Wallcrossing

Wall-crossing is a concept in algebraic geometry and mathematical physics that describes the behavior of certain algebraic objects when they are intersected by a divisor on a complex manifold. It is a tool used to study the moduli spaces of algebraic varieties and the cohomology rings of complex manifolds.

In simple terms, wall-crossing is the process of transforming the algebraic structure of a variety when it passes through a "wall" of singularities in the space of all varieties. This can occur when the variety intersects a divisor, a geometric object that divides the manifold into different regions. The wall consists of points where the degenerate variety has the same dimension as the ambient space, and the wall-crossing corresponds to a change in the algebraic properties of the variety.

The concept of wall-crossing was first developed in the context of moduli spaces of sheaves on a curve, and has since been generalized to more general settings. It has applications in a wide range of areas, including algebraic geometry, geometric representation theory, and theoretical physics.

The motivation behind wall-crossing is to study the variation of algebraic invariants under deformations of the geometric data. This can lead to a better understanding of the moduli spaces of algebraic varieties, which describes the space of all possible geometric shapes. Wall-crossing can be used to compute cohomology rings and other invariants of the manifold, and it has implications for the study of mirror symmetry and other areas of mathematical physics.

The study of wall-crossing continues to be an active area of research in algebraic geometry and mathematical physics, with many open questions and challenging problems to be solved.