Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.
2
votes
1answer
97 views
Periodic points in C^2
I came up with a problem which is similar to the following quesitons:
Consider a map: $f(x,y)=(y^2-2,xy-2)$. It is seems that the number of periodic points of given period is bounded.
I want to ...
3
votes
2answers
302 views
Why can we not always take a Kähler class to be in rational cohomology?
Given a Kähler manifold $(X,\omega)$ we know that its Kähler class lies in an open cone of $H^{1,1}(X) \cap H^2 (X,\mathbb{R})$. Since $\mathbb{Q}$ is dense in $\mathbb{R}$ we should be able to find a ...
1
vote
1answer
106 views
Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?
Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n_1:= ...
2
votes
1answer
172 views
How does one compute the first Chern class of a Line bundle defined as the Kernel of a linear map?
Let $M$ and $N$ be compact complex manifolds of the same dimension ($m$) and
$\mu: M \rightarrow N$ a holomorphic map. Let $D \subset M$ be the subset of
points of $M$, where $d\mu|_p$ fails to be ...
1
vote
1answer
29 views
About convex combinations of real-stable multivariable complex polynomials
Say $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$ is a real stable multivariable polynomial on the variables $(z,w_1,w_2,...,w_n)$. (a "real-stable" polynomial is one which has no zeroes in the open ...
8
votes
2answers
248 views
Implicit Function Theorem on Singular Varieties
Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function
$$f\colon X \to Y $$
and assume that it is bijective at the level of ...
1
vote
2answers
203 views
Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?
Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V ...
0
votes
1answer
165 views
Artin approximation of a diagram
Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods ...
2
votes
1answer
225 views
Triviality of holomorphic vector bundles over contractible Stein manifolds
If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows ...
1
vote
1answer
268 views
Non-proper intersection of surfaces
I'm interested in the first basic case of excess intersection in intersection theory:
Let $X$ be a smooth projective 4-fold and let $S,T$ be two surfaces in $X$. Assume that the intersection $S\cap ...
2
votes
1answer
110 views
Extending holomorphic forms
Let $X$ be a normal variety over $\mathbb{C}$ and $\pi:\tilde{X}\rightarrow X$ a log resolution with (reduced) exceptional divisor $E$. Let $U$ be the smooth locus of $X$ and $\omega$ a holomorphic ...
1
vote
2answers
295 views
Line bundles over Kähler–Hodge manifolds
A Kähler–Hodge manifold $M$ can be defined as a Kähler manifold whose Kähler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists a Hermitian ...
2
votes
1answer
194 views
An identity for Futaki-Donaldson invariant
Let $(X,L)$ be a polarized projective variety
Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of :
a scheme $\mathfrak X$ with a $\mathbb C^*$-action
a ...
1
vote
0answers
98 views
Can we find a torus on $K^3$ surface
Suppose in $P^3$ we have $K3$ surface defined by $x^4+y^4+z^4+w^4=0$ can we find a complex subvariety that is a torus?
6
votes
2answers
306 views
Hard Lefschetz Theorem for the Flag Manifolds
In the case of a generalized flag manifold $G/P$, we have an explicit description of their cohomology groups due to Borel.(See herehere for a description.) I would like to know what the hard Lefschetz ...
1
vote
1answer
82 views
Lifting quadratic forms on the cotangent bundle to higher level forms
Backround
In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.
If $\alpha$ is a $(p,q+1)$ form on a domain ...
2
votes
1answer
164 views
An explicit formula for Weil pairing on a complex torus
I begin by defining the Weil pairing in general (as in Oda's 1969 paper). My question is about an explicit formula for this pairing in the case of an elliptic curve over complex numbers.
Let ...
1
vote
0answers
91 views
Complex but not Symplectic
For every $n$ there exist a smooth manifold $M$ of $dim M = n$ that admits a complex structure but not a symplectic one?
8
votes
1answer
312 views
Automorphisms of generic complete intersections
This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under certain assumptions.
To state the question, let $r \geq ...
4
votes
0answers
260 views
Obstructions to deformations of complex manifolds
Roughly, a deformation of a compact complex manifold $M$ (in the sense of Kodaira-Spencer) is a triple $(\mathcal{M},w,B)$ where $w:\mathcal{M}\to B$ is a holomorphic map over domain $0\in B\subset ...
