# Bivectors and the canonical bundle

Given a smooth projective variety $X$, together with an ample line bundle $L \to X$, we can consider the sheaf $\mathcal{D}^1(L)$ of first-order differential operators on $L$. The corresponding symbol map defines a short-exact sequence (with $\mathcal{T}_X$ denoting the tangent sheaf of $X$)

$$0 \to \mathcal{O}_X \to \mathcal{D}^1(L) \overset{\sigma^{(1)}}{\to} \mathcal{T}_X \to 0$$

which induces a map on cohomology $$\sigma^{(1)}: H^1(X,\mathcal{D}^1(L)) \to H^1(X,\mathcal{T}_X).$$ One can interpret this map as a forgetful map in terms of deformation theory: the first cohomology of first order differential operators parametrizes infinitesimal deformations of the pair $(X,L)$, and any such infinitesimal deformation is mapped to the infinitesimal deformation of the variety $X$ alone by the symbol.

We can similarly look at the short exact sequence for second order differential operators:

$$0 \to \mathcal{D}^1(L) \to \mathcal{D}^2(L) \overset{\sigma^{(2)}}{\to} \text{Sym}^2\mathcal{T}_X \to 0 .$$ Combining the connecting homomorphism with the first-order symbol yields the map

$$\mu_L : H^0(X,\text{Sym}^2\mathcal{T}_X) \to H^1(X,\mathcal{T}_X) ,$$

Question:

Can one prove (perhaps using Cech cohomology) that $$\mu_L =\cup\ \Big(\ \frac{1}{2}[\omega_X]-[L]\ \Big),$$ where $[L] \in H^1(X,\Omega_X^1)$ denotes the first Chern class of $L$, and $[\omega_X]$ is the Chern class of the canonical bundle of $X$.

Context:

This map arises in the context of the Hitchin connection, a flat projective connection on the bundles of non-abelian theta functions over the moduli space of curves. The main reference for this is Hitchin's 1990 paper Flat Connections and Geometric Quantization), but it actually goes back to a topic discussed in its precursor, Welters's Polarized Abelian Varieties and the Heat Equations.

In general, there is way to come up with first-order deformations (of pairs $(X,L)$ of a variety and a line-bundle, as well as triples $(X,L,[s])$ where $s$ is a section of $L$,) from symmetric quadratic tangent fields using the long-exact sequence in cohomology associated to the second-order symbol map above.

Welters proves (Lemma (1.16)), using Cech cohomology, that

$$\mu_L = \mu_{\mathcal{O}_X} - \cup [L].$$

(He then goes on to show that for abelian varieties, which is the case he is interested in, $\mu_{\mathcal{O}_X} = 0$).

Hitchin, concerned with moduli spaces of higher rank bundles on an underlying curve that is being deformed, uses differential-geometric methods to show (on pg. 364, though the fact is stated somewhat implicitly) that actually the term involving the trivial bundle contributes half the canonical class as stated above:

$$\mu_L = \cup (\frac{1}{2}[\omega_X]-[L]) .$$

I would like to reproduce this result in purely algebro-geometric terms.

• Are you certain that your formula is correct? I realize that you specify that $L$ is ample, but consider the case when $L$ is a degree $0$ line bundle on a Riemann surface. In that case, both of the short exact sequences defining $\sigma^{(1)}$ and $\sigma^{(2)}$ are split, since there is an integrable connection on $L$. Thus, $\mu_L$ is the zero map and $[L]$ is zero, but $[\omega_X]$ need not equal zero. – Jason Starr Nov 28 '17 at 11:53
• Thanks for the comments. The formula is my understanding (which I think is correct) of what Hitchin obtains, but the reason I am asking is that I haven't been able to reproduce this in algebraic geometry. Hitchin is working manifestly in a Kahler setting, which is why it seems the ample condition must play a role, but I don't quite understand where it comes into play. – Johan Nov 28 '17 at 12:14
• I think that I was wrong about the splitting of those exact sequences. I was assuming that they were (up to taking duals and twists) the same as the Atiyah extension of the line bundle. The Atiyah extension is split if $[L]$ equals $0$. The first exact sequence does appear to equal the Atiyah extension. However, the second exact sequence is not the Atiyah sequence. So now I believe the formula is correct. – Jason Starr Nov 28 '17 at 12:21
• The first sequence is indeed the corresponding Atiyah algebroid. One could even generalize the question, and try to understand the connecting homomorphism for the short exact sequence given by the symbol map of higher order differential operators (even if the Atiyah algebroid splits, the same need not be true for any of these), but it is degree two that is of relevance for the Hitchin connection. In any case, because of the rigidity of the Riemann sphere the situation is not so instructive there. – Johan Nov 28 '17 at 12:32
• Have you looked at "A proof of Jantzen conjectures", Corollary 2.4.6 or its precursor, "Projectively flat connections", Corollary A2.6? – Pavel Safronov Nov 28 '17 at 12:43