Questions tagged [complex-geometry]

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.

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2 votes
1 answer
225 views

Holomorphic Spectrum

My question: is it true that we can define a spectrum of $\mathbb{C}$-algebra $A$ in such a way that it becomes a complex manifold with the algebra of holomorphic functions $A$? Maybe it will work if ...
7 votes
1 answer
512 views

Holomorphic Weinstein Lagrangian neighborhood theorem

The Weinstein Lagrangian neighborhood theorem says that if $(M,\omega)$ is a symplectic manifold and $L\subset M$ is a Lagrangian submanifold, then there are neighbourhoods $U$ of $L$ in $M$, and $U'$ ...
-3 votes
1 answer
202 views

Conformal map from a 7-sided polyhedron to a square pyramid

I have a right-angled square pyramid, $A$, whose height and base-length are $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
8 votes
2 answers
1k views

Generalising the Penrose Twistor Fibration

As is well known, there exists a fibration $\mathbb{CP}^3 \to S^4$, of the four sphere by complex projective $3$-space, called the Penrose twistor fibration. Does this fibration admit a "canonical" ...
1 vote
0 answers
198 views

Riemann's bilinear relations

I am reading the paper [1], which states Haupt showed that a vector with complex entries $(w_1, \cdots, w_g, z_1, \cdots, z_g)$ is the period row of some holomorphic differential with respect to a ...
2 votes
0 answers
149 views

Construct Torsion element in $H^2(X,\mathbb{Z})$ with Ambrose–Singer theorem

Let $X$ be a Kahler manifold. Using the exponential sequence one obtains a homomorphism $c_1:H^1(X,\mathcal{O}_X^*)\rightarrow H^2(X,\mathbb{Z})$. This is associating to a holomorphic line bundle $L$ ...
1 vote
0 answers
612 views

Questions on Néron–Severi group

$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$I have two questions on a comment from Daniel Hyubrechts's Complex Geometry on pages 133/134. Let $X$ be a compact Kähler manifold. Consider ...
2 votes
0 answers
91 views

What's the behavior of the laplacian of dbar operator w.r.t. a singular metric of a holomorphic line bundle?

What's the behavior of the laplacian of dbar operator w.r.t. a singular metric of a holomorphic line bundle or other holomorphic vector bundle over a complex manifold ?Do we have anything similar ...
1 vote
0 answers
292 views

Sectional curvature in complex manifold

Let $(X, \omega)$ be a Hermitian manifold .Say that the sectional curvature of X is negative is the same to say that the sectional curvature of the Hermitian metric $\omega$ is negative, otherwise, ...
10 votes
2 answers
499 views

Is there a Kähler manifold with no anti-holomophic involution?

That is, is there a Kähler manifold $X$ on which there is no map $$ \tau:X\to X $$ such that $$ d\tau\circ I=-I\circ d\tau $$ and $$ \tau\circ \tau=\mathrm{Id}_X? $$
2 votes
0 answers
142 views

Bounded holomorphic section

Is it possible that there exists a multivalued holomorphic function $\phi\not \equiv 0$ from $\mathbb {CP}^1$ to $\mathbb C^k$ that has monodromies $A_1, \dots A_n \in GL_k(\mathbb C)$ around points $...
9 votes
0 answers
824 views

Is there a Calabi-Yau threefold with $h^{1,1}=1$ and $h^{1,2}=0$?

Is there a Calabi-Yau threefold with $h^{1,1}=1$ and $h^{1,2}=0$?
5 votes
1 answer
183 views

Can harmonic maps with immersive boundary conditions have singular points?

Let $\mathbb D^2$ be the closed unit disk in $\mathbb R^2$. Let $f:\mathbb D^2 \to \mathbb{R}^2$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^2 \to \mathbb{R}^2$ be ...
8 votes
2 answers
4k views

A question on Ricci curvature and Ricci form.

It seems to me that there are two definition of Ricci flatness; real and complex ones. The real Ricci flatness claims vanishing of Ricci curvature. To define complex Ricci flatness, one needs to ...
1 vote
1 answer
217 views

Some simple algebra of rational functions by André Weil

In André Weil's dissertation, he considers two meromorphic functions $x,y$ on a complex curve.  He assumes every pole of $y$ is a pole of $x$, and its multiplicity as a pole of $y$  is no greater than ...
2 votes
0 answers
160 views

geometrical or physical interpretation of second Chern classes of Calabi-Yau threefold

It's my first post. Consider Calabi-Yau threefold $M$ and its tangent bundle $TM$. I know $c_1(TM)=0$ means metric on $M$ is a solution of vacuum Einstein equation. Then my question is "are there any ...
4 votes
0 answers
190 views

Do we have $D^b_{coh}(X)\simeq D^b(coh(X))$ for a compact complex manifold $X$?

