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
votes
2answers
250 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
274 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
276 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
99 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
154 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
123 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
103 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
202 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
178 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
108 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
159 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
2answers
320 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
101 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
166 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
95 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
190 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 ...
2
votes
1answer
327 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
147 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
159 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
275 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
141 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 ...
0
votes
0answers
77 views

N=2 Dualities; k-differentials on the riemann sphere and spectral curves

Currently I am working on my masters thesis about dualities in QFT and their geometric realizations. As of now, I am trying to understand the article 'N=2 Dualities" by Davide Gaiotto. On the internet ...
2
votes
0answers
268 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 ...
3
votes
1answer
264 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 ...
4
votes
1answer
191 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 ...
3
votes
2answers
217 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 $M$, ...
4
votes
1answer
140 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 ...
5
votes
1answer
251 views

degrees of complex projective spaces and quadrics

A well-known result of Kobayashi and Ochiai says that an $n$-dimensional Fano maniofold $M$ is biholomorphic to $\mathbb{C}P^n$ or complex quadrics if its index is $n+1$ or $n$ respectively. In these ...
3
votes
1answer
117 views

Existence of a map between automorphism group of universal covers

Let $f:X\to Y$ be a holomorphic map of holomorphic manifolds. You can assume that $dimY=1$. Let $\tilde X$ and $\tilde Y$ be universal covers of $X$ and $Y$ with group of holomorphic automorphisms ...
2
votes
0answers
169 views

Weakened jacobian conjecture for entire functions

A rudin's theorem is the assertion that any polynomial injection between affine spaces of the same dimension has a polynomial inverse, and the inverse is also given by polynomials. The jacobian ...
10
votes
4answers
550 views

What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
0
votes
1answer
126 views

Inverse “Riemann mapping” [closed]

The Riemann mapping theorem states, that any simply connected domain $U \subset \mathbb C$ can be conformally mapped to the open unit disk $D$. I.e. there is a Diffeomorphism $\Psi: D \to U$ such that ...
3
votes
1answer
116 views

Linear Complex Structure and Kahler Angles

I am trying to read Donaldson's paper on symplectic submanifolds http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214459407 and am getting a bit ...
1
vote
0answers
145 views

Hyperkaehler Structures on the cotangent bundle

Let $M$ be a symplectic manifold (not Kaehler). Does there exists in a neighbourhood of the zero section in the cotangent bundle $T^{*}M$ a Hyperkaehler structure? I know that by the paper by Feix on ...
1
vote
1answer
104 views

Off-diagonal holomorphic extension of real analytic functions on $\mathbb{C}^n \times\mathbb{C}^n$

I am struggling trying to understand an statement in a paper I am reading: Let $M$ be a complex manifold of dimension $2n$. Let's consider a function $\xi$: $M$ $\rightarrow$ $\mathbb{C}$ whose ...
1
vote
1answer
293 views

Holomorphic vector field on Fano Kähler–Einstein manifold

Let $M$ be a compact Fano Kähler–Einstein manifold, and $V$ a holomorphic $(1,0)$ vector field on $M$. The Fano conditions say that $V = \nabla^{1,0} f$ for some smooth complex-valued function. By ...
5
votes
1answer
181 views

Comparison of two traces

Suppose $X$ is a smooth quasi-projective variety over $\mathbb{C}$ and $Z$ a proper subscheme, there is a formal duality isomorphism (here we consider the Zariski topology) due to Hartshorne: $$ tr: ...
9
votes
1answer
268 views

Local polynomial form of holomorphic functions

It is well-known that a germ of a holomorphic function of $n\geq 2$ variables, with at most an isolated singularity (i.e. the singularity of the analytic variety defined by the zero locus of the ...
2
votes
1answer
73 views

How a matrix of C^1 functions on a domain Ω in Cn generates a C^1 distribution in Ω

I am reading the paper Complex foliation generated by (1,1)-forms by M. Klimek. I have a problem understanding why is true a detail in one of the theorems: Let $\Omega$ be an open connected subset ...
3
votes
0answers
139 views

Serre duality for compactly supported sheaves

Given a smooth quasi-projective variety $X$ over $\mathbb{C}$ and bounded complexes of vector bundles $(P,d)$ and $(P',d')$ with compactly supported cohomology. It is well-known that such complexes ...
2
votes
1answer
143 views

Determine complex analytic germ along a smooth compact curve via normal bundle?

Let $X_1, X_2$ be two smooth complex manifold and $C_1 \subset X_1, C_2 \subset X_2$ be two smooth projective curves. Assume that $C_1 \simeq C_2$ as complex curves and their normal bundles are ...
1
vote
1answer
184 views

Coherent sheaf with Vanishing chern classes

If one can construct a coherent sheaf over a smooth projective variety with all chern classes vanishing which is not locally free? or for any such coherent sheaf $E$ , it must be locally free?
1
vote
1answer
193 views

pull back of a Kähler form by a smooth circle group action on a compact Kähler manifold

Suppose a compact Kähler manifold $(M,\omega)$ admits a smooth circle action $g_t,~t\in S^1$. So the pull back of the Kähler form $g_t^{\ast}(\omega)$ is a nondegenerate two-form. Since the circle ...
4
votes
1answer
295 views

When is normalization functorial?

Let $X$ and $Y$ be two irreducible, affine $\newcommand{\C}{\mathbb C}\C$-varieties. Let $f:X\to Y$ be a morphism. Denote by $u:\tilde X\to X$ and $v:\tilde Y\to Y$ their normalizations. Now, if $f$ ...
7
votes
1answer
326 views

Do all varieties have only finitely many etale covers of fixed degree

I've been wondering about the following "finiteness statement" concerning etale covers for a while. Let $K$ be a field of characteristic zero, not necessarily algebraically closed. A variety over $K$ ...
9
votes
1answer
245 views

Variety $X$ such that $TX$ is ample on any curve in $X$

Let $X$ be a smooth complex projective variety such that the restriction of $TX$ on any curve $C$ in $X$ is ample. Is true in this case that $X$ is isomorphic to $\mathbb CP^n$? I guess the above ...
3
votes
1answer
229 views

Do Deligne-Mumford curves also have rational functions

If $X$ is a curve over a field of characteristic zero, then $X$ has a rational function, i.e., a finite morphism to the projective line. Question. Suppose that $X$ is a Deligne-Mumford (or just ...
3
votes
2answers
437 views

Is there a “by hand” proof on the symmetry of the Atiyah class of $TX$?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
4
votes
1answer
216 views

global section of local system from direct image

Deligne has a theorem in "Theorie de Hodge II" as follows: Let $S$ be a smooth separated scheme, and $f:X\to S$ be a smooth proper morphism. Let $\bar{X}$ be a non singular compactification of ...
0
votes
0answers
48 views

intersecting irreducible components of complex analytic sets invariant under group action

Let $\Gamma$ be a group acting freely by automorphisms on a complex analytic space. Under what assumptions the following is always true: Let $X$ and $Y$ be irreducible components of closed ...