All Questions
10 questions
2
votes
0
answers
100
views
Deformations of invertible sheaves admitting global sections
We follow Sernesi's treatment of algebraic deformations, working over the complex numbers.
Given a pair $(X,L)$ consisting of a compact complex manifold $X$ and an invertible sheaf $L$ on $X$, we ...
- 518
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
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
2
votes
0
answers
212
views
Elementary questions about vanishing cycles and emerging cycles
Let $X\to D$ be a proper $C^\infty$ map with $D$ an open disk about the origin in some Euclidean space. Suppose $0\in D$ is the only singular value, i.e that over $D^\times=D\setminus \left\{ 0 \right\...
- 10.5k
3
votes
1
answer
987
views
Lie algebras : Deformations and Rigidity
I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie ...
13
votes
3
answers
1k
views
DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons
As proposed by Quillen, Drinfeld, and Deligne and other important mathematicians, there is supposed to be a philosophy that, at least over a field of characteristic zero, assigns to every "deformation ...
- 2,816
2
votes
1
answer
636
views
Tangent complex of dgla/ twisted dgla
I am looking for a theorem which says if we twist an $L_\infty$ quasi-isomorphism between dgla's by a Maurer-Cartan element, we'll get a new $L_\infty$ quasi-isomorphism.
Let $(\mathfrak{g}, d, [-,-]...
- 151
7
votes
2
answers
1k
views
References for the moduli space of complex structures
I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...
- 2,816
8
votes
0
answers
463
views
On the cohomology of Kontsevich graph complex
Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...
- 1,609