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Results tagged with complex-geometry
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user 82179
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.
12
votes
Isomorphism of varieties in $\mathbb{C}$ implies isomorphism over finite fields
I think your post contains a lot of the right ideas, but as in my comment, the situation is much more clear if you allow field extensions.
Here is a positive result:
Lemma. Let $S$ be a scheme and let …
- 28.8k
12
votes
Accepted
Algebraic vs. homological equivalence for curves on a smooth complex projective surface
Super vast generalisation: for divisors on a smooth projective variety over an algebraically closed field of any characteristic, the notions of algebraic, homological (for any Weil cohomology theory), …
- 28.8k
11
votes
Accepted
Do all simple factors of jacobians of curves come from correspondences?
This argument is mostly contained in t3suji's comments, but with some of the proofs somewhat expanded.
As a convention (consistent with that of the theory of Chow motives), all actions of corresponde …
- 28.8k
10
votes
Accepted
Alterations and smooth complete intersections
As you guessed, there are cohomological obstructions. Indeed, if $f \colon Y \twoheadrightarrow X$ is a surjective morphism of smooth projective varieties and $H$ is a Weil cohomology theory (with coe …
- 28.8k
8
votes
Accepted
When are two resolutions of a coherent sheaf homotopic
If this were true, then any short exact sequence of vector bundles would split. Indeed, if $0 \to \mathscr E_1 \to \mathscr E_2 \to \mathscr E_3 \to 0$ is a short exact sequence of vector bundles, the …
- 28.8k
8
votes
Accepted
Infinitely small intersections with nef $\mathbb R$-Cartier divisors
This is possible. Basically, if $N$ is a point on the boundary of the nef cone that is not in the span of the rational points on the boundary, then we can approximate $N$ arbitrarily closely by ration …
- 28.8k
7
votes
Accepted
When is the pullback of a coherent analytic sheaf again coherent?
This is always true. By Oka's coherence theorem, $\mathcal O_X$ and $\mathcal O_Y$ are coherent. Therefore, a quasi-coherent sheaf $\mathscr F$ is coherent if and only if it is of finite presentation …
- 28.8k
6
votes
Accepted
Universal covering of symmetric product
In fact, the universal cover of $C^{(n)}$ will not be $\mathcal H^n$ once $n \gg 0$. Indeed, if $C$ is a compact Riemann surface of genus $g \geq 2$ (so the universal cover is $\mathcal H$) with a bas …
- 28.8k
5
votes
Automorphism induced by an automorphism of the base
In this post I will work in the greatest generality I can think of. In particular, fundamental group means étale fundamental group, but for varieties over $\mathbf C$ the same argument carries through …
- 28.8k
5
votes
Accepted
Splitting a trivial bundle over punctured $\mathbb C^n$
The decomposition $\mathscr E|_U = V_1 \oplus V_2$ gives projectors $\pi_i \colon \mathscr E|_U \to \mathscr E|_U$ such that $\operatorname{im}(\pi_i) = V_i$. By Hartog's lemma, the restriction $\Gamm …
- 28.8k
5
votes
reference quest: surface whose $1$-forms have a common zero
If $S$ is any surface with $h^{1,0}(S) = 1$, then the Albanese morphism is a surjection $a \colon S \to E$ to an elliptic curve, and the pullback $H^0(E,\Omega_E^1) \to H^0(S,\Omega_S^1)$ is an isomor …
- 28.8k
5
votes
Finding global sections of a sheaf of sets using (some kind of) sheaf cohomology?
It is indeed true that the sequence
$$1 \to \mathscr F_H \to \mathscr F_G \to \mathscr F_M \to 1\tag{1}\label{1}$$
of sheaves of pointed sets is exact. The only nontrivial thing to check is surjectivi …
- 28.8k
5
votes
Accepted
Relative Kodaira Vanishing?
The natural analogue of Kodaira vanishing is $R^q\pi_*(K_{X/S} \otimes \mathscr L) = 0$ for $q > 0$, which follows from Kodaira vanishing plus 'cohomology and base change' [Hartshorne, Thm. III.12.11( …
- 28.8k
4
votes
When is bijective map between closed point of varieties a morphism?
By Nakayama's lemma, the support of a coherent sheaf $\mathscr F$ on a Noetherian scheme $X$ is the set of $x \in X$ such that $\mathscr F \otimes_{\mathcal O_X} \kappa(x) \neq 0$ (as opposed to $\mat …
- 28.8k
4
votes
Accepted
Self-intersection of the diagonal on a surface
Here is a reasonably geometric way to move $\Delta_X$ to some other (non-effective) divisor.
Strategy. For $\mathbf P^1$, we know how to do this (see below for an alternative method). In general, take …
- 28.8k