules)

$$\phi: E \rightarrow E$$

with the property that when you restrict it to the complex 2-sphere $$Z_K$$ and $$F$$

$$\phi: F \rightarrow F$$

it is an idempotent and defines a vector bundle on $$K\otimes_k A$$. Let $$L_1:=ker(\phi)$$ and $$L_2:=Im(\phi)$$. It follows there is an exact sequence of $$K\otimes_k A$$-modules

$$0 \rightarrow L_2 \rightarrow F \rightarrow L_1 \rightarrow 0,$$

and it seems $$L_i$$ are rank one vector bundles on $$K\otimes_k A$$. This is phrased in the language of commutative algebra/algebraic geometry.

Example: It seems you may construct a family of examples as follows: If $$f_{\beta}:=x_1^2+x_2^2+x_3^2-\beta^2$$ with $$\beta \in k^*$$ and let $$A_{\beta}:=k[x_i]/(f_{\beta})$$. Let $$K\otimes_k A_{\beta}$$ be the complexification. If you define the matrix

$$\phi_{\beta}:=\frac{1}{2}(\frac{1}{\beta}h(x)+I(2))\in Mat(K\otimes_k A_{\beta},2)$$

you get an idempotent matrix

$$\phi_{\beta} \circ \phi_{\beta}=\phi_{\beta}$$

and corresponding linebundles $$L(\beta)_i$$ on $$A_{\beta}$$. How do you phrase this in terms of "smooth manifolds"?

Note: If $$X$$ is any scheme and $$E$$ is a finite rank locally trivial sheaf (or coherent module) with an endomorphism $$\phi$$ you may consider the loci of points $$x \in X$$ where the induced map at the fiber

$$\phi(x): E(x) \rightarrow E(x)$$

satisfies various conditions. Hence for a scheme $$X$$ it may be your question may be phrased in terms of degeneracy locies of such morphisms of coherent sheaves.

Example: For simplicity if $$X:=Spec(A)$$ and $$\mathcal{E}$$ is a finite rank locally trivial $$\mathcal{O}_X$$-module, you may view $$\phi$$ as an element

$$\phi \in H^0(X, \mathcal{End}(\mathcal{E}))$$

and to $$\phi$$ you sometimes get a surjection

$$\phi^* : \mathcal{End}(\mathcal{E})^* \rightarrow \mathcal{O}_X \rightarrow 0$$

and a section $$\phi^*: X \rightarrow \mathbb{P}(\mathcal{End}(\mathcal{E})^*)$$

of the projection morphism. Here $$\mathbb{P}(\mathcal{End}(\mathcal{E})^*)$$ is the projective space bundle of $$\mathcal{End}(\mathcal{E})$$. Similar constructions exist for differentiable manifolds and complex manifolds.

References: For complex manifolds and holomorphic vector bundles you find a treatment of degeneracy locies and Chern classes in the section on the Gauss Bonnet formulas (page 413) in Griffiths/Harris book "Principles of algebraic geometry". Similar constructions can be done for real smooth manifolds and real smooth vector bundles.