DeformationTheoryAG · Deformation Theory and Moduli Spaces in Algebraic Geometry

Deformation Theory and Moduli Spaces in Algebraic Geometry

Executive Summary: The main topic of the proposal was to prove a number of different results in Deformation Theory and Moduli Spaces in algebraic geometry, using the modern language of algebraic geometry, based on DGLAs and L-infinity algebras. The first objective of this project was to give new explicit descriptions of DGLAs (or L-infinity algebras) controlling classical deformation problems of geometric objects. Since only few explicit results were known, we aimed to analyse new cases, with particular reference to the deformations of maps. A good strategy to “produce” DGLAs is via semi-cosimplicial objects and homotopy constructions. In other words, a deformation of a geometric object consists in deforming the object locally and then glue back together these local deformations. Then, from the algebraic point of view, we have to find the algebraic objects (DGLAs) that control locally the deformations and then glue them together. In this setting, we analysed the problem of infinitesimal deformations of pairs (X,D), where X is a smooth projective variety and D a normal crossing or smooth divisor. The deformations of the pair (X,D) are nothing else that the deformations of the closed embedding of D in X. Using the semi-cosimplicial language, we were able to provide an explicit description of a DGLA controlling the locally trivial deformations of the pair. Moreover, whenever D is a smooth divisor, this DGLA controls all the infinitesimal deformations. A second objective was the investigation of the obstructions problem. Indeed, we aimed to use deformation theory to understand local properties of the moduli spaces. More precisely, we sought to prove the existence of a semi-regularity map, containing all the obstructions in the kernel, providing a better control of the smoothness problem of the associated moduli spaces. In this setting, we proved various results about the obstruction and smoothness of the infinitesimal deformations of pairs. First of all, we described a new semi-regularity map containing all obstructions to the locally trivial deformation of the pair (X,D) with D a simple normal crossing divisor. In the case of D smooth, this map annihilates all obstructions to all infinitesimal deformations. The strategy performed to define this map was completely formal and algebraic. Then, we applied this semi-regularity map to prove that in many cases the DGLA controlling the deformations of the geometric object is homotopy trivial. This implies that there are no obstructions and the geometric problem is smooth. In particular, we proved the unobstructedness whenever (X,D) is a log Calabi-Yau pair, in the case of a smooth divisor D in a Calabi Yau variety X and whenever D is a smooth divisor in the linear system |-m K_X|, for some positive integer m. These results are all contained in the preprint: D.Iacono “Deformations and obstructions of pairs (X,D)”, submitted in June 2013 and available at http://arxiv.org/abs/1302.1149 Abstract: We study infinitesimal 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. The third objective was concerned with the derived algebraic geometry. The fundamental idea was the following. Once there are new examples of DGLAs controlling deformation problems, then one can aim to construct a derived structures on the associated moduli spaces. Indeed, this could provide a technique to describe new examples of dg schemes. For this purpose, the participant and host scientist focused their attention on the case of all the infinitesimal deformations of pairs (X,D) with X smooth projective variety and D a normal crossing divisor. The idea was to consider the pair (X,D) as a pair of dg-scheme so that the divisor D can be considered as a dg-smooth divisor. As a preliminary result, we obtained an explicit description of a DGLA controlling all the infinitesimal deformations, and not only the locally trivial ones. Because of the singularity of D, we were led to use resolvent and dg-sheaves. Now, the participant and host scientist are investigating the obstruction problem. This approach seems very promising and we expect to define a semi-regularity map also for this general case, providing information on the smoothness problem (collaboration in progress). We underline that the investigation was also motivated by the fact that the understanding of infinitesimal deformations of pairs (X,D) is also interesting for understanding the infinitesimal deformations of the variety X itself. More precisely, it is well defined a forgetting morphism between the infinitesimal deformations of the pairs and the infinitesimal deformations of X. Then, once the unobstructedness for a pair (X,D) is established, the smoothness of the forgetting morphism implies that also the deformations of X are unobstructed.

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