Hodge index rate

The index set of Hodge is a theorem from the mathematical field of algebraic geometry . It calculates the signature of algebraic surfaces .

It says: Let be an ampler divisor on an algebraic surface . Then the is -sectional shape negative definite on . ${\displaystyle H}$ ${\displaystyle X}$ ${\displaystyle H^{\perp }}$

This is especially true if the divisor of the hyperplane intersection is an embedding . In this case, is , which gives the index of inertia of the cut form as , where the dimension of the vector space is rational divisors modulo algebraic equivalence (equivalent to the rank of the Néron-Severi group ). ${\displaystyle H}$ ${\displaystyle X\to \mathbb {C} P^{n}}$ ${\displaystyle H\cdot H=\deg(X)>0}$ ${\displaystyle (1,d-1)}$ ${\displaystyle d}$

The theorem was proven by Hodge as an application of Lefschetz's topological methods in complex algebraic geometry. In textbooks today it is mostly derived as a consequence of the Riemann-Roch theorem or from the Kähler identities . It holds more generally over any algebraically closed fields .

literature

• Hartshorne: "Algebraic Geometry", Berlin, New York: Springer-Verlag 1977, ISBN 978-0-387-90244-9
• Voisin: "Hodge Theory and complex algebraic geometry", Cambridge University Press, Cambridge et al. 2002, ISBN 0-521-80260-1

Web links

• Akhil Mathew: The Riemann-Roch and Hodge Index Theorems on surfaces