In the first part of the proposal PI will work on the framework he and his coauthors have introduced for varieties of general type. In this part, PI will illustrate the striking new results that emanate from this framework for algebraic surfaces. The methods developed are not restricted to the case of surfaces alone, but apply to varieties of general type of arbitrary dimension. Many of the questions have been open for a long time even for the case of algebraic surfaces. They are compelling for many reasons that include applications to moduli problems. One such long standing open question, posed by Enriques in the 1940's, is to find methods of constructing the so-called canonical surfaces, that is surfaces of general type with birational canonical map. This question has attracted a lot of attention from geometers down the years. We accomplish this task. The new methodology, involving very current mathematical technology, has already yielded several new results on the deformation, construction and the moduli, of varieties of general type. For instance, the methods allow us to describe components of infinitely many moduli spaces of surfaces of general type. The question of description of the moduli components is so basic, these are found in graduate texts on algebraic surfaces. The PI proposes to pursue this project. Recent results indicate that the methods PI has developed with his coauthors are capable of handling deformation of canonical maps of varieties of general type of arbitrary dimensions. This in turn will have implications to their moduli, which are being explored. Applications include new results for linear series on Calabi-Yau threefolds and their deformations, settling a 14 year old conjecture and on the moduli space of CY-threefolds.
The topic of holomorphic convexity of the universal cover of a projective variety has attracted considerable interest and is usually referred to as Shafarevich conjecture. One of the reasons is due to the fact that it is a higher dimensional analogue of the much celebrated uniformization theorem of Riemann and Kobe. The uniformization theorem says that any simply connected Riemann surface is conformally equivalent to one of the three: the open unit disk, the complex plane, or the Riemann sphere. Examples show that a pursuit of exact analogues of this great theorem is out of question. The best one can hope for in algebraic geometry is the holomorphic convexity of the universal covering of a projective variety, which goes under the name of Shafarevich conjecture. There are two approaches to this problem, one is constructing a general theoretical framework that shows interconnections and the other involves solving hard open cases that also have a compelling presence in different aspects of algebraic geometry. Even for the case of surfaces only elliptic surfaces has had a satisfactory and complete answer. With his co authors PI deals with these problems for much larger class of surfaces fibered over a curve, open questions for which there was until now no solution, via the fundamental groups of algebraic surfaces. The methods of PI and his coauthors, which are new and conceptual in the context, yield a stronger form of Shafarevich conjecture on holomorphic convexity and also provide an affirmative answer to a conjecture of Nori on fundamental groups of surfaces fibered over a curve. Besides this, the results have application to second homotopy groups of these varieties. It must be mentioned here that the classes of surfaces the PI and coauthors deal with for which Shafarevich conjecture has been very much open have a huge moduli and appear ubiquitously in the geometry of surfaces of general type and higher dimensions.. The results of the PI and a more recent one in progress also contain surprising elements concerning the bounds on the multiplicity of the multiple fibers. This result contrasts sharply with the famous results of Kodaira on the multiple fiber of an elliptic fibration. PI's recent results show the power of the methods that is used to attack these circles of problems and the PI wants to pursue the new ideas to more general fibrations for surface and higher dimensional varieties.
