Sheaves on higher dimensional varieties

该项目的中心主题是应用派生范畴技术来解决源于出生几何和弦理论的问题。特别地，P.I.有三个主要目的：(1)利用关于Bridgeland稳定性条件的交叉法，研究曲面上稳定层的模空间；(2)证明某些三层稳定复的Chern类的一个猜想的界，推广了Bogomolov和Gieseker的一个经典结果；(3)研究射影空间、三次超曲面和稳定n点有理曲线的Grothendieck-Knudsen模空间上的层。更广泛的应用包括Le Potier的奇异对偶猜想，Calabi-Yau三层结构上Bridgeland稳定性条件的存在性，三层结构的联线性级数的Fujita猜想以及在计数不变量理论方面的新结果。这个项目的更广泛的背景是代数几何领域。代数几何的主要研究对象是代数簇，即多元多项式方程的解的轨迹。粗略地说，这个想法是通过使用与几何对象相关的某些“同调”不变量--例如，微分形式--来“间接”研究代数簇。派生范畴的技术是Verdier在1967年Grothendieck的指导下在他的论文中发展起来的。最初的动机是需要为Grothendieck的对偶理论找到一个适当的基础，该理论提供了上述不变量之间的非平凡关系。最近，派生范畴的理论在最初的动机之外，甚至在代数几何之外，已经发现了重要和深入的应用；特别是在表示论、辛几何和高能物理中。当前的建议确实建立在与这些学科的相互作用的基础上--特别是基于康采维奇、邦达尔、奥尔洛夫、布里奇兰、库兹涅佐夫等人的工作--在代数几何和其他领域推导出新的结果。

The central theme of the project is to apply derived category techniques to solve questions arising from Birational Geometry and String Theory. In particular, the P.I. has three main goals:(1) To study moduli spaces of stable sheaves on surfaces, by using wall-crossing with respect to Bridgeland stability conditions.(2) To prove a conjectural bound on Chern classes of certain stable complexes on threefolds, which generalizes a classical result by Bogomolov and Gieseker.(3) To study sheaves on projective spaces, cubic hypersurfaces, and the Grothendieck-Knudsen moduli space of stable n-pointed rational curves.Far-reaching applications would include Le Potier's Strange Duality Conjecture, the existence of Bridgeland stability conditions on Calabi-Yau threefolds, the Fujita Conjecture on adjoint linear series for threefolds, and new results in the theory of counting invariants.The broader context of this project is the area of Algebraic Geometry. The central objects of interest in Algebraic Geometry are algebraic varieties, namely the loci of solutions of polynomial equations in many variables. Roughly, the idea is to study algebraic varieties "indirectly," by using certain "homological" invariants associated to geometric objects on them -- for example, differential forms. The technique of the derived category was developed in Verdier's thesis, under the guidance of Grothendieck, in 1967. The original motivation was the need to find a proper foundation for Grothendieck's duality theory, which provides non-trivial relations among the above mentioned invariants. More recently, the theory of derived categories has found important and deep applications beyond the original motivation and even outside Algebraic Geometry; in particular, to Representation Theory, Symplectic Geometry, and High Energy Physics.The present proposal builds indeed on the interaction with these disciplines -- notably on the work of Kontsevich, Bondal, Orlov, Bridgeland, Kuznetsov among others -- to deduce new results in Algebraic Geometry and other areas.