Complex Algebraic Geometry

Serre in 1956 proved GAGA, the theorem connecting algebraic and analytic geometry on projective varieties. The theory of K3 surfaces with h^{1,1}=20 became the foundation of mirror symmetry and string theory.

• Mirror symmetry: exchanges h^{p,q} and h^{n-p,q} for pairs of Kahler manifolds
• String theory: compactifications on K3 x T^2 and CY3 are the basis of superstring theory
• Arithmetic geometry: Hodge numbers connect to L-functions through the Langlands program

Предварительные знания

• Previous lesson

Serre's GAGA

Serre in 1956 proved: on a smooth projective variety X in P^n, the category of algebraic coherent sheaves is equivalent to the category of analytic coherent sheaves. This is GAGA (Geometrie Algebrique et Geometrie Analytique). Consequence: every analytic bundle on P^n is algebraic -- it is O(d) for some d.

Bezout's Theorem and Hodge Numbers

Bezout's theorem in P^n: the number of intersection points of hypersurfaces of degrees d_1,...,d_n equals the product d_1*...*d_n (counted with multiplicity). Hodge numbers of K3 surface: h^{1,1}=20, h^{2,0}=h^{0,2}=1, all others zero -- universal structure for the 22-dimensional cohomology group.

Key Ideas

• GAGA: Coh_alg(X) ~= Coh_an(X) for projective X
• Hodge decomposition: H^k(X,C) = direct sum_{p+q=k} H^{p,q}(X)
• h^{p,q} = h^{q,p} = h^{n-p,n-q} (Hodge symmetries)
• Bezout: #(V(f1) cap ... cap V(fn)) = deg f1 * ... * deg fn in P^n
• K3: h^{1,1}=20, chi=24, H^2 ~= E8^2 + U^3

Further Directions

These ideas open paths to deeper mathematics.

• ca-28-stein-manifolds — extends

Вопросы для размышления

• Give a concrete example.
• How does this connect to other areas of mathematics?