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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 ...
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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 ...
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 ...
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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 $...
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 ...
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 ...
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1 vote

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 ...
3 votes
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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 ...
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. ...
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 ...
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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 $\...
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 ...
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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 ...
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2 votes
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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 ...

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