We extend Kontsevich’s formality morphism to a homotopy braces morphism and to a homotopy Gerstenhaber morphism. We show that this morphism is homotopic to Tamarkin’s formality morphism, obtained using formality of the little disks operad, if in the latter construction one uses the Alekseev–Torossian associator. Similar statements can also be shown in the “chains” case, that is, on Hochschild homology instead of cohomology. This settles two well-known and long-standing problems in deformation quantization and unifies the several known graphical constructions of formality morphisms and homotopies by Kontsevich, Shoikhet, Calaque, Rossi, Alm, Cattaneo, Felder, and the author.

Given a quasiprojective variety $X$ with only Kawamata log terminal singularities, we study the obstructions to extending finite étale covers from the smooth locus $X_{\mathrm{reg}}$ of $X$ to $X$ itself. A simplified version of our main results states that there exists a Galois cover $Y\to X$, ramified only over the singularities of $X$, such that the étale fundamental groups of $Y$ and of $Y_{\mathrm{reg}}$ agree. In particular, every étale cover of $Y_{\mathrm{reg}}$ extends to an étale cover of $Y$. As a first major application, we show that every flat holomorphic bundle defined on $Y_{\mathrm{reg}}$ extends to a flat bundle that is defined on all of $Y$. As a consequence, we generalize a classical result of Yau to the singular case: every variety with at worst terminal singularities and with vanishing first and second Chern class is a finite quotient of an abelian variety. As a further application, we verify a conjecture of Nakayama and Zhang describing the structure of varieties that admit polarized endomorphisms.