Given an algebraic variety defined by a set of equations, an upper bound for its dimension at one point is given by the dimension of the Zariski tangent space. The infinitesimal deformations of a variety $X$ play a somehow similar role, they yield the Zariski tangent space at the local moduli space, when this exists, hence one gets an efficient way to estimate the dimension of a moduli space.
It may happen that this moduli space consists of a point, or even a reduced point if there are no infinitesimal deformations. In this case one says that $X$ is rigid, respectively inifinitesimally rigid.
A basic example is projective space, which is the only example in dimension 1. In the case of surfaces, infinitesimally rigid surfaces are either the Del Pezzo surfaces of degree $\ge$ 5, or are some minimal surfaces of general type.
As of now, the known surfaces of the second type are all projective classifying spaces (their universal cover is contractible), and have universal cover which is either the ball or the bidisk (these are the noncompact duals of $P^2$ and $P^1 \times P^1$ ), or are the examples of Mostow and Siu, or the Kodaira fibrations of Catanese-Rollenske.
Motivated by recent examples constructed with Dettweiller of interesting VHS over curves, which we shall call BCD surfaces, together with ingrid Bauer, we showed the rigidity of a class of surfaces which includes the Hirzebruch-Kummer coverings of the plane branched over a complete quadrangle.
I shall also explain some results concerning fibred surfaces, e.g. a criterion for being a $K(\pi,1)$-space; I will finish mentioning other examples and several interesting open questions.[-]

Let $X$ be a compact Kähler manifold. The so-called Kodaira problem asks whether $X$ has arbitrarily small deformations to some projective varieties. While Kodaira proved that such deformations always exist for surfaces. Starting from dimension 4, there are examples constructed by Voisin which answer the Kodaira problem in the negative. In this talk, we will focus on threefolds, as well as compact Kähler manifolds of algebraic dimension $a(X) = dim(X) -1$. We will explain our positive solution to the Kodaira problem for these manifolds.[-]

I will discuss the existence of hyperkähler structures on local symplectic groupoids integrating holomorphic Poisson manifolds, and show that they always exist when the base is a Poisson surface. The hyperkähler structure is obtained by constructing the twistor space by lifting specific deformations of the Poisson surface adapted from Hitchin's unobstructedness result. In the special case of the zero Poisson structure, we recover the Feix-Kaledin hyperkähler structure on the cotangent bundle of a Kähler manifold.[-]

W. Thurston's theorems almost all aim to give a purely topological problem an appropriate geometry, or to identify an appropriate obstruction.. We will illustrate this in two examples:
--The Thurston pullback map to make a rational map from a post-critically finite branched cover of the sphere, and
--The skinning lemma, to find a hyperbolic structure for a Haken 3-manifold.
In both cases, either the relevant map on Teichmüller space has a fixed point, solving the geometrization problem, or there is an obstruction consisting a multicurve.[-]

I am going to define toroidal crossing singularities and toroidal crossing varieties and explain how to produce them in large quantities by subdividing lattice polytopes. I will then explain the statement of a global smoothing theorem proved jointly with Felten and Filip. The theorem follows the tradition of well-known theorems by Friedman, Kawamata-Namikawa and Gross-Siebert. In order to apply a variant of the theorem to construct (conjecturally all) projective Fano manifolds with non-empty anticanonical divisor, Corti and Petracci discovered the necessity to allow for particular singular log structures that are known by the inspiring name 'admissible'. I will explain the beautiful classical geometric curve-in-surface geometry that underlies this notion and hint at why we believe that we can feed these singular log structures into the smoothing theorem in order to produce all 98 Fano threefolds with very ample anticanonical class by a single method.[-]