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Results tagged with complex-geometry
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user 40297
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
21
votes
Accepted
GAGA for stacks
For your "special" question, the answer is negative, already when $A$ is an elliptic curve. In fact, a principal $A$-bundle over a smooth projective curve $B$ which is not topologically trivial is nev …
- 38k
18
votes
Accepted
Vector bundles on vector bundles
This is definitely false in the algebraic or holomorphic setting, even in dimension 1. There is a well-known example (see this post) of a rank 2 vector bundle $E$ on $\mathbb{C}\times \mathbb{P}^1$ su …
- 38k
17
votes
Accepted
What is the role of projective spaces in GAGA?
The Serre comparison theorems are valid for complete (= proper) varieties over ${\Bbb C}$, with no relation to projective space. See this talk by Grothendieck (Séminaire Cartan 9 (1956-1957), Exposé N …
13
votes
Accepted
Must a canonical line bundle be associated to a cartier divisor?
The answer is no. The group $H^0(X, \mathcal{K}_X^*/\mathcal{O}_X^*)$ is isomorphic to the divisor group $\mathrm{Div}(X)$ (see e.g. Huybrechts' Complex Geometry, Prop. 2.3.9), so its image in $\mathr …
- 38k
12
votes
Accepted
Singularities of the moduli stack of Calabi-Yau threefolds
Yes, Calabi-Yau manifolds have unobstructed deformations. This is due to Tian and Todorov; there is a nice algebraic proof in a paper by Kawamata, J. Algebraic Geom. 1 (1992), no. 2, 183–190.
- 38k
11
votes
Automorphism group of flag manifolds?
Claudio's answer settles the determination of $\DeclareMathOperator\Aut{Aut}\Aut^{\mathrm{o}}(F)$; that of $\Aut(F)$ is more subtle. For Grassmannians this is a classical result of Chow (On the geomet …
- 38k
11
votes
Accepted
Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau
The diagonal $\Delta $ is linearly equivalent to $\{p\}\times \mathbb{P}^1 +\mathbb{P}^1\times \{p\} $ for any $p$ in $\mathbb{P}^1$. Therefore $X$ is the zero locus in $S\times S'$ of a section of $L …
- 38k
10
votes
Accepted
Existence of holomorphic retraction
No, this is actually very rare. Indeed the existence of such a retraction implies that the exact sequence $$0\rightarrow T_X\rightarrow T_{M|X}\rightarrow N_{X/M}\rightarrow 0$$ splits. In particular, …
- 38k
9
votes
Accepted
Negative curves on surface of general type
Yes. Start with a rational surface $S$ containing infinitely many $(-1)$-curves $E_n$ (for instance $\mathbb{P}^2$ blown up along 9 general points). Choose a very ample divisor $H$ on $S$, a smooth cu …
- 38k
9
votes
Accepted
Holomorphic structures for line bundles over projective manifolds
The group of isomorphism classes of complex line bundles on $M$ is isomorphic to $H^2(M,\mathbb{Z})$, via the map $L\mapsto c_1(L)$. The analogous group $\operatorname{Pic}(M) $ for holomorphic line b …
- 38k
9
votes
Accepted
Atiyah classes of holomorphic vector bundles with trivial Chern classes
No. A counter-example : the vector bundle $\mathcal{O}_{\mathbb{P}^1}(p)\oplus \mathcal{O}_{\mathbb{P}^1}(-p)$ on $\mathbb{P}^1$ has zero Chern class, but does not admit a holomorphic connection if $p …
- 38k
8
votes
Accepted
3-folds with "simple" Betti numbers and positive Kodaira dimension
Let me just mention that the non-existence of such a threefold is an immediate consequence of Yau's inequality. First, as explained in the above comment, the conditions $b_2=1$ and $\mathrm{Kod}(X)\ge …
- 38k
8
votes
Accepted
Deformation invariance of Chern classes
This is actually true for all Chern classes, but you must first say how you identify $H^*(X,\mathbb{Z})$ and $H^*(X_t,\mathbb{Z})$. There is no problem for small deformations, that is if $B$ is a ball …
- 38k
8
votes
Accepted
Is there a formula for the total Chern Class of the tangent space of a projectivized vector ...
No, your formula is not correct. You have to take into account the Chern classes of $V$. The relative tangent bundle $T_{\mathbb{P}V/M}$ is given by the so-called Euler exact sequence
$$0\rightarrow …
- 38k
8
votes
Fixed points under a finite group action on projective variety
The answer to the edited question is no. Take a double covering $\pi :X\rightarrow \mathbb{P}^2$ branched along a smooth quartic curve $C$ (so $X$ is a Del Pezzo surface), and $G=\langle \sigma \ran …
- 38k