Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

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3
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1answer
315 views

top chern class

Let $E\to M$ be a holomorphic vector bundle ($M$ is compact) and assume $s\colon M\to E$ is a non-trivial section transverse to the zero section. Is it be possible that $s^{-1}(0)\neq \emptyset$, ...
2
votes
0answers
45 views

Is the Szego projection on a codim-$k$ CR manifold an integral operator?

The Szego projection on a CR manifold $M$ is defined to be the orthogonal projection from $L^2(M)$ to the closed subspace $H^2(M),$ where $$H^2(M) = \{f \in L^2(M)\ |\ \bar{\partial}_{b}f = 0\ ...
1
vote
1answer
131 views

SU(2) Lefschetz decomposition for cohomology of Riemann surface Jacobian

Start with a closed Riemann surface with $g$ handles $\Sigma_g.$ I'm interested in the cohomology of its Jacobian $Jac(\Sigma_g)=T^{2g},$ in particular how the $SU(2)$ or $SL(2,\mathbb{R})$ Lefschetz ...
3
votes
2answers
341 views

The holomorphic version of Galois theory

We identify the space of polynomials of degree n with $\mathbb{C}^{n+1}-\mathbb{C}^{n}$, that is an $n+1$ tuple $(a_{n},a_{n-1},\ldots,a_{0})$ with $a_{n} \neq 0$ is identified with $p(z)=a_{n}z^{n} ...
-2
votes
2answers
229 views

topology on varieties

Let $X, Y$ be varieties over $\mathbb{C}$, and the topology I am talking about is the Eucliden topology. I am not sure if the following two results are true, and where can I find the references: (1) ...
2
votes
2answers
181 views

compute the automorphism of Iwasawa manifold

An Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup. We can also refer to Griffiths and Harris's Principles of Algebraic Geometry ...
1
vote
1answer
91 views

Unicity of a vector field on $S^1$-bundle

Let M be a complex smooth manifold,and let $\zeta $ be a vector filed on $M$, why always there exists a unique vector field $\hat{\zeta }$ on $L^{\times}$ which project down to $\zeta $ and $\alpha( ...
-1
votes
4answers
293 views

An isomorphism on space of smooth sections

Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to $$A=\{f:L^{\times}\to ...
1
vote
1answer
106 views

Legal potentials on delzant polytopes

Let $P \subset \mathbb R^n$ be a Delzant polytope defined by inequalities $\ell_i(x) \geq 0, i=1, \ldots, d$. Of course, from the symplectic point of view, the inequalities $a_i \ell_i \geq 0$ still ...
3
votes
1answer
179 views

Question about the Aganagic-Vafa A-brane

According to Aganagic-Vafa (hep-th/0012041) and Fang-Liu (arXiv:1103.0693), for a semi-projective toric Calabi-Yau 3-manifold $X$, the Aganagic-Vafa A-brane $L_{AV}\subset X$ is defined by the ...
0
votes
1answer
77 views

A subspace of the algebra of infinitesimal CR automorphisms

Let $(M^{2n+1}, D, J)$ be a strictly pseudoconvex CR sturcture on a compact $2n+1$-dimensional manifold, where $D$ is a nonintegrable distribution of codimension 1. The algebra of the infinitesimal CR ...
3
votes
0answers
72 views

Monodromy along strata of a pushforward

Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived. Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X ...
6
votes
0answers
89 views

What classes of complex manifolds are known to be definable in an o-minimal expansion of the real field?

It is a widely known (perhaps slightly folkloric) fact that compact complex manifolds, understood as first-order structures with a predicate for each analytic subset, are definable in an expansion of ...
4
votes
1answer
211 views

Toric Fano Kahler manifolds and Delzant polytopes

Let $P$ be a Delzant polytope in $\mathbb R^n$, given by a set of inequalities $\ell_i(x) > 0$ where $\ell_i(x) = \sum_k \mu_k^i x_k - \lambda_i$. In his paper http://arxiv.org/abs/0803.0985 ...
2
votes
1answer
178 views

Properness of Ding-functional independent of the chosen Kahler metric

On Page 62 of "canonical metrics in Kahler geometry" written by Gang Tian, the author pointed out that properness of Ding-functional $F_\omega$ is independent of $\omega$, without proof. What I want ...
2
votes
1answer
165 views

A continuous version of Teichmuller uniqueness

By the Teichmuller uniqueness theorem, given a homeomorphism $f:X \rightarrow X$ where $X$ is the $n$-punctured sphere, there is a unique quasiconformal homeomorphism $g$ fixing $0$, $1$, and ...
3
votes
0answers
210 views

