Topological properties of algebraic varieties

本项目进一步研究了光滑复射影簇的拓扑性质与几何性质之间的关系。更确切地说，我们调查了几个具体的问题，在那里我们期望的拓扑结构和几何代数簇之间的相互作用揭示了一个特别丰富和富有成效的图片。我们的项目可以简单地概括如下：项目A：解决光滑复射影簇的积分Hirzebruch问题，即对任意正整数m，确定光滑复射影簇的Chern数和Hodge数的Z/m-线性组合中哪些是光滑流形的拓扑不变量。也解决自然推广的希泽布鲁赫的问题的混合霍奇数的奇异和可能的非紧品种。项目B：证明紧致Kähler流形允许无零点的全纯1-形式当且仅当它在圆上光滑纤维。项目C：证明所有具有给定拓扑不变量的一般类型的（极小）复射影簇形成一个有界族。项目D：利用拓扑性质和方法攻击曲面上曲线的有界负猜想。

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.