Sheaves on higher dimensional varieties

该项目的中心主题是应用派生类技术来解决双有理几何和弦理论所产生的问题。特别是，P.I.主要有三个目的：（1）利用Bridgeland稳定性条件的跨壁方法研究曲面上稳定层的模空间。(2)证明了一类三重稳定复形的Chern类的一个上界，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，其深远的应用包括Le Potier的奇异对偶猜想、Calabi-Yau三重上Bridgeland稳定性条件的存在性、三重上伴随线性级数的Fujita猜想、以及计数不变量理论的新成果。这个项目更广泛的背景是代数几何领域。代数几何中感兴趣的中心对象是代数簇，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”研究代数簇，通过使用某些与几何对象相关的“同调”不变量-例如，微分形式。导出范畴的技术是在1967年格罗滕迪克的指导下，在Verdier的论文中发展起来的。最初的动机是需要找到一个适当的基础格罗滕迪克的对偶理论，它提供了非平凡的关系，上述不变量。近年来，导出范畴理论在代数几何之外也有了重要而深刻的应用;特别是表示论，辛几何和高能物理。本建议确实建立在与这些学科的相互作用上--特别是在Kontsevich，Bondal，奥尔洛夫，Bridgeland，库兹涅佐夫等人-推导出代数几何和其他领域的新成果。

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.