## Gauss-Manin Connection 2

It’s probably been awhile since you read the first post in this series, so I’ll quickly remind you of the key point. ${S}$ is a smooth scheme over a field ${k}$. We fixed connection ${\rho}$ on ${\mathcal{E}}$. Then given a derivation ${\delta}$ corresponding to ${D: \Omega\rightarrow \mathcal{O}_S}$, then for any element of ${\mathcal{D}er_k(\mathcal{O}_S)}$, the sheaf of germs of ${k}$-derivations we can compose the maps we have and we get ${\overline{D}\in \mathcal{E}nd_k(\mathcal{E})}$.
So every connection gives a map ${\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})}$, ${\delta\mapsto \overline{D}}$ and we had a relation between ${D}$ and ${\overline{D}}$, and any such map satisfying the relation comes from a connection.

Now we’ll go back to the construction of the Gauss-Manin connection from last time. We haven’t actually checked that it is a connection or that it is flat. Recall that it is just one of the maps we get from the spectral sequence associated to the filtration of the complex ${\Omega_{X/k}^\cdot}$. Now the filtration is compatible with taking wedge products (${F^i\wedge F^j\subset F^{i+j}}$) and the functors ${\mathbf{R}^q\pi_*}$ are multiplicative, so we have a product structure on the terms of the spectral sequence as follows.

If we take sections of the sheaves over an open, then ${E^{p,q}_r\times E^{p',q'}_r\rightarrow E^{p+p', q+q'}_r}$ by ${(e,e')\mapsto e\cdot e'}$. If you want the actual construction see Godement. The product satisfies a few important properties. We have a type of anti-commutativity ${e\cdot e'=(-1)^{(p+q)(p'+q')}e'\cdot e}$. Also we know how it behaves under the differential map: ${d_r(e\cdot e')=d_r(e)\cdot e'+ (-1)^{p+q}e\cdot d_r(e')}$.

In particular, let’s look at what this product rule for the differential is for the Gauss-Manin map. For ${\nabla=d_1^{0,q}:E_1^{0, q}\rightarrow E_1^{1,q}}$ which is really mapping ${\mathcal{H}^q_{dR}(X/S)\rightarrow \Omega_{S/k}\otimes \mathcal{H}^q_{dR}(X/S)}$, the differential is really just ${d_{S/k}\otimes Id}$. Thus that rule says that ${\nabla(\omega\cdot e)=d\omega\cdot e+(-1)^0\omega\cdot \nabla(e)}$. So it is a connection!

The curvature is easily seen to be ${d_1^{1,q}\circ d_1^{0,q}}$ and since the ${d_1}$‘s are maps of a complex we get that it is ${0}$, and hence ${\nabla}$ is flat and hence the Gauss-Manin connection is integrable. We’ve now proved the theorem that any smooth ${k}$-morphism of smooth ${k}$-schemes gives rise to a canonical integrable connection on the relative de Rham cohomology sheaves that is compatible with the cup product.

If you want a more explicit way to see what the map is see the paper, but it is kind of tedious since writing out how it appears in the spectral sequence you will quickly find that it is the connecting homomorphism when taking the long exact sequence after applying the functor ${\mathbf{R}^q\pi_*}$ to the exact sequence ${0\rightarrow \mathrm{gr}^{p+1}\rightarrow F^p/F^{p+2}\rightarrow \mathrm{gr}^p\rightarrow 0}$.

This is the third post on this topic, and I haven’t given you a reason to care yet. Here’s why we should care. One would hope (via a conjecture of Grothendieck) that there is some sort of relative de Rham Leray Spectral Sequence: ${E_2^{p,q}=\mathbf{H}^p(S, \Omega_{S/k}^\cdot \otimes_{\mathcal{O}_S} \mathcal{H}^q(X/S))\Rightarrow H_{dR}^{p+q}(X/k)}$. For the ${E_2}$-term to make any sort of sense we needed ${\Omega_{S/k}^\cdot \otimes_{\mathcal{O}_S} \mathcal{H}^q(X/S)}$ to be a complex, and since the Gauss-Manin connection is integrable it is a complex. Also, ${H_{dR}^{p+q}(X/k)}$ is defined to be ${\mathbf{H}^{p+q}(X, \Omega_{X/k}^\cdot)}$.

It turns out that when ${S}$ is affine such a spectral sequence exists. In case you’re wondering, affineness is needed for a nice proof of this because it makes certain cohomologies vanish. Deligne has proved it in a more complicated way when ${S}$ is not affine (but still with our standing assumptions). This is of great importance in proving every K3 surface lifts from characteristic ${p}$ to characteristic ${0}$.

