Lefschetz Theorem Using Real Morse Theory

TL;DR

This paper introduces a simplified approach to the Lefschetz hyperplane section theorem using real Morse theory, avoiding traditional methods like vanishing cycles, and extends to proofs of classical formulas for curves.

Contribution

It presents a novel proof of the Lefschetz hyperplane section theorem via real Morse functions, providing new insights and methods in algebraic geometry.

Findings

Proved the Lefschetz hyperplane section theorem using real Morse theory.

Derived the genus formula for plane curves with Morse theory and Bézout's theorem.

Established the Riemann-Hurwitz formula for ramified maps between curves.

Abstract

We prove the Lefschetz hyperplane section theorem using a simpler machinery by making the observation that we can compose the Lefschetz Pencil with a Real Morse function to get a map from the variety to $\mathbb{R}$ which is "close" to being a Real Morse function. The proof uses a new method unlike the conventional one which uses vanishing cycles, thimbles and monodromy. We prove the genus formula for plane curves using Morse theory, Lefschetz pencil and B\'ezout's theorem. And then we prove the Riemann Hurwitz formula for ramified maps between curves by employing techniques from deformation theory. Lastly, we prove the Lefschetz Hyperplane Section Theorem solely using Real Morse Theory and exact sequences.