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Results tagged with complex-geometry
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user 4428
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.
3
votes
Accepted
Picard group of complex orbifolds
I guess that for reductive $G$ one has $H^1(X//G,O) = H^1(X,O)^G$ (the $G$-invariants). So, the condition you want is $H^1(X,O)^G = 0$. For nonreductive $G$ there is a spectral sequence $H^q(G,H^p(X,O …
- 39.3k
4
votes
Accepted
Restriction of a linear system to a divisor
Any effective divisor in $|L\vert_D|$ is the zero locus of a global section of $L\vert_D$. If $s_D$ is such a section, by the surjectivity assumption there is a global section $s$ of $L$ on $X$ that r …
- 39.3k
4
votes
Accepted
Rational smooth complex projectives three fold with non-rational deformation
A conjecture of Iskovskikh says that this never happens. To be more precise, it says that if there is a family of smooth projective threefolds with general threefold nonrational then all these threefo …
- 39.3k
4
votes
Sections of the canonical bundle of a blow-up
Use exact sequences
$$
0 \to \sigma^*K_X \to \sigma^*K_X(E) \to \sigma^*K_X(E)_{|E} \to 0,
$$
$$
0 \to \sigma^*K_X(E) \to \sigma^*K_X(2E) \to \sigma^*K_X(2E)_{|E} \to 0,
$$
$\dots$
$$
0 \to \sigma^*K_ …
- 39.3k
1
vote
Higher ramification loci for finite maps
The cohomology class of the expected corank 2 locus for a morphism of vector bundles is given by the Porteous formula (there is also a version of that formula for symmetric maps). If this class is non …
- 39.3k
7
votes
Accepted
How do we write a locally free resolution for...
In general, there is no straight way to write such a resolution (note by the way, that a resolution is in no way unique!). However, in some cases there is a distinguished resolution. For example, if $ …
- 39.3k
2
votes
Irreducible components of a general singular fiber correspond to irreducible components of t...
The next example shows that this is not true without shrinking to an analytic neighborhood of $b$ (see the comment of Jason Starr below).
Consider the universal conic --- the incidence hypersurface
$$ …
- 39.3k
4
votes
Accepted
topological Euler characteristic of canonical divisor
This is not true. Consider, for instance a Calabi--Yau threefold $Y$ with $h^{2,1}(Y) = h^{1,1}(Y) + 1$ (an example of such can be found in https://arxiv.org/abs/1602.06303, see page 29) and let $X$ b …
- 39.3k
7
votes
Accepted
Torelli theorem for smooth complex cubic surfaces?
The Hodge structures of cubic surfaces do not allow to distinguish them. But there are some ways around. For instance, given a cubic surface $X \subset \mathbb{P}^3$, you can associate with it a cycli …
- 39.3k
3
votes
Accepted
Isomorphic direct images of sheaves on product space
In general no. For instance take $Y = P^1$, $X = P^1 \times Y$, and take $S$ to be 4 different points. Then $Z$ is an elliptic curve and $W = P^1 \times Z$. Take $F$ to be (the pullback to $W$ of) a n …
- 39.3k
2
votes
Accepted
Connectedness of a section of an algebraic bundle
Let $Z = \{s = 0\}$. It is connected if and only if $H^0(Z)$ is 1-dimensional. You can compute $H^0(Z)$ by using Koszul resolution
$$
0 \to \Lambda^n E^* \to \Lambda^{n-1}E^* \to \dots \to E^* \to O_X …
- 39.3k
3
votes
Two questions on complex geometry
If $(x_0:x_1:\dots:x_n)$ are the homogeneous coordinates on $P^n$ then the map $O(-1) \to O^{n+1}$ is given by $s \mapsto (sx_0,sx_1,\dots,sx_n)$.
- 39.3k
1
vote
Cycle class of zeroes of a global section
Assume the zero locus has positive dimension, but is smooth (or is, at least, locally complete intersection). Then it comes with a vector bundle (so-called "excess intersection" bundle), whose top Che …
- 39.3k
5
votes
Accepted
Cycle class of zeroes of a global section
Let $V = H^0(X,\mathcal{F})$ be the space of global sections of $\mathcal{F}$ and let
$$
V \otimes \mathcal{O}_X \to \mathcal{F}
$$
be the evaluation morphism. If it is surjective (i.e., $\mathcal{F}$ …
- 39.3k
7
votes
Accepted
Does a projective variety have only finitely many associated Hilbert polynomials?
Yes for the first question, by Riemann--Roch.
No for the second --- even in the simplest case of a projective line, the polynomial $td + 1$ is the Hilbert polynomial (with respect to $L = O(d)$).
- 39.3k