0
votes
0answers
66 views
Wang's C-subgroups and M-manifolds
Let $K$ be a semisimple compact Lie group.
In here H.C. Wang defines a C-subgroup as a closed subgroup $U$ of $K$ such that the semisimple part of $U$ equals the semisimple part of the centralizer ...
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vote
0answers
47 views
An explicit description of the Torelli spaces of pointed genus 2 Riemann surfaces
In [N], there is a nice and very explicit description of the Torelli space ${\rm Tor}_{1,n}$ of $n$-pointed elliptic curves, for any $n\geq 1$:
$$
{\rm Tor}_{1,n}=\left\{
\big(\tau, ...
1
vote
0answers
114 views
Hermitian metric on conic Kaehler-Einstein setting
I have a technical question :
Consider the triple $(M,D,\omega)$ where $M$ is a Fano manifold, $D$ is a smooth divisor whose Poincare dual is $\lambda c_1(M)$ and $\omega$ is a conic Kaehler ...
0
votes
1answer
132 views
Continuations of holomorphic functions on submanifolds to the total space
I have the following question:
I know that every holomorphic function $f$ defined on a closed complex submanifold $M$ of the space $\mathbb C^d$ can be extended to a holomorphic function on the total ...
9
votes
0answers
192 views
Examples of Brody hyperbolic affine varieties which are not Kobayashi hyperbolic
Let $X$ be a complex space.
We say that $X$ is Brody hyperbolic if there is no non-constant holomorphic map $f\colon\mathbb C\to X$.
We say that $X$ is Kobayashi hyperbolic if the Kobayashi ...
3
votes
2answers
254 views
Fano manifold admit an smooth anti-canonical divisor?
Let $M$ ba a compact Kaehler Fano manifold. Under which conditions $M$ admit a smooth anti-canonical divisor $D$
7
votes
1answer
309 views
Examples of Symplectic Questions Solved by ``Mirror Symmetry Translation'' to Complex Questions
According to the proponents of homological mirror symmetry, when a complex and symplectic manifold are mirror symmetric, we can take difficult questions about the symplectic space and transfer them ...
7
votes
1answer
168 views
A question on the twistor space of a manifold
Let $M$ be either (a) self-dual conformal 4-manifold, or (b) hypercomplex $4n$-manifold.
In either case one can construct the twistor space $Z$ (in the case (b) $Z=\mathbb{C}\mathbb{P}^1\times M$ as a ...
0
votes
0answers
81 views
What does the equation $\tau \tau^* = \sigma^* \sigma$ represent in the ADHM construction of vector bundles?
I'm looking at the explicit construction of vector bundles with Anti-Self-Dual (ASD) connections on them via the ADHM construction of instantons. At the heart of this is the complex
$$
V ...
0
votes
0answers
103 views
Coordinate charts on converging Riemann surfaces
Let $S$ be a $2-$dim manifold and $q \in S$. Furthermore, let $j_{n}$ be a sequence of complex structures on $S$ converging in $C^{\infty}_{\text{loc}}$ to a complex structure $j$ on $S$ as ...
10
votes
1answer
209 views
Is there a solvable point on any variety over the field of complex rational functions?
Let $K = \mathbb{C}(T)$ be the field of complex rational functions in one variable, and let $V$ be a variety defined over $K$.
Must $V$ have a solvable point?
The variety $V$ is assumed ...
2
votes
1answer
123 views
Flatness of a morphism of complex analytic spaces
Let $f\colon X\to D$ be a morphism of a complex analytic space $X$ into the 1-dimensional disk $D$. Assume for simplicity that $X$ has a single irreducible component which may not be reduced.
...
7
votes
1answer
302 views
Intuitive Aproach to Dolbeault Cohomology [closed]
(Duplicated from math.stackexchange)
I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. ...
2
votes
0answers
50 views
Computing Dolbeault cohomology of some simple domains
I have seen computations of the Dolbeault cohomology groups on compact Kahler manifolds using Hodge theory.
I have never seen the computation of Dolbeault cohomology for simple domains in ...