## Gauss-Manin Connection 1

We’ll shift gears for a second and define the relative de Rham cohomology of a scheme. The stuff from the last post will eventually play a role in this. Let ${\pi: X\rightarrow S}$ be a smooth map of (smooth) ${k}$-schemes. We define the relative de Rham cohomology sheaf to be the quasi-coherent sheaf ${\mathcal{H}^q_{dR}(X/S):=\mathbf{R}^q\pi_*(\Omega_{X/S}^\cdot)}$. Note that the R is bolded to mean we take hypercohomology, so that the ${\mathbf{R}^q\pi_*}$ is the hyperderived functor of ${\pi_*}$. Note that ${\mathcal{H}^q_{dR}(X/S)}$ are sheaves of graded anticommutative ${\mathcal{O}_S}$-algebras.

We now will describe a canonical integrable (${0}$ curvature) connection on these sheaves called the Gauss-Manin Connection denoted ${\nabla :=\nabla(X/S, q)}$. First notice that whenever we have a smooth map the fundamental sequence ${0\rightarrow \pi^*(\Omega_{S/k})\rightarrow \Omega_{X/k}\rightarrow \Omega_{X/S}\rightarrow 0}$ is exact. We now consider the filtration on the complex ${\Omega_{X/k}^\cdot}$.

${\displaystyle\Omega_{X/k}^\cdot=F^0(\Omega_{X/k}^\cdot)\supset F^1(\Omega_{X/k}^\cdot)\supset F^2(\Omega_{X/k}^\cdot)\supset \cdots}$ where ${\displaystyle F^i(\Omega_{X/k}^\cdot)=\mathrm{im}\left(\Omega_{X/k}^\cdot [-i]\otimes_{\mathcal{O}_X}\pi^*\Omega^i_{S/k}\rightarrow \Omega_{X/k}^\cdot\right)}$

The smoothness tells us that ${\Omega_{X/k}^i}$ and ${\Omega_{S/k}^i}$ are locally free, so the exactness of the fundamental sequence allows us to compute the associated graded with respect to this filtration as ${\mathrm{gr}^i:=\mathrm{gr}^i(\Omega_{X/k}^\cdot)=F^i/F^{i+1}=\pi^*(\Omega_{S/k}^i)\otimes_{\mathcal{O}_X} \Omega^\cdot_{X/S}[-i]}$.

Now the functor ${\mathbf{R}^0\pi_*}$ takes complexes of (abelian) sheaves on ${X}$ to complexes of sheaves on ${S}$, and the derived functors are just ${\mathbf{R}^q\pi_*}$. Now we can take the spectral sequence associated to a filtration. It takes the form ${E_1^{p,q}=\mathbf{R}^{p+q}\pi_*(\mathrm{gr}^p)\Rightarrow \mathbf{R}^q\pi_*(\Omega_{X/k}^\cdot)}$.

We can just work out that ${E_1}$ term more explictly using our previous calculation.

${\mathbf{R}^{p+q}\pi_*(\mathrm{gr}^p)=\mathbf{R}^{p+q}\pi_*(\pi^*(\Omega_{S/k}^p)\otimes_{\mathcal{O}_X} \Omega^\cdot_{X/S}[-p])}$, then just by shifting for the first equality, and then projection formula for next iso that whole thing is ${=\mathbf{R}^{q}\pi_*(\pi^*(\Omega_{S/k}^p)\otimes_{\mathcal{O}_X} \Omega^\cdot_{X/S})\simeq \Omega^p_{S/k}\otimes_{\mathcal{O}_S}\mathbf{R}^q\pi_*(\Omega_{X/S}^\cdot)}$

Which is just ${\Omega^p_{S/k}\otimes_{\mathcal{O}_S}\mathcal{H}^q_{dR}(X/S)}$

Let’s consider the ${E_1}$ terms. The degree of ${d_1}$ is ${(1,0)}$ so we get maps (for every ${q}$) of the following form ${0\rightarrow \mathcal{H}^q_{dR}(X/S)\rightarrow \Omega^1_{S/k}\otimes \mathcal{H}^q_{dR}(X/S)\rightarrow\Omega^2\otimes \mathcal{H}^q_{dR}(X/S)\rightarrow \cdots}$. The maps shown here are ${d_1^{0,q}}$ and ${d_1^{1,q}}$.

It turns out that ${\nabla=d_1^{0,q}}$ is actually a flat connection. We’ll check this next time, but taking this to be true, we get the Gauss-Manin connection on ${\mathcal{H}^q_{dR}(X/S)}$. I just thought this was a really neat way to construct a canonical flat connection. There is another way to construct this by using Čech computations, and this paper actually goes through and checks that it is actually the same map, but we probably won’t do this.

## Connections and Curvature … on Schemes! … in Characteristic p?!

I feel bad about my absence. I lasted posted during winter break, and now winter quarter is completely over. I kept meaning to do a series on “well-known” algebraic geometry results and constructions that don’t appear with any amount of thoroughness in the references. I thought it would be good to get that information out there. Unfortunately, I had already written these things down into a notebook and just couldn’t motivate myself to type something up that I already had. Anyway, one thing led to another and I didn’t do any posts. I’m not sure why I’m trying to justify my absence with an excuse.