Let $X$ be a compact complex manifold and $\mathcal{O}_X$ be the structure sheaf of holomorphic functions. We call a sheaf of $\mathcal{O}_X$-module $\mathcal{F}$ coherent if it satisfies the ...
0 votes
1 answer
84 views

Regarding definition of Kobayashi length [closed]

I am totally new in the area of complex geometry. I have been reading this paper by Steven Krantz The Carath ́eodory and Kobayashi Metrics and Applications in Complex Analysis. In the definition of ...
1 vote
0 answers
128 views

Essential Steinness of projective manifold

As we all know, a projective manifold is an essentially Stein manifold. Here, we use the definition as follows: A Kähler manifold Y is said to be essentially Stein if there exists an analytic ...
9 votes
3 answers
971 views

$\mathbb CP^k$ bundles over $\mathbb CP^n$ are projectivisations of vector bundles. Any reference?

Statement. Let $X$ be a smooth complex projective variety that is a $\mathbb CP^k$ bundle over $\mathbb CP^n$ in analtytic topology. It is well known that there exists a rank $k+1$ complex vector ...
5 votes
1 answer
295 views

Can the automorphism group vary too much in families of complex projective varieties?

In a family of smooth projective curves over a reduced complex scheme of finite type the list of isomorphism classes of automorphism groups of the fibers is finite. This follows from the Hurwitz's ...
6 votes
2 answers
828 views

A harmonic function

Consider the vertical strip of angle $\alpha=\frac{\pi}{2}$ In this case, the harmonic function which is $0$ on the left line and $1$ on the right line is given by $$f(a+ib)=\frac{a}{T}.$$ Now, when ...
4 votes
0 answers
95 views

Generalising definition of Hurwitz number of compactified moduli space of curve

I am asking mostly for reference if such a definition exists in the literature. I am also interested in the count if it appears somewhere. Let $\mu:=(\mu_1,\ldots , \mu_n)\vdash d$ for positive ...
2 votes
4 answers
2k views

Learning roadmap for complex geometry

I am interested to pursue my graduate studies in complex geometry but sadly I did not find a lot of references regarding the learning roadmap for complex geometry on the website. (most of them are ...
3 votes
1 answer
759 views

Constructing a very ample line bundle on a projective bundle

Let $X$ be a smooth complex projective variety and $p:Y\to X$ be a locally trivial in analytic topology $\mathbb CP^k$-bundle. Suppose we have a line bundle $L$ on $Y$, restricting to $\mathcal O(1)$ ...
3 votes
0 answers
125 views

differential map of a flow

I am trying to understand this passage of the paper "APPLICATIONS OF THE HOLOMORPHIC LEFSCHETZ FORMULA" from Kosniowski 1) Just to make sure I understand the notion correctly: Let $n$ be the complex ...
5 votes
1 answer
326 views

$h^{p,q} = h^{q,p}$ on complex smooth projective scheme

I know that for compact Kähler manifolds $M$ there is an isomorphism: $$ H^p(M, \Omega_M^q) = H^q(M, \Omega_M^p) $$ where $\Omega_M$ is the sheaf of holomorphic $1$-forms. It is because $H^p(M, \...
5 votes
0 answers
664 views

Wedge product on cohomology groups

I have a complex smooth projective scheme $X$ with the sheaf of Kähler differentials $\Omega_{X/\mathbb{C}}$ (or only $\Omega$). Denote its analytification $X^{an}$ with analytification morphism $h:X^{...
5 votes
0 answers
159 views

multiplication in spectral sequence

I am trying to understand this paper. Let $M$ be a compact Kaehler manifold of dimension $n$, $X$ is a holomorphic vector field, $i_X$ the contraction operator, i.e. for $\alpha$ a $p$-form, then $i_X(...
8 votes
1 answer
409 views

Holomorphic deformation of complex structure on the real plane

It is known that each complex structures on $\mathbb{R}^2$ is biholomorphic to either $\mathbb{C}$ or the open unit disk $\Delta$. One can continuously deform one complex structure to the other as is ...
6 votes
1 answer
389 views

Almost complex structures on cotangent bundles of almost complex manifolds

Let $M$ be a smooth manifold and $I:TM\to TM$ an integrable almost complex structure. Then, the cotangent bundle $T^*M$ admits a canonical complex structure, which can be built from holomorphic charts ...
1 vote
3 answers
415 views

Complex structure on product of two $n$-dimensional real manifolds

Let $M$, $N$ be $n$-dimensional real manifolds. Does $M\times N$ admits a complex structure? If not, are there known condidtions ensuring that $M\times N$ admits a complex structure?
4 votes
1 answer
623 views

What is the local structure of a general Artin stack?

