Problems in Higher Dimensional Algebraic Geometry

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As a link between algebra and geometry, the main goal of algebraic geometry is to understand geometrically certain objects called algebraic varieties (i.e., common zero sets of multivariable polynomials). These very natural and fundamental algebraic objects appear frequently in many branches of mathematics and science. Hence, algebraic geometry sees many applications in other parts of mathematics such as number theory or complex geometry, and also in other scientific or engineering disciplines such as physics, coding theory, robotics, and computational biology. In particular, any general structure theory of these algebraic objects can be very useful in many fields of study. The part of algebraic geometry devoted to this structure theory is called higher dimensional algebraic geometry and it is the main subject of this research project. The project aims to further develop the theory of higher dimensional algebraic geometry. For studying varieties over a field of characteristic zero and of positive characteristic, there are very different tools available. Examples of the former are Hodge theory and the related vanishing theorems, while examples for the latter are Frobenius lifting techniques or quotients by p-closed foliations. In particular, the above dichotomy yields two remarkably different cases to most conjectures of higher dimensional algebraic geometry. The particular flavor of this project is that it concerns standard topics and conjectures of higher dimensional algebraic geometry both in characteristic zero and in positive characteristic, such as higher dimensional moduli spaces, minimal model program, subadditivity of Kodaira dimension, and Shafarevich type hyperbolicity conjectures.