Complex Projective Varieties

Complex projective varieties are geometric shapes defined by polynomial equations in complex projective space—a mathematical setting where points are represented by lines through the origin in complex coordinate space. These varieties generalize familiar shapes (like curves and surfaces) to higher dimensions, capturing complex solutions that remain invariant under scaling. They are fundamental objects in algebraic geometry, providing a framework to study the properties of solutions to polynomial equations with complex numbers, and serve as a bridge between algebra, geometry, and complex analysis.