# Tag Info

## New answers tagged complex-geometry

1 vote

### Hermitian holomorphic line bundle and curvature Chern form in Demailly's book

Thanks to the aid of @Gunnar Þór Magnússon, I will write down my understanding of Demailly's proof, if there is anything unclear, please comment below. Let $\cup_{i\in I}U_i$ be a covering of $X$ such ...
• 147
1 vote

### Gorenstein varieties: why the two definitions are equivalent?

As Donu mentioned, Gorenstein can be defined as Cohen-Macaulay and such that the canonical=dualizing sheaf is a line bundle. The point is that the dualizing complex is quasi-isomorphic to a sheaf if ...
• 41.1k
1 vote
Accepted

### Global sections of a line bundle on a reducible complex space

In any case $H^0(S,L)$ is a subspace in $\oplus H^0(V_i,L)$, but the way it sits there depends on the way the components $V_i$ are glued together. In the simplest case where they for a simple normal ...
• 32.2k
Accepted

• 8,313

### Oka-Grauert principle, up to the boundary

This is not a full answer, but sketches an approach that works for $n=1$ and likely generalises to higher dimensions with a finite amount of work. Note however, that for $n=1$ there is also an easy ...
• 593

### Can we define Whitney stratification algebraically?

There is a purely algebraic characterisation of Condition (B) due to Le and Teissier, see Proposition 1.3.8 of the paper Lê Dũng Tráng; Teissier, Bernard, Limites d’espaces tangents en géométrie ...
• 14.8k
We think of the cycle $\alpha_i$ as coming from a point, specifically, the point we need to add to compactify the puncture $p_i$. Here "come from" refers to the excision exact sequence in ...
Does this work? Let $U\subset Z$ be the set of smooth points. This is open and dense in $Z$. Let $Z'\subset M$ be the smallest closed complex analytic variety containing $Z$. Let $U'$ be the set of ...