Semistable minimal model of a $K3$-surface and the special fibre

Suppose that $K$ is a $p$-adic field, that is a field of characteristic $0$ whose ring of integers is a complete discrete valuation ring $\mathcal O_K$ and with residue field $k$ (algebraic closed) of ...
5
votes
1answer
310 views

What is the moduli of an algebraic torus

Given an algebraic torus $(\mathbb{C}^\ast)^n$, what's the moduli space of complex structures? Even for $\mathbb{C}^\ast$, since it's a non-compact Riemann surface with puncture, it doesn't seem ...
1
vote
0answers
138 views

The Leibniz Rule for Vector Valued Holomorphic Forms

I am at the moment re-reading Voisin's book on complex geometry, from which I take the following: Let $M$ be a Kahler manifold, and let $E$ be a smooth vector bundle over $M$. Moreover, let us ...
3
votes
1answer
244 views

Do there exist double points on an algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all. I'm familiar with the du val singularities on surfaces, also known as rational double points. In http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf, 2.1, they are ...
5
votes
1answer
393 views

Are the Kahler Identities for a Holomorphic Vector Bundle Actually Interesting?

For a Kahler manifold $M$, and a smooth vector bundle $E$ over $M$, let us denote by $A^{(p,q)} := \Omega^{(p,q)} \otimes E$ the bundle of forms with values in $E$. Now with respect to a choice of ...
2
votes
1answer
177 views

Questions on the Hodge Dual of the Kähler Class

Let $M$ be a compact complex surface, i.e., a complex manifold with complex dimension two. Also, let us denote its Kähler metric by $g$ and its Kähler form by $K$. Let us denote the Kähler class by ...
2
votes
3answers
271 views

Solutions of the $\overline{\partial}$ equation in the upper half-plane

I am looking for references on solutions of the $\overline{\partial}$ equation in the upper half-plane (with conditions on the boundary). But this subject is completely outside my area of expertise ...
4
votes
1answer
271 views

Any non-constant surjective holomorphic map between connected compact complex manifolds of equal dimension is a ramified finite-sheet covering

I'm reading this site:holomorphy of inverse map There is a statement made by Colin Tan at the last answer made by himself. Any non-constant surjective holomorphic map between connected compact ...
3
votes
2answers
270 views

Varieties that do not extend to flat families

Is it easy to give an example of a function field $K$ and a smooth proper variety $X$ over $K$ that does not extend to a flat scheme over $B$, where $B$ is a smooth proper variety with function field ...
0
votes
2answers
341 views

global sections of canonical line bundle of a projective variety

Given a smooth projective variety $X \subset \Bbb{CP}^k$ why is it true that global sections of $O(l)|_X, l >> 0 $ are just global sections of $O(l)$ on $\Bbb{CP}^k$ restricted to $X$? Here ...
5
votes
1answer
381 views

Geometric structure of flag manifolds, Borel -Weil-Bott theorem

I want to know if there is proof of Borel Weil Bott theorem, that is as geometric as it can be. Let $G$ be a semisimple compact Lie group and $T$ be a maximal torus. We know that $G/T$ is a ...
2
votes
1answer
116 views

Dual of a Complex 2-Torus

Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?
6
votes
1answer
167 views

Intuition for holomorphic bisectional curvature

I would like to know the intuition behind the holomorphic bisectional curvature of Hermitian manifolds. I already know that the classical sectional curvature of a Riemannian (not necessarily complex) ...
1
vote
0answers
129 views

k-amplitude of the Hodge bundle on $\overline M_g$

Let $\mathbb E$ denote the rank $g$ Hodge bundle on $\overline M_g$, the moduli space of stable curves of genus $g$. It is a nef vector bundle. I wonder what other positivity properties the Hodge ...
3
votes
0answers
123 views

Calabi-Yau structure on cotangent bundle?

Let $M$ be a smooth (compact) manifold, my question is when the cotangent bundle $T^*M$ has a Calabi-Yau structure. Certain constructions are known, for instance, if $M=\Sigma\times S^1$ or $M$ is a ...
2
votes
1answer
216 views

Push forward of a Vector bundle is a coherent sheaf?

Let $X$ be a smooth compact Kahler manifold and let $Y\subset X$ be a smooth complex submanifold of complex codimension at least $2$. Let $$j:Y\hookrightarrow X$$ the natural (holomorphic) embedding ...
2
votes
1answer
205 views

Complement of Donaldson's symplectic submanifold

I am just starting to learn more symplectic geometry on Stein manifolds. I understand that an important class of Stein manifolds arises as the complement of a Donaldson's codimension two symplectic ...
2
votes
1answer
117 views

Are harmonic maps quasiconformal at the boundary of hyperbolic spaces?

