Abstract:
We study deformations of pairs (X,D), with X smooth projective variety and D a smooth or a normal crossing divisor, defined over an algebraically closed field of characteristic 0. Using the differential graded Lie algebras theory and the Cartan homotopy construction, we are able to prove in a completely algebraic way the unobstructedness of the deformations of the pair (X,D) in many cases, e.g., whenever (X,D) is a log Calabi-Yau pair, in the case of a smooth divisor D in a Calabi Yau variety X and when D is a smooth divisor in |-m K_X|, for some positive integer m.

Abstract:
This paper is devoted to the study of various aspects of deformations of log pairs, especially in connection to questions related to the invariance of singularities and log plurigenera. In particular, using recent results from the minimal model program, we obtain an extension theorem for adjoint divisors in the spirit of Siu and Kawamata and more recent works of Hacon and McKernan. Our main motivation however comes from the study of deformations of Fano varieties. Our first application regards the behavior of Mori chamber decompositions in families of Fano varieties: we prove that, in the case of mild singularities, such decomposition is rigid under deformation when the dimension is small. We then turn to analyze deformation properties of toric Fano varieties, and prove that every simplicial toric Fano variety with at most terminal singularities is rigid under deformations (and in particular is not smoothable, if singular).

Abstract:
We identify dglas that control infinitesimal deformations of the pairs (manifold, Higgs bundle) and of Hitchin pairs. As a consequence, we recover known descriptions of first order deformations and we refine known results on obstructions. Secondly we prove that the Hitchin map is induced by a natural L-infinity morphism and, by standard facts about L-infinity algebras, we obtain new conditions on obstructions to deform Hitchin pairs.

Abstract:
We show that some properties of log canonical centers of a log canonical pair (X,D) also hold for certain subvarieties that are close to being a log canonical center. As a consequence, we obtain that if one works with deformations of pairs (X, D) where all the coefficients of D are bigger than 1/2, then one need not worry about embedded points on D. May 20: Results strengthened using recent work of Birkar and Hacon and Xu.

Abstract:
We study complex plane projective sextic curves with simple singularities up to equisingular deformations. It is shown that two such curves are deformation equivalent if and only if the corresponding pairs are diffeomorphic. A way to enumerate all deformation classes is outlined, and a few examples are considered, including classical Zariski pairs.

Abstract:
We prove that for every compact K\"ahler manifold $X$ there exists an $L$-infinity morphism, lifting the usual cup product in cohomology, from the Kodaira-Spencer differential graded Lie algebra to the suspension of the space of linear endomorphisms of the singular cohomology of $X$. As a consequence we get an algebraic proof of the principle ``obstructions to deformations of compact Kaehler manifolds annihilate ambient cohomology''.

Abstract:
Let C:L ￠ ’L ˉ be a projective deformation of the second order of two totally focal pseudocongruences L and L ˉ of (m ￠ ’1)-planes in projective spaces Pn and P ˉn, 2m ￠ ’1 ￠ ‰ ¤n<3m ￠ ’1, and let K be a collineation realizing such a C. The deformation C is said to be weakly singular, singular, or ±-strongly singular, ±=3,4, ￠ € |, if the collineation K gives projective deformations of order 1, 2 or ± of all corresponding focal surfaces of L and L ˉ. It is proved that C is weakly singular and conditions are found for C to be singular. The pseudocongruences L and L ˉ are identical if and only if C is 3-strongly singular.

Abstract:
We revisit the theory of deformations of pairs $(X, E)$, where $X$ is a compact complex manifold and $E$ is a holomorphic vector bundle over $X$, from an analytic viewpoint \`{a} la Kodaira-Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer-Cartan equation and DGLA governing the deformation problem, and express them in terms of differential-geometric notions such as the connection and curvature of $E$, obtaining a Chern-Weil--type refinement of the classical results that the tangent space and obstruction space of the moduli problem are respectively given by the first and second cohomology groups of the Atiyah extension of $E$ over $X$. We also investigate circumstances where deformations of pairs are unobstructed using our analytic approach.

Abstract:
We generalize the notions of dual pair and polarity introduced by S. Lie and A. Weinstein in order to accommodate very relevant situations where the application of these ideas is desirable. The new notion of polarity is designed to deal with the loss of smoothness caused by the presence of singularities that are encountered in many problems. We study in detail the relation between the newly introduced dual pairs, the quantum notion of Howe pair, and the symplectic leaf correspondence of Poisson manifolds in duality. The dual pairs arising in the context of symmetric Poisson manifolds are treated with special attention. We show that in this case and under very reasonable hypotheses we obtain a particularly well behaved kind of dual pairs that we call von Neumann pairs. Some of the ideas that we present in this paper shed some light on the so called optimal momentum maps.

Abstract:
We give the characterization of Arnol'd-Mather type for stable singular Legendre immersions. The most important building block of the theory is providing a module structure on the space of infinitesimal integral deformations by means of the notion of natural liftings of differential systems and of contact Hamiltonian vector fields.