Questions Related to Curves on Complex Projective Manifolds

这个项目的目的是加深对复射影流形X上的复曲线族的行为的理解。这将通过解决两个列举问题来完成(例如，如何存在具有给定的次数、亏格和偶然性的曲线？)以及变形理论问题(例如，曲线的哪些变形存在于X的一般变形中？)。攻击这类问题的一些重要工具包括Atiyah-Bott局部化定理和Kuranishi的形式变形理论。除了研究部分，调查人员将重新启动成功的WAGS(西方代数几何)会议。这项研究与计数问题有关。例如，给出一个多变量多项式方程的解集，有多少特定类型的对象(如直线)完全位于该解集内？我们进一步研究对象集合的“形状”，即使在对象集合不再是有限的情况下也是如此。这项研究与现代物理学中一种著名的理论--弦理论有关，该理论提出通过在六个独立方向上略微加厚时空来解决统一自然基本力的问题。我们的研究从这六个新的方向研究了宇宙(微小的)横截面的几何学。

The aim of this project is to further the understanding of thebehavior of families of complex curves on a complex projective manifold X.This will be done by addressing both enumerative questions (e.g. How may curves are there with given degree and genus and incidences?) and deformation-theoretic questions(e.g. What deformations of curves lie in a generic deformation of X?). Some of the important tools for attacking such questions include the Atiyah-Bott localization theorem and Kuranishi's formal deformationtheory. In addition to the research component,the investigators will restart the successful WAGS(Western Algebraic Geometry) conferences.This research relates to counting problems. For example, given the set of solutions of a polynomial equation in several variables, how many objects of a certain type (e.g. lines) lie entirely inside the solution set? We further study the 'shape' of the collection of objects even in cases in which that collection of objects is no longer finite. This research is related to a celebrated modern theory in physics, called 'string theory,' which proposes to resolve the problem of unifying the fundamental forces of nature by slightly 'thickening' space-time in six independent directions. Our research studies the geometry of the (tiny) cross-section of the universe in those six new directions.