Abstract
This is a relatively self-contained introduction to recent developments in the $$\bar{\partial }$$ -equation, Ohsawa–Takegoshi extension theorem and applications of pluripotential theory to the Bergman kernel and metric. The main tools are the Hörmander $$L^2$$ -estimate for $$\bar{\partial }$$ and Bedford–Taylor’s theory of the complex Monge–Ampère operator.
Highlights
Holomorphic functions of several variables are precisely solutions to the homogeneous Cauchy–Riemann equation ∂ ̄ u = 0. (1.1)Here both sides are forms of type (0,1) which is a rather special case of the ∂ ̄ -equation because all solutions, even in the distributional sense, have to be smooth, in contrast to the general case of the equation for ( p, q)-forms
The original Hörmander estimate was slightly weaker: the right-hand side depended on the minimal eigenvalue of the complex Hessian of φ but his method gives this slightly stronger version. This turns out to be an extremely powerful result as will be again demonstrated here. What makes this approach so useful is a big abundance of plurisubharmonic functions: they are usually much easier to construct than holomorphic functions and this is where pluripotential theory comes into play
The following theorem proved by Ohsawa and Takegoshi [98] turned out to be one of the most important results in complex analysis and complex geometry
Summary
Holomorphic functions of several variables are precisely solutions to the homogeneous Cauchy–Riemann equation (often called the ∂ ̄ -equation). One of the most important results in several complex variables has been the Ohsawa– Takegoshi extension theorem [98] It states that holomorphic functions can be extended from lower dimensional sections with L2-estimates. We can state Hörmander’s estimate as follows: Theorem 2.1 Assume that is a pseudoconvex open subset of Cn and φ ∈ P S H ( ). Plurisubharmonic functions satisfying (2.3) are precisely of the form ψ = − log(−v) for some v ∈ P S H −( ) It was shown by Berndtsson [5] that Theorem 2.2 is a formal consequence of Hörmander’s estimate: Proof of Theorem 2.2 By standard approximation we may assume that ψ is smooth, strongly plurisubharmonic and that , φ, ψ are bounded. Ψ and a are as in Theorem 2.3 and φ := φ + ψ |∂ ̄ ψ|i2∂∂ ̄φ ≤ |∂ ̄ ψ|i2∂∂ ̄ψ and Theorem 2.4 immediately gives Theorem 2.3
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