3
votes
1answer
149 views
A weak analytic version of the valuative criterion of properness
EDIT: Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that
(a) $f$ is injective on points;
(b) $f$ is local imbedding near each point $x\in ...
11
votes
1answer
282 views
Thom conjecture in CP3
Thom conjecture, that was originally asked in $\mathrm{CP}^2$, and is now proven for symplectic 4 manifolds, states that complex curves (symplectic surfaces) are genus minimizing in their homology ...
1
vote
2answers
138 views
there exists a hypersurface H ⊂ X such that X \ H is Stein and L is trivial over X \ H
"Suppose that X is a compact projective manifold
equipped with a K¨ahler metric ω. Let L be a holomorphic line bundle
In general, there exists a hypersurface H ⊂ X such that
X \ H is Stein and L is ...
0
votes
1answer
155 views
existence of positive curved line bundles on a compact Riemann surface
Can anyone suggest a proof of the existence of positive line bundles on a compact Riemann surface, avoiding Hodge decomposition. (I am aware of the method in Dror Varolin's book, but I consider that ...
3
votes
1answer
153 views
Discriminant of a singular conic bundle
In the context of the Minimal Model Program it can arise that we need to deal with contractions of extremal rays that are conic bundles $\pi:X\to Y$ with relative Picard number 1 (with possibly ...
0
votes
0answers
51 views
A question about bergman kernel
This is lemma 3.3 in B.Berndtsson's paper "Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains":
Let $\Omega$ be a bounded domain and $\phi_j$ a ...
4
votes
2answers
178 views
Projective curves of constant curvature
A nodal projective curve in $\mathbb{CP}^2$ inherits a Kähler metric from the Fubini-Study metric, and hence a Riemannian metric. In particular, with respect to this metric, a line has constant ...
7
votes
0answers
140 views
Complex structures on $\Bbb R^4$
Calabi & Eckmann proved that $S^{2p+1} \times S^{2q+1}$ admits an integrable complex structure fibred by holomorphic tori, and this implies that $\Bbb R^{2p+2q+2}$, obtained by removing a point in ...
4
votes
1answer
239 views
Torsors in the analytic topology versus torsors in the etale topology
Let $S= \mathbb A^1_{\mathbb C}$ be the affine line, and let $G$ be a smooth connected reductive group over $S$, e.g., $G = \mathbb G_m, \mathrm{SL}_n$ or $SO_n$.
Is every analytic $G$-torsor over ...
0
votes
0answers
44 views
on the existence of holomorphic coordinates under bounded curvature
Let $(X,g)$ be a complete Kahler manifold with the Riemann curvature and its derivatives bounded. Suppose also the injectivity radius of this manifold is bounded below by a constant $i_0>0$. My ...
0
votes
0answers
105 views
Kodaira-Spencer theory in d=1
Consider genus $g$ Riemann surfaces, and thier moduli space $\mathcal M_g$.
To determine dimension of $T\mathcal M_g$,
start with a complex structure, which in some coordinates can be written
...
1
vote
1answer
115 views
Decomposing quasi-finite separated group schemes
Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)
Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...
0
votes
0answers
60 views
A question od being algebraic stable for birational map
Recently I need to read a paper related to complex algebraic geometry and several complex variable. I think I may need some criterion of a function being algebraic stable.
Let $f$ be a birational ...
3
votes
0answers
125 views
Properties of finite quotients of quasi-projective varieties
Let $G$ be a finite group acting on a (smooth) quasi-projective variety over $\mathbb C$.
One can consider the stacky quotient $[X/G]$ or the "classical" quotient $X/G$. In general, $[X/G]$ is not a ...
3
votes
0answers
108 views
Toroidal compactifications
Does anyone know if there is an intrinsic definition of a toroidal compactification (over $\mathbb{C}$)?
Something like: Let $X$ be an algebraic variety over $\mathbb{C}$. Then $X \subset \bar{X}$ ...
1
vote
2answers
385 views
Different definitions of spin structures
This is the definition of spin structure according to Wikipedia:
which is supposed to be the standard definition. But in the book The Geometry of Four-Manifolds (Donaldson-Kronheimer, page 76) one ...
