Topological properties of algebraic varieties

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In this project, we study further the relationship between topological and geometric properties of smooth complex projective varieties. More precisely, we investigate several specific questions, where we expect that the interplay between the topology and the geometry of algebraic varieties reveals a particularly rich and fruitful picture. Our projects can briefly be summarized as follows.Project A: Solve the integral Hirzebruch problem for smooth complex projective varieties.That is, determine for any positive integer m, which Z/m-linear combinations of Chern and Hodge numbers of smooth complex projective varieties are topological invariants of the underlying smooth manifold. Solve also the natural generalization of Hirzebruch's problem for the mixed Hodge numbers of singular and possibly non-compact varieties. Project B: Prove that a compact Kähler manifold admits a holomorphic 1-form without zeros if and only if it fibres smoothly over the circle. Project C: Prove that all (minimal) complex projective varieties of general type and with given topological invariants form a bounded family. Project D: Use topological properties and methods to attack the bounded negativity conjecture for curves on surfaces.