Let $X$ be an Artin stack over the complex numbers. What can one say about the local structure of $X$, i.e. what is the simplest class of stacks by which were can always find a cover of $X$ by open ...
1 vote
0 answers
126 views

Almost complex structure commuting with symplectomorphism

Let $(V,\omega)$ be a symplectic vector space with symplectic form $\omega$. Furthermore, let $\varphi : V \rightarrow V$ be a linear symplectomorphism. Consider the set $$ \mathcal{I}_{\varphi} := \{ ...
22 votes
1 answer
1k views

Relationship between the signs of different notions of curvature in complex geometry

Let $(X,\omega)$ be a complex hermitian manifold, and call $\Theta$ its Chern curvature tensor. Out of this we can consider different notions of curvature, namely the holomorphic bisectional curvature ...
1 vote
0 answers
115 views

3rd Cohomology of a fibration with Flag varieties as fibers

Let $X$ be a smooth projective rational variety over $\mathbb{C}$, let $Y$ be another smooth projective variety, both of dimension bigger than 2, and let $\pi : Y \rightarrow X$ be a locally trivial ...
2 votes
1 answer
218 views

Cup Product with Ample Line Bundles

I am wondering if the following argument is true: Let $X$ be a $\dim n$ compact projective complex manifold, let $\alpha\in H^{2n-2}(X,\mathbb{Q})$ be a cohomology class. If for any ample line bundle ...
7 votes
1 answer
549 views

Grauert's Contractibility Theorem

I am interested in reading the proof of Grauert's Contractibility Theorem, asserting that an integral compact curve in a smooth compact surface (without the projectivity assumption - this is the case ...
2 votes
1 answer
159 views

The Monge- Ampère equation with a non positive right hand side

Let $\Omega$ be a domain, $u$ and $f$ are real valued functions on $\Omega$, $(u_{ij})$ is the Hessian matrix of $u$. The function $f$ may change sign: that said, do there exist solutions for the ...
5 votes
0 answers
226 views

Showing that a certain level set of a continuous family of holomorphic maps is locally path connected

I'm working with a continuous function $P: [0,1] \times W \to \mathbb{C}^n$, where $W \subset \mathbb{C}^n$ is an open, relatively compact ball centred at the origin. The map $P$ satisfies the ...
5 votes
0 answers
197 views

cohomology of dual intersection complex of a k3 surface

Let $\Delta \subset \mathbb{C}$ be a small disc and let $f: X \to \Delta$ be a flat morphism of complex manifolds such that $X_t$ is a smooth K3 surface for $t \neq 0$, and $X_0$ is an snc divisor. ...
8 votes
1 answer
606 views

Automorphism group of compact hyperkähler manifolds

Let $M$ be a compact simply-connected hyperkähler manifold, and let $$ \mathrm{Aut}(M) $$ be the automorphism group of $M$, i.e. the group of tri-holomorphic diffeomorphisms preserving the metric. ...
3 votes
0 answers
575 views

English reference for Fischer-Grauert theorem and its generalization by Schuster

From this MSE question and its answer, and from this MO question I have learned of the following remarkable theorem of Wolfgang Fischer and Hans Grauert. Theorem. A proper holomorphic submersion with ...
1 vote
0 answers
53 views

A non-Hermitian-Einstein vector bundle over a compact homogeneous Kahler manifold?

An Hermitian-Einstein $V$ vector bundle over a compact Kahler manifold $M$ is an Hermitian holomorphic vector bundle whose Chern connection $\nabla$, with curvature $F_{\nabla}$, satisfies $$ \Lambda ...
0 votes
0 answers
67 views

Is the sum of a Kähler metric and a balanced metric balanced?

Let X be a compact complex manifold, and let $u$ be a Kähler metric and $v$ a balanced metric. My question is about the metric $u+v$, is it balanced? More precisely, is the operator $*$ of Hodge on ...
2 votes
0 answers
54 views

A holomorphic map into a Hilbert space with prescribed orthogonality

This is a variation of my previous question. Let $X\subset \mathbb{C}^n$ be a domain, and let $L:X\times X\to \mathbb{C}$ be such that $L(x,x)>0$, $L(y,x)=\overline{L(x,y)}$ and $L(\cdot,y)$ is ...
9 votes
1 answer
724 views

Quotients of $K3$ surfaces by finite groups

Let $G$ be a finite subgroup of the group of automorphsims of a $K3$ surface $S$. Consider the quotient $S/G$. I am interested in the collection of such qutients: $$\{ S/G \mid S\text{ is a K3 ...
4 votes
0 answers
220 views

Quotients of Kähler manifolds

Let $X$ be a Kähler manifold and $G$ a complex semisimple Lie group acting freely on $X$ by biholomorphisms and such that the Riemannian metric is preserved by a maximal compact subgroup $K$ of $G$. ...
0 votes
0 answers
150 views

covariant derivative of a function

Let $f$ be a smooth function such on a compact kahler manifold $(M, w)$, and the component of $w$ is denoted by $g_{ij}$, assume there is a constant $s$ such that $sf = -g^{ij}\sqrt{-1}\partial_{j}\...
4 votes
1 answer
2k views

Laplace spectrum of the $2$-Sphere [closed]

The $2$-sphere $S^2$ endowed with usual round metric, we have a Laplacian operator $\Delta_{\mathrm{d}} = \mathrm{d}^*\mathrm{d} + \mathrm{d}\mathrm{d}^*$ acting on functions. The eigenvalues of this ...

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