## New answers tagged complex-geometry

1
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### 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 ...

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1
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### 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 ...

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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

5
votes

Accepted

### The kernel of $H^{\bullet,\bullet}_{\bar\partial}(X)\to H_A^{\bullet,\bullet}(X)$

Let $\alpha$ be a $\bar{\partial}$-closed form. Denote its Dolbeault cohomology class by $[\alpha]_{\bar{\partial}}$ and its Aeppli cohomology class by $[\alpha]_A$; note that the map $g$ is given by $...

- 17.4k

1
vote

Accepted

### Can deformation equivalent Kähler manifolds always be obtained by a deformation where all the fibers are Kähler?

I don't think this is known. For hyperkahler manifolds, conjecturally,
all smooth complex deformations are class C and birational to hyperkahler.
If this is true, your conjecture would follow ...

- 8,313

2
votes

### Complex quadric as a symmetric space

So while the projective quadric is indeed a homogeneous space of the form $G/P$ for $P$ parabolic in $G$ (marking it as a generalised flag manifold) this is not the realisation of it as a symmetric ...

- 637

1
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### Different algebraic structures on complements to divisors

Do you know other examples of non-isomorphic algebraic
structures on complements to square-zero curves
The easiest example is
the twisted cotangent bundle to an elliptic curve.
This space can be ...

- 8,313

3
votes

Accepted

### On the stack of semistable curves

I am posting an answer to correct my wrong comments above. The answer is based on the comment by user @johan.
For every $g\geq 0$,
denote by $\mathfrak{M}_g$
the stack of genus-$g$ curves that are ...

Community wiki

11
votes

### Mixing solids and liquids

Good question!
I think the real context for the question was whether certain objects that are implicit in work of Darmon (and collaborators) could exist within this framework of analytic geometry. ...

- 14.7k

2
votes

### Understand the proof that rational resolution is independent of the resolution

This is not exactly what is asked for, rather it is a different proof of the fact in question.
I assume you are working in characteristic zero with varieties. I'm also going to assume you want $X$ to ...

- 19.3k

3
votes

Accepted

### Bott-Chern cohomology for singular complex spaces

closed (1,1)-forms and currents on X
are not necessary locally $dd^c$-exact in general
What makes it different when X is singular?
The obstruction to local $dd^c$-lemma
is $R^1\pi_*(O_{X'})$, where
$\...

- 8,313

3
votes

### 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

11
votes

### 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

2
votes

Accepted

### Example motivating mixed Hodge structures

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 ...

- 119k

0
votes

### Real analytic subvariety in complex manifold which is complex outside of its singular set

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 ...

- 49k

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