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.

notthe Atiyah sequence. So now I believe the formula is correct. $\endgroup$ – Jason Starr Nov 28 '17 at 12:21