Birational Methods in Topology and Hyperkähler Geometry

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The goal of this proposal is to make significant progress on the following two distinct problems. Our approach to them would rely on methods coming from birational geometry. 1. Chern numbers and algebraic structures. To any complex manifold X, one can associate the Chern classes of its tangent bundle. Such classes are elements of the integral cohomology groups of X. For instance, the first Chern class of X is the class of the canonical bundle of X. If the dimension of X is n, any product of Chern classes of total degree 2n is called a Chern number of X. The study of Chern numbers is a classical and important topic which is across-the-board in algebraic geometry, differential geometry and topology. Generalising a question of Hirzebruch, Kotschick asked the following basic question: which Chern numbers are determined up to finite ambiguity by the underlying smooth manifold? Together with S. Schreieder, we treated this question in dimension higher than 3, proving that most Chern numbers are unbounded. The main aim of this project is to prove that on any smooth complex projective 3-fold X, the Chern number given by the cube of its first Chern class is bounded by a constant depending only on the topology of X. Results in this direction have been obtained in a recent preprint with P. Cascini, where tools from the Minimal Model Program have been used, combined with techniques from topology and arithmetic. This is a joint project with P. Cascini (Imperial College London) and S. Schreieder (University of Bonn). 2. SYZ conjecture on hyperkähler manifolds. Bauville-Bogomolov's decomposition theorem asserts that up to a finite étale cover any compact Kähler manifold with numerically trivial canonical bundle is the product of compact complex tori, strict Calabi-Yau varieties and hyperkähler manifolds. In this sense, hyperkähler manifolds are among the most important examples of varieties with zero scalar curvature. More precisely, a compact Kähler manifold X of even dimension is said to be hyperkähler if it is simply-connected and if the space of holomorphic two-forms is generated by a nowhere degenerate form. In dimension 2 they are nothing but K3 surfaces. The aim of this second project is to investigate the SYZ conjecture (named after Strominger-Yau-Zaslow) on projective hyperkähler manifolds, which states that any nef line bundle on a hyperkähler has a multiple which is base-point free. This is considered one of the most important open problems in the theory of hyperkähler manifolds. This is a joint project with V. Lazic (University of Bonn).