Sheaves on higher dimensional varieties

该项目的中心主题是应用衍生范畴技术来解决由几何和弦理论引起的问题。特别地，P.I.有三个主要目标：(1)研究稳定滑轮在表面上的模空间，通过使用桥地稳定性条件下的墙壁穿越。(2)推广了Bogomolov和Gieseker的经典结果，证明了三折上某些稳定复合体的Chern类的一个猜想界。(3)研究稳定n点有理曲线的投影空间、三次超曲面和Grothendieck-Knudsen模空间上的束。深远的应用包括Le Potier的奇异对偶性猜想，Calabi-Yau三倍上桥地稳定条件的存在性，三倍上伴随线性级数的Fujita猜想，以及计数不变量理论的新结果。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变量，即多变量多项式方程的解的轨迹。粗略地说，这个想法是“间接地”研究代数变量，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。派生范畴的技术是在1967年Verdier的论文中，在Grothendieck的指导下发展起来的。最初的动机是需要为Grothendieck的对偶理论找到一个合适的基础，该理论提供了上述不变量之间的非平凡关系。最近，派生范畴理论已经发现了重要而深刻的应用，超越了最初的动机，甚至超出了代数几何；特别是表征理论、辛几何和高能物理。目前的建议确实建立在与这些学科的相互作用上——特别是在Kontsevich， Bondal, Orlov, Bridgeland， Kuznetsov等人的工作上——推导出代数几何和其他领域的新结果。

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.