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Let $S$ be a reducible compact complex analytic space, thus we have the decomposition $S=\bigcup_{i=1}^n {V_i}$ where $V_i$ is the irreducible component of $S$. Let $L$ be a line bundle on $S$, I wond …

Let $Y$ be a smooth submanifold of codimension $r$ in a complex manifold $X$. By virtue of the adjunction formula, we always have the isomorphism $$K_Y\simeq (K_X\otimes \det N_Y){\,|\,}_Y.$$ Recall t …

In Coherent analytic sheaves, one has the following theorem due to Grauert: Let $f: X \rightarrow Y$ be a holomorphic family of compact complex manifolds with connected complex manifolds $X, Y$ and $V …

I am reading this paper:A SIMPLIFIED PROOF OF OPTIMAL L2-EXTENSION THEOREM AND EXTENSIONS FROM NON-REDUCED SUBVARIETIES by Hosono. https://arxiv.org/pdf/1910.05782.pdf. The setting is as follows.Let …

Recently, I am reading D. Popovici's article $L^2$ extension for jets of holomorphic sections of a Hermitian line bundle, https://arxiv.org/pdf/math/0409170.pdf where some parts possibly confuse me. I …

Consider the following setting: suppose that $X$ is a smooth variety and let $f:X\rightarrow \Delta$ be a smooth morphism outside the origin $0$. Let the central fiber $X_0$ be a reduced (Cartier) d …

Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it. Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\ …