# Example of non-holonomic D-module and explicit computation of characteristic variety

I'm currently trying to have a better understanding of the concepts of characteristic variety and holonomic $D$-modules (let us assume that they are coherent) on a holomorphic manifold $X$. I know that for a system of differential equations $P$, the holonomicity of the $D$-module $D_X / D_X P$ means that the system $P$ is maximaly overdetermined. In order to see that, we compute the characteristic variety $\text{char}(D_X/D_X P)$ and if it is lagrangian in $T^* X$, the $D$-module is holonomic by definition.

I'd like to compute explicitely the charactistic variety of two "easy" coherent $D$-modules in order to see the holonomicity. So to be precise, my question is

Find two systems of differential equations $P$ and $Q$ on, let's say $\mathbb{C}^2$, such that $\text{char}(D_X/D_X P)$ and $\text{char}(D_X/D_X Q)$ can be explicitely computed in a short time and such that $\text{char}(D_X/D_X P)$ is lagrangian and $\text{char}(D_X/D_X Q)$ is not.

For example, I've also studied the notion of regularity and for that concept, the examples are easy : $z\partial_z -z$ is regular and $z\partial_z -1$ is not. But I couldn't find such easy examples for holonomicity. Perhaps it is linked with the complexity of the computation of characteristic varieties. (I know it is linked to Gröbner basis, but I don't know very well this theory)

Any examples or literature recommendations will be highly appreciated.

• If you choose coordinates $z_1,z_2$ for $\mathbb{C}^2$ then $Q=0$ and $P=\{\mathrm{d/d}z_1,\mathrm{d/d}z_2\}$ seem to satisfy your requirements. Then $\mathrm{char}(D_X/D_XP)$ is just the zero section of $T^\ast X$ and $\mathrm{char}(D_X/D_XQ)$ is the whole of $T^\ast X$. Do you want to apply further constraints or do these satisfy your requirements? – Simon Wadsley Jan 16 '17 at 12:16
• Mmm, yes I would prefer less trivial examples. But, I have nonetheless a question. How could $\text{char}(D_X/D_X Q)$ be $T^*X$ since we know in general that $\text{dim}\text{char}(A) \leq \text{dim} X.$ ? – C. Dubussy Jan 16 '17 at 12:31
• Can you be more precise about how non-trivial? With regards the inequality, it goes the other way. The dimension of the characteristic variety is at least dim $X$ not at most. – Simon Wadsley Jan 16 '17 at 12:38
• Oh yes, sorry, you're right. It is difficult to precise how complex must be the system, but perhaps it would be very helpful to compute the characteristic variety of something like $$\{f(z,w)\partial_z + g(z,w)\partial_w + h(z,w), f'(z,w)\partial_z + g'(z,w)\partial_w + h'(z,w)\}$$ with $f,g,h,f',g',h'$ polynomials of low degree.Of course some of these polynomials can be constant but not all, if possible. – C. Dubussy Jan 16 '17 at 12:44

Holonomic: $D\cdot\{z_1\partial_2,z_2\partial_1\}$. This left ideal has characteristic ideal of dimension 2, so it is holonomic.
Non-holonomic: let $f=(z_1^3-z_2^2)$; $D\cdot\{f\partial_1+\partial f/\partial z_1, f\partial_2+\partial f/\partial z_2\}=D\cdot\{\partial_1 f,\partial_2 f\}$. This left ideal has char.ideal of dimension 3, so it isn't holonomic.
• @KetilTveiten Isn't the term holonomic rank usually reserved for the dimension of the (holimorphic) solution space near a generic point? I've only ever heard the dimension of the characteristic variety of $M$ referred to as the dimension of $M$. – Avi Steiner Jan 17 '17 at 15:24