Research in higher dimensional algebraic geometry

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The investigator will work on several problems in higher dimensional algebraic geometry. In a project, joint with Alexeev, Hassett, and Kollár, the PI plans to complete the proof of existence of a coarse moduli space of stable log surfaces and higher dimensional varieties, an analog of the moduli space of stable curves. This an open problem whose solution is crucial to many applications in the field. Another project, pursued jointly by Kollár and the PI, will provide crucial results for the above projects. The main theme of the project is to understand deformations of singularities that appear on stable log varieties. In a related project the PI is planning to prove various generalizations of the classical Kodaira vanishing theorem. In another project the PI is going to work on the refined Viehweg conjecture regarding subvarieties of moduli stacks of canonically polarized smooth projective varieties. This conjecture evolved from a landmark conjecture of Shafarevich, and its solution by Arakelov and Parshin, which played an important role in Faltings' proof of the Mordell Conjecture. Part of this project is joint work with Kebekus. In another project the PI and Hacon are going to study the impact of the existence of nowhere vanishing differential forms on the geometry of the underlying variety. Their goal is to prove several outstanding conjectures in the area. In another project the PI is planning to prove a a characterization of the projective space and quadric hypersurfaces that will give a far reaching common generalization of Mori's theorem (earlier known as Hartshorne's conjecture) and Beauville's conjecture. The latter was settled by Araujo, Druel and the PI recently. In yet another project the PI hopes to prove a strong rational resolution theorem. One application of this result would be that varieties with rational singularities admit a compactification with only rational singularities themselves.This research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one that blossomed to the point where it has solved problems that have stood for centuries. Originally, and still in its simplest form it treats figures defined in the plane by polynomials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. A central problem in algebraic geometry is the classification of all geometric objects. In turn, an important part of classification theory is the theory of moduli. The latter's core idea is that one does not only want to understand these objects, but also understand the way they can be deformed. Moduli spaces play a very important role in theoretical physics. Studying curves on moduli spaces provides information on how an object is changing in space-time. One of the focuses of this project is on compact moduli spaces. Those are extensions of moduli spaces in general and they give additional information about singular deformations, ones that are essentially different from others. Other goals of the project involve a better understanding of certain higher dimensional varieties.