All Questions
24 questions with no upvoted or accepted answers
14
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0
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709
views
Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?
A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...
- 19.3k
7
votes
0
answers
506
views
A general definition of an equisingular family of singular varieties?
This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions.
Let $X$ be a ...
- 7,789
7
votes
0
answers
215
views
"Tubular neighbourhood" for non-reduced curves
I have a manifold $X$ covered by a family of elliptic curves, some of which have
non-reduced structure (like multiple fibers on elliptic surfaces; such non-reduced curves $C$ are members of my family, ...
- 71
6
votes
0
answers
227
views
Deformation of Complex Spaces
I am trying to learn about deformations of complex spaces from the paper of Palamodov. I am particularly interested in the relative tangent cohomology.
Is there any other modern reference to this ...
- 727
5
votes
0
answers
284
views
Formal neighborhood of stable curves
For a smooth projective curve $X/\mathbb{C}$, every (infinitesimal) deformation is trivial when restricted to $X \setminus x$ for any $x \in X$. In particular, all deformations can be obtained by “...
- 834
5
votes
0
answers
218
views
Reducible surface as a degeneration
I am interested in the following situation. If $S_1\cup_D S_2$ is a union of two irreducible smooth projective surfaces over $k=\overline{k}$(over $k=\mathbb{C}$ is enough, if it's relevant) glued ...
- 71
5
votes
0
answers
189
views
Extension of holomorphic maps to smooth family of holomorphic maps
Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a ...
- 1,409
4
votes
0
answers
218
views
Why does a deformation modify the complex structure *holomorphically*?
This is a question regarding Chapter 9.1 of Claire Voisin's book [1]
Let $\phi: \mathcal X \to B$ be a family of compact complex manifolds, that is a proper holomorphic submersion, with central fiber $...
- 1,286
4
votes
0
answers
130
views
Stability of Ricci-flat Fujiki class $\mathcal C$ by small deformations
As we know, a compact Kähler manifold remains Kähler after any infinitesimal deformations. Since a compact complex manifold in Fujiki class $\mathcal C$ is bimeromorphic to a Kähler manifold, it was ...
- 471
4
votes
0
answers
236
views
Deform a non-Kähler manifold to a Kähler one
Let $X$ be a compact complex non-Kähler manifold, then what conditions do we need to make it has a Kähler deformation? that is to say it can be deformed to a Kähler manifold.
Obviously not all the ...
- 471
4
votes
0
answers
207
views
Rigid non-algebraic manifolds
The famous Kodaira problem asks: whether a compact Kähler manifold can always be deformed to a projective manifold? In order to provide a counterexample, one way is trying to construct a rigid compact ...
- 471
4
votes
0
answers
254
views
Deformation space and Kodaira-Spencer map of cyclic Galois coverings
This question concerns a statement from the paper by Ben Moonen "Special subvarieties arising from families of cyclic covers of the projective line. Documenta Math. 15 (2010)", Lemma 5.5. ii).
More ...
- 2,221
4
votes
0
answers
245
views
Deformations of the moduli space of ppav's
Consider the complex algebraic moduli space $X:=\mathcal A_g^n$ of ppav's of dimension $g$ with some high enough level $n$ structure (so that it represents the corresponding functor).
Can one compute ...
- 193
3
votes
0
answers
186
views
$H^2(X,T_X)=0$ implies the Frölicher spectral sequence degenerates at $E_1$?
Let $X$ be a compact complex manifold, if $X$ satisfies $H^2(X,T_X)=0$, then it is well-known that the Kuranishi space of $X$ is smooth by Kodaira and Spencer's deformation theory. On the other hand, ...
- 471
2
votes
0
answers
129
views
Hodge coniveaux of Calabi-Yau manifolds
Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
- 279
2
votes
0
answers
110
views
Invariance of plurigenera: singular surface case
The invariance of plurigenera is one most central problems in deformation theory. I mainly concern about the (singular) surface setting in this post since
Iitaka had proved that the deformation ...
- 295
2
votes
0
answers
366
views
Deformation over small disk and deformation over complex disk
Let $D=\mathop{Spec}(\mathbb C[[t]])$ be the algebraic small disk and let $\Delta= \{z\in \mathbb C: |z|<1\} $. Let $X_0$ be an algebraic surface over $\mathop{Spec}\mathbb C$
Suppose now that I ...
1
vote
0
answers
96
views
Birational deformations of holomorphic symplectic manifolds
Let $X$ and $X'$ be birational holomorphic symplectic manifolds. Then the birational morphism between them identifies $H^2(X)$ with $H^2(Y)$. The period space of $X$ is defined to be a subset of $\...
- 178
1
vote
0
answers
240
views
Unexpected holomorphic tubular neighborhood
While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular ...
- 1,761
1
vote
0
answers
176
views
Obstruction map for holomorphic line bundle $\operatorname{Ob}_L:H^1(X,\mathcal O)\to H^2(X,\mathcal O)$
$\DeclareMathOperator\Ob{Ob}$In Chan & Suen's paper A differential-geometric approach to deformation of pairs $(X, E)$ p.20, the authors define a Kuranishi map (or obstruction map) for the ...
- 471
1
vote
0
answers
257
views
When the whole space $H^2(X,\mathbb Q)$ can be represented by $c(L_t)$ of $X_t$?
Let $X$ be a compact complex manifold, $\pi:\mathcal X\to B$ be a holomorphic family of $X$ with $X_t=\pi^{-1}(t),t\in B$, and $X_0=X$.
Let $L_t$ be a holomorphic line bundle over $X_t$, then its ...
- 471
1
vote
0
answers
161
views
Manifolds with $[,]:H^1(X,T_X)\times H^1(X,T_X)\rightarrow 0$
Let $X$ be a compact complex manifold, for arbitrary $\phi_1,\phi_2\in H^1(X,T_X)$, if the Lie bracket $[,]:H^1(X,T_X)\times H^1(X,T_X)\rightarrow H^2(X,T_X)$ always maps $\phi_1,\phi_2$ to zero, i.e.$...
- 471
1
vote
0
answers
125
views
Explicit resolution of $\Omega^1_C$ for prestable curve $C$
Suppose $C$ is a complex projective curve (or a compact $1$-dimensional connected reduced complex space). If $C$ is smooth, then its module of differentials $\Omega^1_C$ is locally free. If $C$ is a ...
- 1,761
0
votes
0
answers
82
views
How geometry changes up to Hermitian inner product on Line bundle (Kodaira embedding)
Riemann metric $g \colon= \Sigma g_{ij} dx_i \otimes dx_j$ on a Kähler manifold $M$ will define the length of a line on $M$, i.e. intrinsic geometry. The line bundle $L$ on $M$ is equipped with a ...
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