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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.
6
votes
Accepted
Deformations of a pair of compact, complex manifolds
The infinitesimal deformations of
$(X,M)$ are controlled by the sheaf $T_X\langle M\rangle$ of vector fields on $X$ which
are tangent to $M$: see for instance this paper, Prop. 1.1. Thus the obstructi …
- 38k
4
votes
Accepted
Short exact sequence of trivial holomorphic line bundles not splitting
For any line bundle $L$, such exact sequence splits. Indeed the map $L\rightarrow L$ induced by the surjection $p:L\oplus L\rightarrow L$ must be nonzero on one of the summands, say the first one; hen …
- 38k
2
votes
A question of direct image of relative canonical bundle
The answer to the first question is yes, this is the Grauert semi-continuity theorem. It applies to any proper map $f:X\rightarrow Y$ of reduced analytic spaces, and any coherent sheaf $\mathcal{F}$ …
- 38k
4
votes
Accepted
Chow lemma for complex spaces
As pointed out in the comments, if a complex compact manifold $X$ of dimension $n$ is bimeromorphic to a projective variety, its field of meromorphic functions must have transcendence degree $n$ (the …
- 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
3
votes
Accepted
Set of sections whose zeroes avoid a given divisor is (Zariski) dense?
Yes. Consider the incidence variety $Z\subset D\times \mathbb{P}(H^0(E))$ of pairs $(x,[s])$ with $x\in D$, $s\in H^0(E)\smallsetminus \{0\} $ and $s(x)=0$. Let $p,q$ be the projections from $Z$ to …
- 38k
5
votes
Accepted
Sections of tangent bundle on hypersurface
This has little to do with hypersurfaces. If $\dim(X)=d$, the only condition you need on $X$ is $H^0(X,\Omega ^{d-1}_X)=0$.
The wedge product gives a non-degenerate pairing $\Omega ^1_X\otimes \Omeg …
- 38k
2
votes
Accepted
Very ample linear systems - intersections with multiplicity >1
A partial answer (too long for a comment): suppose $G$ is sufficiently ample so that $H^1(\mathbb{F}_{n},\mathcal{O}(G-S))=0$. Then the restriction map $H^0(\mathbb{F}_{n},\mathcal{O}(G))\rightarrow H …
- 38k
6
votes
Accepted
Constancy of Hodge numbers in a family of compact complex manifolds
Nakamura has constructed a family of compact complex threefolds such that the Hodge number $h^{p,q}$ is not deformation invariant for $p+q>0$. They are obtained by deforming an "Iwasawa manifold" $\ma …
- 38k
4
votes
Accepted
Cup Product with Ample Line Bundles
No. If $\alpha $ is of type $(n-2,n)$, its product with any class of type $(1,1)$ is zero, but $\alpha $ is not necessarily zero (you can take for instance $\alpha = c_1(L)^{n-2}[\omega ]$, where $L$ …
- 38k
6
votes
Accepted
Projective variety of general type such that $S^m \Omega_X^1$ is globally generated
If $X$ is a surface it is true. In general, a smooth projective variety with $S^m\Omega ^1_X$ globally generated does not contain any smooth rational curve $C$. Indeed $\Omega ^1_C$ is a quotient of …
- 38k
7
votes
Accepted
Roots of line bundles in a family
The locus you consider is either empty, or equal to $B$.
This can be seen as follows. Line bundles on a fiber $F$ of $\pi $ are parameterized by $H^1(F,\mathscr{O}^*_F)$. This group fits into an exact …
- 38k
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
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