JournalsjncgVol. 9, No. 1pp. 161–184

The Dolbeault dga of the formal neighborhood of the diagonal

  • Shilin Yu

    University of Pennsylvania, Philadelphia, USA
The Dolbeault dga of the formal neighborhood of the diagonal cover
Download PDF

Abstract

A well-known theorem of Kapranov states that the Atiyah class of the tangent bundle TXTX of a complex manifold XX makes the shifted tangent bundle TX[1]TX[-1] into a Lie algebra object in the derived category D(X)D(X). Moreover, he showed that there is an LL_\infty-algebra structure on the shifted Dolbeault resolution (AX1(TX),)(\mathcal A^{\bullet-1}_X(TX),\overline{\partial}) of TXTX and wrote down the structure maps explicitly in the case when XX is Kähler. The corresponding Chevalley-Eilenberg complex is isomorphic to the Dolbeault resolution (AX0,(JX),)(\mathcal A^{0,\bullet}_X(\mathcal{J}^\infty_X),\overline{\partial}) of the jet bundle JX\mathcal{J}^\infty_X via the construction of the holomorphic exponential map of the Kähler manifold. In this paper, we show that (AX0,(JX),)(\mathcal A^{0,\bullet}_X(\mathcal{J}^\infty_X),\overline{\partial}) is naturally isomorphic to the Dolbeault dga (A(XX×X(),)(\mathcal A^\bullet(X_{X \times X}\scriptscriptstyle {(\infty)},\overline{\partial}) associated to the formal neighborhood of the diagonal of X×XX \times X which we introduced in [15]. We also give an alternative proof of Kapranov's theorem by obtaining an explicit formula for the pullback of functions via the holomorphic exponential map, which allows us to study the general case of an arbitrary embedding later.

Cite this article

Shilin Yu, The Dolbeault dga of the formal neighborhood of the diagonal. J. Noncommut. Geom. 9 (2015), no. 1, pp. 161–184

DOI 10.4171/JNCG/190