Complex algebraic varieties are subsets of complex projective space $\mathbf{CP}^n$ defined by polynomial equations. These subsets are not manifolds in general, but the types of singularities that can arise in this algebraic context are not arbitrary. A fundamental problem in algebraic geometry is therefore to determine which types of singularities may arise and, more generally, which special topological properties are exhibited by complex varieties.
In this course, we will study methods first introduced by Solomon Lefschetz to explore topological properties of complex algebraic varieties. The key idea is to exploit the rigidity of polynomial equations to reduce questions about varieties in dimension $n$ to questions about varieties of dimension $n -1$. More precisely, Lefschetz pencils allow us to regard a complex algebraic manifold $X$ of dimension $n$ as a family of complex algebraic varieties $X_t$, $t \in \mathbf{CP}^1$, of dimension $n-1$ parametrized by a complex projective line. All but finitely many fibers $X_t$ will be manifolds. It was Lefschetz' beautiful insight that knowledge of these fibers and of how they vary with respect to $t$ determines a great deal of topological data pertaining to $X$.
The essential tools to be applied in this course are complex analysis and algebraic topology. Familiarity with single-variable complex analysis will be assumed. The necessary techniques from algebraic topology, including singular homology and Poincaré duality, will be recalled.