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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.
4
votes
Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
The strict transform of $W$ in the blowup $\tilde{Q}_3$ of $Q_3$ at $O$ is equal to the blowup of $W$ at $O$, and since $W$ has a node at $O$, this blowup is the standard nonsingular model.
To const …
- 39.3k
4
votes
Negative definite of exceptional curve in higher dimension
All the components of $C$ are orthogonal to the pullback $f^*H_T$ of an ample class of $T$. One has
$$
(f^*H_T, f^*H_T) = (H_T,H_T) > 0,
$$
hence its orthogonal complement is negative definite.
A ge …
- 39.3k
5
votes
Accepted
Is the associated G/B fibration to a G-torsor projective?
A similar, but shorter answer. Choose a $G$-equivariant projective embedding $G/B \to \mathbb{P}(V)$. It gives a closed embedding
$$
G/B \times_G P \hookrightarrow \mathbb{P}(V) \times_G P
$$
over $X …
- 39.3k
1
vote
Nonequidimensional birational Mori contractions
Let
$$
Y = \left\{ A = \left(\begin{smallmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23}
\end{smallmatrix}\right)
\right\} \cong \mathbb{A}^6.
$$
Let also
$$
X = \{ (A,v) \in Y \times …
- 39.3k
2
votes
Accepted
Degree three, codimension one subvarieties lying on a quadratic hypersurface
If the linear span of $V$ has dimension $n-1$, then $V$ is a cubic hypersurface in a hyperplane. Otherwise, $V$ is a variety of minimal degree, hence it is a cone over a linear section of $\mathbb{P}^ …
- 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
8
votes
Given a smooth hyperplane section Y of a variety X there exists a Lefschetz pencil of hyperp...
A pencil of hyperplane sections of $X$ corresponds to a line in the dual projective space $\check{\mathbb{P}}^N$. It is a Lefschetz pencil if and only if the line is transverse to the projectively dua …
- 39.3k
12
votes
Accepted
Do non-projective K3 surfaces have rational curves?
Some of them do, and some don't.
Indeed, by global Torelli theorem, there is a K3 surface $X$ with $\mathrm{Pic}(X) = 0$. Such $X$ has no curves, in particular no rational curves.
On the other hand, t …
- 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
3
votes
Accepted
One-dimensional family of complex algebraic K3 surfaces
Choose any ample class in $\mathrm{Pic}(X)$, assume its degree is $d$. Let $M_d$ be the moduli space of polarized K3 surfaces of degree $d$ with appropriate level structure so that it has a universal …
- 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
1
vote
Accepted
Global sections of a line bundle on a reducible complex space
In any case $H^0(S,L)$ is a subspace in $\oplus H^0(V_i,L)$, but the way it sits there depends on the way the components $V_i$ are glued together. In the simplest case where they for a simple normal c …
- 39.3k
6
votes
Accepted
Intersecton form of complete smooth Toric surface
The Picard group of a toric surface is generated by the simple toric divisors $D_1,D_2,\dots,D_n$, and if $v_1,v_2,\dots,v_n$ are the generators of the rays of the corresponding fan, there are relatio …
- 39.3k
2
votes
Stability of sheaves of non-constant rank
The rank of a coherent sheaf is defined as its rank at the general point (equivalently, as the rank on a dense open subset where the sheaf is locally free). So, yes, the definition applies.
- 39.3k
4
votes
Blow up and critical points of the projection map
The following simple observation is quite useful: if $D \subset Y$ is a Cartier divisor and $D$ is smooth, then $Y$ is smooth along $D$. Indeed, for each point $y \in D$ one has
$$
\dim Y - 1 = \dim D …
- 39.3k