Multiplier ideal sheaf

From Conservapedia

Jump to: navigation, search

This article or section needs to be written in plain English, using plain English that most of our readers can understand. Articles that depend excessively on technical terms accessible only to specialists are useless for our purposes, so writers are admonished to avoid jargon
This article has been nominated for deletion. See Conservapedia:AFD Multiplier_ideal_sheaf

Let φ be a plurisubharmonic function on a complex manifold M. The multiplier ideal sheaf Iφ is a coherent sheaf defined as follows: The stalk over a point p is the set of locally defined holomorphic functions f such that

f(z)e − φ

is locally in L2.

Multiplier ideal sheaves are of enormous use in algebraic geometry primarily due to the Nadel vanishing theorem:

Let (M,ω) be a weakly pseudo-convex Kaehler manifold with a holomorphic line bundle F. Suppose F has a singular metric with associated curvature current i\partial\overline{\partial}\phi \geq \epsilon\omega. Then

H^i(M,\mathcal{O}(K_M+F)\otimes I_{\phi})) = 0

for i > 0.

The following (admittedly overly-simple) application illustrates the use of multiplier ideal sheaves: Let C be a smooth algebraic curve with positive line bundle L. We will show, using multiplier ideal sheaves, that \mathcal{O}(mL) is generated by global sections for m sufficiently large.

Fix a point p \in C. It suffices to prove that we can find a section s \in H^0(C,mL) that does not vanish at p, for m sufficiently large.

Let z denote the section associated to the Cartier divisor p. Then | | z | | 2 defines a singular metric for the line bundle \mathcal{O}(p) whose associated multiplier ideal sheaf is clearly the ideal sheaf of p.

Choose m so that mL − KC + p is positive. By Nadel vanishing, H1(C,mL + p) = 0. Now consider the exact sequence:

0\to I_p \to \mathcal{O}_C \to \mathcal{O}_p\to 0

After tensoring this sequence with mL and taking the long exact sequence in cohomology, we get H^0(C,mL)\rightarrow H^0(p,mL)\rightarrow H^1(C,mL+p). Since this last term is zero, it follows that the global sections of mL over p lift to the whole curve. This completes the proof.

The machinery applied to the above proof, while unnecessary, illustrates the basic techniques involved in applying multiplier ideal sheaves and Nadel vanishing to the generation of global sections of line bundles.

Retrieved from "http://www.conservapedia.com/Multiplier_ideal_sheaf"
Personal tools