Complex analysis notes

# Complex analysis notes ## Plan * (Pluri)subharmonic functions * Regularisation of qpsh * First-order differential operators * Review of Kaehler geometry * The Ohsawa-Takegoshi extension theorem * Review of blowups and valuations * Multiplier ideal sheaves ## (Pluri)subharmonic functions We follow [DeAG]. $\Omega\subset \mathbb R^d$ * Green-Riesz representation theorem: Let $\Delta$ be the usual neg definite Laplacian $\mathbb R^d$. * Green kernel, Poisson kernel and Newton kernel satisfies: $$u(x) = \int_\Omega G_\Omega(x,y)\Delta(y)+\int_{\partial\Omega}P_\Omega(x,y)u(y)d\sigma(y)$$ and $G_\Omega(x,y)\equiv N_d(x-y) \mod \mathcal C^\infty$. * Defn: $u\colon\Omega\to \mathbb R\cup\{-\infty\}$ is **sh** if it is usc, not identically $-\infty$, and for each ball $\bar B(a,r)\subset \Omega$, we have 'mean value' ineq $$u(a)\le \strokedint_{S(a,r)}u$$ * Example: This is also some sort of Laplacian...: [FIGURE: [ラプラス](https://www.youtube.com/channel/UCENwRMx5Yh42zWpzURebzTw)] ## Refs * [Be] B. Berndtsson. An introduction to things $\overline \partial$ * [Br] Haïm Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations * [DeAG] J.-P. Demailly. Complex analytic and differential geometry * [De2] J.-P. Demailly. Analytic methods in algebraic geometry, * [Ei] D. Eisenbud. Commutative algebra. * [GH] Griffiths-Harris. * [GR] Gunning-Rossi * [Ha] Hartshorne. * [La] R. Lazarsfeld. Positivity * [Stacks] Stacks Project. http://stacks.math.columbia.edu, * [Vo] Voisin