Summer Lectures in Geometry 

26/06/2001, 16:30 — 17:30 — Room P3.10, Mathematics Building
Yum-Tong Siu, Harvard University

Introduction to the application of \(\overline{\partial}\) estimates to complex geometry (I)

The series of three lectures will discuss the recent applications of \(L^2\) estimates of to geometric problems. The main technique of these applications is the use of multiplier ideal sheaves. The use of \(\overline{\partial}\) multiplier ideal sheaves is a completely new way of deriving a priori estimates of partial differential equations. So far the technique has been developed only for the equation but should be adaptable to other systems of partial differential equations arising from geometric problems.

When a priori estimates cannot be readily derived by the usual methods of integration by parts, one multiplies the quantity to be estimated by a function to make the a priori estimate hold. The set of all such multipliers form an ideal sheaf. Global geometric conditions are studied which can force the ideal to be the unit ideal, thereby making the desired a priori estimate automatically hold. This method is a powerful tool for many geometric problems.

Without the assumption of any pre-requisites, this series of lectures starts with the derivation of \(L^2\) estimates of \(\overline{\partial}\). Then the two kinds of multiplier ideal sheaves, Kohn's and Nadel's, are introduced, along with the problems and motivations from which they originate.

As examples of the geometric application of multiplier ideal sheaves, the following kinds of problems in algebraic geometry are discussed:

1. Fujita's conjecture which says that if L is a positive holomorphic line bundle over a compact complex manifold \(X\) of complex dimension n with canonical line bundle \(K_X\), then \(m L+K_X\) is generated by global holomorphic sections when \(m\geq n+1\) and is very ample when \(m\geq n+2\) (in the sense that any basis of the space of global holomorphic sections define a holomorphic embedding of \(X\) into a complex projective space).
2. Effective Matsusaka's Theorem which says that if \(L\) is a positive holomorphic line bundle over a compact complex manifold \(X\) of complex dimension \(n\), then \(m L\) is very ample when \(m\) is greater than some explicit effective function of the Chern numbers \(L^n\) and \(L^{n-1} K_X\).
3. Deformational invariance of plurigenera which says that \(m\)-genus of a compact complex projective algebraic manifold \(X\) ( i.e., the dimension of the space of global holomorphic sections of \(m K_X\) over \(X\)) is unchanged when \(X\) is holomorphically deformed.