Recently I’ve been typing up a translation of Deligne’s argument (written down by Illusie) that every K3 surface in characteristic ${p>0}$ lifts to characteristic ${0}$. I’m not to the point of trying to understand it, but I wanted a typed version, so that when I get the background material (namely crystalline cohomology!) and go to understand it, I can just fill in the details into my typed notes quickly and easily. I also was curioius as to the overall format of the argument.

This led me to the 1968 paper by Katz and Oda called On the Differentiation of de Rham Cohomology Classes with Respect to Parameters. The next few posts will be about the main result from this paper. It is really quite amazing.

First, some definitions. We’ll always be working with a smooth scheme ${S}$ over a field ${k}$ (no assumptions here!). Let ${\mathcal{E}}$ be a quasi-coherent sheaf of ${\mathcal{O}_S}$-modules. We’ll write ${\Omega}$ for ${\Omega^1_{S/k}}$ and unless otherwise noted, all tensor products will be over ${\mathcal{O}_S}$. We say that ${\nabla}$ is a connection on ${\mathcal{E}}$ if it is a homomorphism ${\nabla: \mathcal{E}\rightarrow \Omega\otimes \mathcal{E}}$ that satisfies the “Leibniz rule”.

In other words, ${\nabla(fg)=f\nabla(g)+df\otimes g}$. This is the standard shorthand meaning ${\nabla(U): \mathcal{E}(U)\rightarrow \Omega(U)\otimes \mathcal{E}(U)}$ satisfies the rule where ${f\in \mathcal{O}_S(U)}$, ${g\in \mathcal{E}(U)}$ and ${df}$ is the image of ${f}$ under the universal map ${\mathcal{O}_S\rightarrow \Omega}$.

Given a connection ${\rho}$, we get homomorphisms for all ${i}$, ${\rho_i: \Omega^i\otimes\mathcal{E}\rightarrow \Omega^{i+1}\otimes \mathcal{E}}$. These are given by ${\rho_i(\omega\otimes e)=d\omega \otimes e+(-1)^i \omega\wedge \rho(e)}$.

The notation is just the one that makes sense: ${\rho(e)\in \Omega\otimes \mathcal{E}}$, so it looks like ${\tau\otimes \epsilon}$. So we define ${\omega\wedge \rho(e)=\omega\wedge(\tau\otimes \epsilon)}$ to be ${(\omega\wedge \tau)\otimes \epsilon\in \Omega^{i+1}\otimes \mathcal{E}}$.

Now we define the curvature of the connection ${K:\mathcal{E}\rightarrow \Omega^2\otimes \mathcal{E}}$ to be the map ${\rho_1\circ \rho}$. The curvature is related to the other ${\rho_i}$ by an easy check ${\rho_{i+1}\circ \rho_i(\omega\otimes e)=\omega\wedge K(e)}$.

This gives some sort of meaning to the curvature now. If the curvature is ${0}$, then the natural de Rham-like sequence we get from a connection by stringing together the ${\rho_i}$ as follows ${0\rightarrow \mathcal{E}\stackrel{\rho}{\rightarrow} \Omega\otimes\mathcal{E}\stackrel{\rho_1}{\rightarrow} \Omega^2\otimes \mathcal{E}\rightarrow \cdots}$ is an honest complex that we can take cohomology with respect to, since ${\rho_{i+1}\circ \rho_i=0}$.

When this happens we call the connection ${\rho}$ integrable. Now let ${\mathcal{D}er_k(\mathcal{O}_S)}$ be the sheaf of germs of ${k}$-derivations of ${\mathcal{O}_S}$ into itself. From the fact that the module of differentials is a representing object, we get that as a sheaf of ${\mathcal{O}_S}$-modules, ${\mathcal{D}er_k(\mathcal{O}_S)\simeq \mathcal{H}om_{\mathcal{O}_S}(\Omega, \mathcal{O}_S)}$.

Let ${\mathcal{E}nd_k(\mathcal{E})}$ be the sheaf of germs of ${k}$-linear endomorphisms of ${\mathcal{E}}$. Given any connection ${\rho}$ on ${\mathcal{E}}$ we get an induced ${\mathcal{O}_S}$-linear map ${\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})}$ as follows. Let ${\delta}$ be a derivation, then it corresponds to a map ${D: \Omega\rightarrow \mathcal{O}_S}$.

So consider the composition ${\overline{D}:\mathcal{E}\rightarrow \Omega\otimes \mathcal{E}\rightarrow \mathcal{O}_S\otimes \mathcal{E}\simeq \mathcal{E}}$, where the first is the connection and the second is ${D\otimes Id}$. This gives the map ${\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})}$ as ${\delta\mapsto \overline{D}}$.

Lastly for today, note that we get a nice relation between ${D}$ and ${\overline{D}}$ as follows ${\overline{D}(fe)=D(f)e+f\overline{D}(e)}$ and that any map ${\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})}$ satisfying this relation comes from a unique connection on ${\mathcal{E}}$.

Today was just a bunch of notation and definitions, but next time it should get more interesting.