Meaning of the determinant of cohomology

Question

The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory that behaves well up to linear equivalence of divisors in the arithmetic case. In particular, we need some analytic data on the fibers at infinity and a new extended concept of divisors, namely Arakelov divisors.

Everything works well and we even have a correspondence between Arakelov divisors and metrized line bundles. The big problem is that, of course, we don't have any cohomology theory for such kind of line bundles, because the data at infinity is somewhat artificial.

The Faltings-Riemann-Roch theorem deals with the so called determinant of the cohomology introduced by Deligne. Formally, it is a way to associate to each coherent module on our surface $X$, a line bundle to the base scheme.

I think that it should be something similar to the concept of dimension of some cohomology group. But the problem is that I'm not able to understand what is the intuition behind this new tool. What information do we want to capture with the determinant of cohomology? What is the analogy with the geometric case?

Many thanks in advance

Answer 1

It doesn't so much represent a dimension of a cohomology group as it does an Euler characteristic.

More precisely, it's based on Grothendieck's generalization of the Riemann-Roch theorem to families. Given a proper map $\pi: Y\to X$ and a line bundle $L$ on $Y$, we could of course expect Riemann-Roch or a generalize to give us a formula for the Euler characteristic of the fiber $\sum_i (-1)^i \dim H^i(Y_x, L)$, which can be calculated using the cohomology sheaves $R^i \pi_* L$. But in fact this is forgetting a lot of information, and its better to calculate the class $\sum_i (-1)^i [ R^i \pi_* L]$ in $K$-theory, which contains this dimensional information, but also other information.

In the case when the base $X$ is a smooth curve, the $K$-theory group is $\mathbb Z \times \operatorname{Pic} X$, where the $\mathbb Z$ comes from calculating the rank of the coherent sheaves $R^i \pi_* L$, and the $\operatorname{Pic} X$ comes from the determinant.

So determinant-of-cohomology is a generalization of the K-theoretic information you get by taking cohomology down to the base curve.

Moreover, one piece of information that is contained in the K-theory classes of the cohomology groups $R^i \pi_* L$ is their Euler characteristics $\chi(X,R^i \pi_* L)$, and we have $\chi(Y,L) = \sum_i (-1)^i \chi(X, R^i \pi_* L)$. So in fact, in the geometric setting of a family of varieties of a curve, the $K$-theory class determines the Euler characteristic of the absolute cohomology groups - it's something like the rank times $1-g$ plus the degree of the determinant-of-cohomology.

The rank is just the usual Euler characteristic of the generic fiber, so it's not so arithmetically interesting, so we can think of the determinant of cohomology as a substitute for the Euler characteristic of the missing arithmetic cohomology theory.

Answer 2

Let us consider the arithemetic surface case, which is already very difficult (see the recent work by Gerald Montplet, for example). In this case Faltings-Riemann-Roch established that $$ \chi(O_{X}(D))-\chi(O_{X})=\frac{1}{2}(D,D-K) $$ Therefore formally in order to evaluate $\chi(O_{X}(D))$ it is suffice to find $\chi(O_{X})$.

There are however at least two issues at here, one is that the intersection pairing $$ (D, D-K) $$ in Arakelov theory depends on the choice of Green functions, which is dependent upon the choice of the Arakelov metric. If we use a different metric, then everything will have to be changed. The other issue is Faltings' construction using the technique of "reduction to the Jacobian" in an essential way and cannot be easily generalized to higher dimensions (see the MathScithnet review, for example).

The one-step "magic solution" of the issue essentially comes from the introduction of the Quillen metric, namely finding a metric on the determinant line bundle over $X_{\infty}$ that satisfies the following properties:

1) It should be smooth on the moduli space of genus $g$ complex algebraic curves. Namely, when we "slide" on the "huge" space parametrizing the whole space of genus $g$ complex algebraic curves, the metric on the determinant line bundle should be a smooth function.

2) It should be compatible with Serre duality.

3) It should not be really dependent upon the metric. Namely if we have two different metrics $d_1, d_2$ on $X$, then the metric on the determinant of cohomology should be independent of $s$ for $d_s=sd_1+(1-s)d_2$. In other words, it should be coming from a topological invariant.

4) It should be naturally generalizable to higher dimensions, not using special nice properties only available for Riemann surfaces.

In Ray-Singer's first paper, they proved (3) for analytic torsion (not for Quillen metric, which did not exist by then). For a survey paper on this, see Pavel Mnev's article. In Quillen's paper, he claimed (1) and Soule provided a detailed proof in his book. I think (2) comes from Deligne's introduction of Deligne pairing and (4) comes from Bismut-Gillet-Soule's work. I think this is partly why analytic torsion considered to be so important for Arakelov theory. I do not know any other analytic invariant that can satisfy (1)-(4) in the same time. If it exists, then it has potential to be the building block for a generalized Arakelov cohomology theory.

Part of the difficulty is that analytic torsion is a "secondary global invariant" that involves all positive time of the heat kernel (you have to deal with $Tr(\int^{\infty}_{0}t^{s}e^{-\Delta t}dt|'_{s=0}$). So any potential good candidate that can define $h^{0}_{Ar}, h^{1}_{Ar}$ would have to involve spectral theory of elliptic operators in an essential way. And it is not just the $\zeta$-function of the operator, but its derivative at $s=0$. Therefore the problem become exceedingly difficult and from what I read even computing $\chi(O_{X})$ for $\mathbb{P}^{N}$ in general is not easy.

I am really a beginner in the field and maybe for you discussing this with experts (Gillet, Soule, Bost, Montplet, Faltings, etc) will be helpful.