We prove a conjecture of Kontsevich, which states that if $A$ is an $E_d-1$ algebra then the Hochschild cochain object of $A$ is the universal $E_d$ algebra acting on $A$. The notion of an $E_d$ algebra acting on an $E_{d-1}$ algebra was defined by Kontsevich using the Swiss cheese operad of Voronov. The degree $0$ and $1$ pieces of the Swiss cheese operad can be used to build a cofibrant model for $A$ as an $E_{d-1}$–$A$–module. The theorem amounts to the fact that the Swiss cheese operad is generated up to homotopy by its degree $0$ and $1$ pieces.

We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound $R\left(X\right)$ to the existence of conical Kähler–Einstein metrics on a Fano manifold $X$. In particular, if $D\in |-K_X|$ is a smooth divisor and the Mabuchi $K$–energy is bounded below, then there exists a unique conical Kähler–Einstein metric satisfying $Ric\left(g\right)=\beta g+\left(1-\beta \right)\left[D\right]$ for any $\beta \in \left(0,1\right)$. We also construct unique conical toric Kähler–Einstein metrics with $\beta =R\left(X\right)$ and a unique effective $\mathbb{Q}$–divisor $D\in \left[-K_X\right]$ for all toric Fano manifolds. Finally we prove a Miyaoka–Yau-type inequality for Fano manifolds with $R\left(X\right)=1$.

The quantum period of a variety $X$ is a generating function for certain Gromov–Witten invariants of $X$ which plays an important role in mirror symmetry. We compute the quantum periods of all $3$–dimensional Fano manifolds. In particular we show that $3$–dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a collection of Laurent polynomials called Minkowski polynomials. This was conjectured in joint work with Golyshev. It suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors.

Our methods are likely to be of independent interest. We rework the Mori–Mukai classification of $3$–dimensional Fano manifolds, showing that each of them can be expressed as the zero locus of a section of a homogeneous vector bundle over a GIT quotient $V∕∕G$, where $G$ is a product of groups of the form ${GL}_n\left(ℂ\right)$ and $V$ is a representation of $G$. When $G={GL}_1{\left(ℂ\right)}^r$, this expresses the Fano $3$–fold as a toric complete intersection; in the remaining cases, it expresses the Fano $3$–fold as a tautological subvariety of a Grassmannian, partial flag manifold, or projective bundle thereon. We then compute the quantum periods using the quantum Lefschetz hyperplane theorem of Coates and Givental and the abelian/non-abelian correspondence of Bertram, Ciocan-Fontanine, Kim and Sabbah.

We study the derived functors of the components ${\Gamma }_{\mathbb{Z}}^d\left(A\right)$ of the divided power algebra ${\Gamma }_{\mathbb{Z}}\left(A\right)$ associated to an abelian group $A$, with special emphasis on the $d=4$ case. While our results have applications both to representation theory and to algebraic topology, we illustrate them here by providing a new functorial description of certain integral homology groups of the Eilenberg–Mac Lane spaces $K\left(A,n\right)$ for $A$ a free abelian group. In particular, we give a complete functorial description of the groups $H_{\ast }\left(K\left(A,3\right);\mathbb{Z}\right)$ for such $A$.

A complex manifold $X$ of dimension $n$ is said to be $q$–complete for some $q\in \left\{1,\dots ,n\right\}$ if it admits a smooth exhaustion function whose Levi form has at least $n-q+1$ positive eigenvalues at every point; thus, $1$–complete manifolds are Stein manifolds. Such an $X$ is necessarily noncompact and its highest-dimensional a priori nontrivial cohomology group is $H^{n+q-1}\left(X;\mathbb{Z}\right)$. In this paper we show that if $q<n$, $n+q-1$ is even, and $X$ has finite topology, then every cohomology class in $H^{n+q-1}\left(X;\mathbb{Z}\right)$ is Poincaré dual to an analytic cycle in $X$ consisting of proper holomorphic images of the ball. This holds in particular for the complement $X=ℂ{\mathbb{P}}^n\setminus A$ of any complex projective manifold $A$ defined by $q<n$ independent equations. If $X$ has infinite topology, then the same holds for elements of the group ${\mathcal{\mathscr{H}}}^{n+q-1}\left(X;\mathbb{Z}\right)=\underset{j}{lim}{H}^{n+q-1}\left(M_j;\mathbb{Z}\right)$, where ${\left\{M_j\right\}}_{j\in \mathbb{N}}$ is an exhaustion of $X$ by compact smoothly bounded domains. Finally, we provide an example of a quasi-projective manifold with a cohomology class which is analytic but not algebraic.

This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner–Weitzenböck-type formula for the norm of the self-dual Weyl tensor and discuss its applications, including connections between geometry and topology. In the second part, we are concerned with the interaction of different components of Riemannian curvature and (gradient and Hessian of) the soliton potential function. The Weyl tensor arises naturally in these investigations. Applications here are rigidity results.

The invariant measured foliations of a pseudo-Anosov homeomorphism induce a natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov monodromy. We show that when the pseudo-Anosov $\varphi :S\to S$ has orientable foliations and does not have 1 as an eigenvalue of the induced cohomology action on the closed surface, then the Sol structure can be deformed to nearby cone hyperbolic structures, in the sense of projective structures. The cone angles can be chosen to be decreasing from multiples of $2\pi$.

We show that a small perturbation of the boundary distance function of a simple Finsler metric on the $n$–disc is also the boundary distance function of some Finsler metric. (Simple metrics form an open class containing all flat metrics.) The lens map is a map that sends the exit vector to the entry vector as a geodesic crosses the disc. We show that a small perturbation of a lens map of a simple Finsler metric is in its turn the lens map of some Finsler metric. We use this result to construct a smooth perturbation of the metric on the standard $4$–dimensional sphere to produce positive metric entropy of the geodesic flow. Furthermore, this flow exhibits local generation of metric entropy; that is, positive entropy is generated in arbitrarily small tubes around one trajectory.

Le groupe modulaire $\Gamma$ du plan privé d’un ensemble de Cantor apparaît naturellement en dynamique. On montre ici que le graphe des rayons, analogue du complexe des courbes pour cette surface de type infini, est de diamètre infini et hyperbolique. On utilise l’action de $\Gamma$ sur ce graphe hyperbolique pour exhiber un quasi-morphisme non trivial explicite sur $\Gamma$ et pour montrer que le deuxième groupe de cohomologie bornée de $\Gamma$ est de dimension infinie. On donne enfin un exemple d’un élément hyperbolique de $\Gamma$ dont la longueur stable des commutateurs est nulle. Ceci réalise un programme proposé par Danny Calegari.

The mapping class group $\Gamma$ of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter and is hyperbolic. We use the action of $\Gamma$ on this graph to find an explicit non trivial quasimorphism on $\Gamma$ and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a hyperbolic element of $\Gamma$ with vanishing stable commutator length. This carries out a program proposed by Danny Calegari.

Using the invariant developed by E Artal, V Florens and the author, we differentiate four arrangements with the same combinatorial information but in different deformation classes. From these arrangements, we construct four other arrangements such that there is no orientation-preserving homeomorphism between them. Furthermore, some pairs of arrangements among this 4–tuple form new arithmetic Zariski pairs, ie a pair of arrangements conjugate in a number field with the same combinatorial information but with different embedding topology in $ℂ{\mathbb{P}}^2$.