# Atiyah class and coboundary map

Let $$L$$ be a line bundle on a smooth algebraic variety $$X$$. Let $$\sigma_i:U_i \times \mathbb{C} \to L_{|U_i}$$ be its local trivializaations and $$u_{ij}$$ the transition functions satisfying $$\sigma_j=u_{ij}\sigma_i$$. Following Welters "Polarized Abelian Varieties and the Heat Equations" (Comp. Math 1983 - Lemma 1.16), let $$L \in H^1(X,\Omega_X^1)$$ be the class given by the cocyle {$$\frac{du_{ij}}{u_{ij}}$$} and consider the short exact sequence

$$0 \to \mathcal{O}_X \to Diff^{(1)}(L) \to T_X \to 0,\ \ \ \ \ \ (*)$$

where $$Diff^{(1)}(L)$$ is the sheaf of 1st order differential operators on $$L$$ and $$T_X$$ is the tangent sheaf. The RHS map is just the symbol map of $$L$$. Welters claims that the coboundary map $$\delta: H^0(X,T_X) \to H^1(X,\mathcal{O}_X)$$ is given by $$- \cup [L]$$, i.e. minus the cup product by the class $$[L]$$.

On the other hand Atiyah (COMPLEX ANALYTIC CONNECTIONS IN FIBRE BUNDLES - 1957, Sect. 4) claims that the extension class of (*), i.e. the Atiyah class of $$L$$ in $$H^1(X,\Omega_X^1)$$, is given by $$[L]$$. But then the coboundary map should be given by $$\cup [L]$$ so there seems to be a contradiction. Does anybody see where the problem is?

• The sign of the coboundary map depends very much on the chosen conventions. I am sure that if you analyze thoroughly the definitions in both papers you'll find out where the difference comes from. – abx Mar 14 at 2:23
• @abx: yeah, that's what I thought as well. unfortunately, browsinf the paper I have found no trace of such change of convention. Maybe they simply tacitely assume opposite conventions? – IMeasy Mar 14 at 16:43