Symplectic Birational Geometry and Almost Complex Algebraic Geometry

The subject of geometry begins with the Greek mathematician Euclid who studied relationships among distances and angles, first in a plane and then in a space. About 200 years ago, Gauss and Riemann opened the door of modern geometry. They studied geometry on the more general notion of "manifold''. This is a space which is not necessarily flat, although locally it is like an Euclidean space, e.g. a sphere. The geometry studied by them is called Riemannian geometry, which is the mathematical foundation of Einstein's general relativity. In the study of Physics, people find that, in some situations, we need modifications of Riemannian geometry. One direction is complex geometry, where the the local model is a complex plane instead of a real plane. Another generalization is symplectic geometry, where we change the notion of metrics, i.e. distances and angles, to a 2-form. On a plane, it is just the area form.

The idea of symplectic geometry made an implicit appearance already in the work of Lagrange on analytical mechanics and later in Jacobi's and Hamilton's formulation of classical mechanics. It is Herman Weyl who first uses the word it symplectic in his book Classical Groups. It is derived from a Greek word meaning complex, a word already used in mathematics with a different meaning. In the study of String Theory, a theory providing a possible model for our universe, these two geometries come together to provide mathematical foundations. The proposed research studies the global property of symplectic manifolds and the interactions with complex manifolds.

Enriques and Kodaira described the birational classification of complex surfaces, i.e. complex 2-manifolds. The surfaces are divided into four categories according to their Kodaira dimensions, which take values negative infinity, 0, 1, and 2. The Minimal Model Program (Mori program) aims to generalize these results to higher dimensional complex projective varieties. This program is complete in dimension 3 in 1980s and is known to work for complex projective varieties of general type recently.

Symplectic topology is a subject concerning important global questions of symplectic manifolds. Comparing to complex manifolds, the topology of symplectic manifolds, even in dimension 4, is far more wild. For example, any finitely presented group can be realized as the fundamental group of a symplectic 4-manifold. Hence in symplectic topology, we have many more objectives to study than complex manifolds.

There are two natural ways to extend the birational classification and other aspects of birational geometry to symplectic manifolds. The first is to fix a symplectic structure. We study how the geometry and topology are changing under simple birational operations like the symplectic blow-up/blow-down and symplectic deformations. This is called the symplectic birational geometry. The techniques and flavours of this subject are more or less topological which gives a lot of flexibility. The other way is to fix an almost complex structure tamed by a symplectic form. This is called the almost complex algebraic geometry, which is more rigid. We plan to use the theory of J-holomorphic curves to generalize the relevant part of algebraic geometry (in particular the Nakai-Moishezon and Kleiman dualities, the cone theorem and linear systems) to symplectic manifolds of dimension 4.

Techniques and interactions from different disciplines, e.g. low dimensional topology, algebraic geometry, differential geometry, complex geometry and symplectic topology, are very crucial for this project.

This is a research project in pure mathematics and so the potential for societal impact is mostly long term.

The project will contribute new results and techniques to mathematics. To maximize the project's impact, the PI will publish papers in high quality journals. To ensure timely availability of the project results to other academics working in the area, the PI will post the preprint versions on arXiv and his personal website. Scientific collaboration with prominent researchers will clearly help increase the prospect of the project yielding high-impact results. At the same time, throughout the duration of the project, the PI will give seminar talks at other Universities, at international conferences, at specialized workshops, and will use these opportunities to disseminate the results of his research. The PI plans to organize an international workshop in the summer of 2017, bringing together an international community of experts from the project-related fields.

The project will also contribute to mathematical education, and so further long-term beneficiaries will be spread across economy and society. Graduate students from the relevant areas will profit directly from interactions with the PI. Undergraduate students at Warwick could write a 3rd or 4th year research essay on a related topic. Moreover, both symplectic topology and birational geometry are very useful tools in theoretical physics, especially in string theory.