Let $M$ be a hyperbolic $3$-manifold and $X$ a negatively curved, simply connected space with geometric boundary $\partial X$. If $\rho:\pi_1M\rightarrow Isom(X)$ does not fix a point in $\partial ...
4
votes
1answer
176 views

Endomorphism of complex tori

The algebra $\mathrm{End}_{\mathbb{Q}}(A)=\mathrm{End}(A)\otimes\mathbb{Q}$ of endomorphisms of an abelian variety (defined over $\mathbb{C}$) is well understood and in particular the following is ...
10
votes
3answers
414 views

orbits of automorphism group for indefinite lattices

I have a question about indefinite lattices. QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice, that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form, not necessarily ...
1
vote
1answer
106 views

moduli space of two-term complexes of vector bundles over a fixed variety

Let $X$ be a fixed algebraic manifold over $\mathbb{C}$ , $\{E_{a}\}$ be vector bundles over $X$. We can construct moduli space of $\{E_{a}\}$ by classical theory. My question is that if we consider ...
2
votes
0answers
175 views

Automorphisms of the affine quadric $z_1^2+\dots+z_n^2=1$

Denote by $V$ the vanishing locus of the affine quadric $z_1^2+\dots+z_n^2-1$ in $\mathbb{C}^n$. Clearly $V$ is a complex manifold. Let $g:V\to V$ be an automorphism of $V$ (a biholomorphism, not ...
0
votes
1answer
112 views

Chow lemma for complex spaces

Is there something like the Chow lemma for complex spaces (stating that every compact complex space is birational to a projective variety, or some variant of this, like: every proper morphism of ...
2
votes
1answer
240 views

Existence of constant scalar curvature Kahler metrics on projective manifolds

It is well known that the blow-up of $\mathbb P^2$ in one or two points does not accept a Kahler-Einstein metric. Kahler-Einstein metrics are particular cases of constant scalar curvature Kahler ...
3
votes
1answer
394 views

A question about Abel-Jacobi map

Let $X$ be a Riemann Surface with genus $g$, $S^g(X)$ be the symmetric power of $X$ (which is naturally identified with the set of effective divisors of degree $g$). Let $A$ be the Abel-Jacobi map ...
0
votes
1answer
159 views

Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me: Is there any open Ricci-flat ALE 4-manifold other than ...
2
votes
0answers
174 views

The relationship of relative differential form

Let $X$ be a compact complex surface and $\omega$ be a holomorphic 1-form. $f,g$ are meromorphic function on $X$ such that $\operatorname{trdeg}_{\mathbb{C}}\mathbb{C}(f,g)=1$ and $\omega =g\,df$. Let ...
3
votes
2answers
289 views

How to determine the transcendence degree of function field

If $X$ is a surface, projective and non-singular. Let $\mathbb{C}(X)$ be the function field of $X$. By a theorem of Siegel, we know that $trdeg_{\mathbb{C}}\mathbb{C}(X)\leq 2$. But how to argue ...
3
votes
1answer
153 views

k-differentials and their residues

I posted a question earlier. This question is smaller, more to the point and more basic I hope. Yet it is at the heart of my problems, I think. In one of my references, degree k meromorphic ...
2
votes
0answers
280 views

Deformation over small disk and deformation over complex disk

Let $D=\mathop{Spec}(\mathbb C[[t]])$ be the algebraic small disk and let $\Delta= \{z\in \mathbb C: |z|<1\} $. Let $X_0$ be an algebraic surface over $\mathop{Spec}\mathbb C$ Suppose now that I ...
2
votes
1answer
284 views

On complex surfaces with Kodaira dimension 1

Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $. What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true ...
3
votes
1answer
234 views

Is there extremal metric on toric Fano manifolds which futaki invariant is nonzero?

According the work by Wang & Zhu, on toric Fano manifolds there exists Kahler-Ricci solitons, if futaki=0 there exist CSCK metric, but if futaki invariant is not vanished, what about extremal ...
2
votes
2answers
229 views

The Szego projector, the dual disc bundle $\overline{D}$ and representation of $S^1$ on $H^2$($\overline{D}$)

This construction arises when constructing the Szego projector. Let's consider the dual disc bundle $\overline{D}$ of a positive Hermitian line bundle ($L$,$h$) over a compact Kahler manifold $M$, ...
3
votes
1answer
151 views

Weighted projective space with rational or real weights

The most common formulation of the weighted projective space is perhaps the global quotient $$ (\mathbb{C}^{n+1} \setminus \{(0,\ldots,0\}) / \mathbb{C}^\ast $$ with the $\mathbb{C}^\ast$ group action ...