Abstract:

When dealing with nonsingular varieties over the complex numbers, an extremely powerful method is to regard them as complex manifolds and use tools from complex analysis and differential geometry to study them. A first step would be to understand their analytic topology, and a crucial result is given by the Lefschetz hyperplane theorem, asserting that a complex projective variety passes on many of its topological properties to any ample divisor sitting inside it. I will present a very quick proof of the result - different from Lefschetz’s original strategy - relying on Morse theory. I will also show how it is closely related to the Kodaira Vanishing theorem and how it can be used to compute the Picard group of smooth complete intersections.