Compactified configuration space of points on a line and homotopies of $A_\infty$ morphisms

L-infinity algebras, Cartan homotopies and period maps

We prove that, for every compact Kaehler manifold, the period map of its Kuranishi family is induced by a natural L-infinity morphism. This implies, by standard facts about L-infinity algebras, that the period map is a "morphism of deformation theories" and then commutes with all deformation theoretic constructions (e.g. obstructions).

Logarithms and deformation quantization

We prove the statement/conjecture of M. Kontsevich on the existence of the logarithmic formality morphism. This question was open since 1999, and the main obstacle was the presence of $dr/r$ type singularities near the boundary $r=0$ in the integrals over compactified configuration spaces. The novelty of our approach is the use of local torus actions on configuration spaces of points in the upper half-plane. It gives rise to a version of Stokes' formula for differential forms with singularities at the boundary which implies the formality property. We also show that the logarithmic formality mo…

高次 $L_{\infty}[1]$-代数的同伦模型

Homotopy models for $L_{\infty}[1]$-algebras in higher degrees

本文推广Fukaya-Oh-Ohta-Ono的$A_{\infty}$同伦概念，提出$L_{\infty}[1]$-态射的高阶同伦理论模型，并证明顶点为拟同构的单纯形满足填充条件。

Operads, configuration spaces and quantization

We review several well-known operads of compactified configuration spaces and construct several new such operads, C, in the category of smooth manifolds with corners whose complexes of fundamental chains give us (i) the 2-coloured operad of A-infinity algebras and their homotopy morphisms, (ii) the 2-coloured operad of L-infinity algebras and their homotopy morphisms, and (iii) the 4-coloured operad of open-closed homotopy algebras and their homotopy morphisms. Two gadgets - a (coloured) operad of Feynman graphs and a de Rham field theory on C - are introduced and used to construct quantized r…

Gröbner cells of punctual Hilbert schemes in dimension two

We begin with a comprehensive discussion of the punctual Hilbert scheme of the regular two-dimensional local ring in terms of the Gröbner cells. These schemes are the most degenerate fibers of the Grothendieck-Deligne norm map (the Hilbert-Chow morphism), playing an important role in the study of Hilbert schemes of smooth surfaces. They are generally singular, but their Gröbner cells are affine spaces; they admit an explicit parametrization due to Conca and Valla. We use this to obtain the Gröbner decomposition of compactified Jacobians of plane curve singularities, which is non-trivial even f…