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.
3,143
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Non-compact hard Lefschetz theorem
For a compact Kaehler manifold $M$, a basic structural result for its de Rham cohomology is the hard Lefschetz theorem. See here or here for an overview of the result.
What happens in the non-...
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2
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Theta functions on an elliptic curve and Serre duality
Given an elliptic curve $E$ (over $\mathbb{C}$) and line bundle $L$, one can identify $H^0(E,L)$ with a particular space of theta functions.
Serre duality gives a perfect pairing between $H^0(E,L)$ ...
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Warped product manifold with real and complex parts
Is possible to define a warped product manifold $M=(N,g_N) \times f(F, g_F)$ where $(N, g_N)$ is a Riemannian manifold with Riemannian metric (i.e., real manifold with real structure) and $(F, g_F)$ ...
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2
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Analytic and algebraic torsor of abelian scheme
Let $M$ be an affine complex manifold, let $A$ be an abelian scheme over $M$. Let $\mathcal{A}$ be the sheaf of local sections of $A$ over $M$. We can equip $M$ with etale topology $M_{et}$ or complex ...
user39380
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reference for the weak compactness of currents
I am trying to follow the arguments in page 22 of the following paper k\"{a}hler currents and null loci
It quotes the weak compactness of currents, I wonder if there is any reference about it. My ...
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How the hyperbolic metric changes when we add a puncture?
Suppose we have a surface $S$ of a finite genus, without boundary with a finite number of punctures. Suppose that this surface comes equipped with a hyperbolic metric of curvature $-1$.
Question 1: If ...
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Negative Definite Fano Manifolds
A complex manifold $M$ is said to be Fano if the Chern curvature $2$-form is a positive definite $(1,1)$-form. What happens if the Chern curvature $2$-form is a negative definite $(1,1)$-form? What ...
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Kodaira Dimension of a Calabi-Eckmann manifold
What is the Kodaira dimension of a Calabi-Eckmann manifold, $X$, when the underlying topological manifold is $S^{2p+1} \times S^{2q+1}$ where $p \geq 1$, $q \geq 1$ and $p \neq q$ ?
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0
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Quotient by finite subgroups are biholomorphic
Let $X$ be a complex manifold and let $G$ and $H$ be two finite subgroups of its automorphism group $Aut(X)$. Suppose we are given that $X/G$ and $X/H$ are bi-holomorphic complex manifolds. What can ...
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Maps between grassmannians with inclusion property
Edit: According to the comment of L. Spice I changed the inclusion sign to the subset sign.
Is there a continuous map $f:\mathbb{C}P^3 \to \textrm{Gr}_{\mathbb{C}}(2,4)$ with $x\subset f(x)$? What ...
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Existence of holomorphic coverings having small degree
Let $\Sigma$ be a closed Riemann surface of genus $g$. In the book of Farkas and Kra, they prove that there exists a holomorphic covering map $F : \Sigma \to \mathbb{S}^2$ of degree less than or equal ...
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Stable vector bundle and Hitchin map
Let $E$ be a stable vector bundle over a curve $X$. $K_X$ the canonical bundle of $X$. $W$ the base of the Hitchin map.
Is the Hitchin map $H:H^0(E\otimes E^*\otimes K_X)\rightarrow W$ surjective?
...
- 1,871
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Structure of the module of derivations on the space of Holomorphic functions
Maybe this is well-known, maybe not.
Let
$\Omega\subset \mathbb{C}$ be connected open and non-empty.
It can be shown that if
$d\in\mathfrak{Der}(\mathcal{H}(\Omega))$
(i.e. $d$ is a derivation of ...
- 3,696
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Constructing certain Global section with prescribed zero locus over Stein manifold
Let $X^n$ be a Stein manifold (complex submanifold in $\mathbb{C}^N$ for some large $N$). Let $D = \{(z,z)\in X\times X: z\in X\}$ be the diagonal in $X\times X$. I'm looking for some holomorphic ...
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Where can I learn about the discrete symmetries of the complex projective plane (or space)?
I understand that $CP^1$ is the Riemann Sphere. I guess all its discrete symmetries were known for a long time and well-classified. (But suggestions or good references where this is worked out in a ...
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Holomorphic functions to complex torus
Let $X$ be a complex algebraic variety and $T$ a complex torus (not necessarily algebraic).
Assume that $X$ is a proper subset of its completion $\bar{X}$.
Let $f:X \to T$ be a holomorphic map.
Are ...
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Locally affine varieties and du Val singularities
Let me start with an apologetic disclaimer: I am very far from an algebraic geometer, so this question might be crudely formulated.
I have a specific question about du Val singularities, but while ...
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Does the $\overline{\partial}$ operator have closed image?
Let $X$ be a complex-analytic manifold, not necessarily compact.
Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
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Disk with punctures and convex geodesical hull of the punctures isomorphic?
Consider a unit disk with marked points $z_i$, $i=1, \dots , n$ on its boundary.
Let us call this surface $X$.
As it is well known, the disk can be equipped with an hyperbolic metric and is then ...
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Spectral gaps for spin manifold Laplace spectrum
For a (compact) spin manifold, we know that the eigenvalues $\lambda_n$ of the Dirac operator are countable, with finite multiplicity, and satisfy
$$
|\lambda_n| \to \infty, ~~~ \text{ as } n \to \...
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Condition for Integrability of an Almost Complex Structure
The following question concerns a remark made in the paper:
Lebrun, C., Complete Ricci-flat Kähler metrics on $\mathbb{C}^n$ need not be flat, Proceedings of Symposia in Pure Mathematics, Volume 52 ...
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Is $h^1(X,O_X)$ always equal to the dimension of the Albanese?
Let $X$ be a projective integral scheme over $\mathbb{C}$.
If $X$ is smooth, then $\mathrm{h}^1(X,\mathcal{O}_X)$ is the dimension of the Albanese variety of $X$. Probably, even if $X$ is normal, ...
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Toric Fan for the Du Val's singularities D_n and E_n
Let us consider the Du Val's singularities.
i.e. https://en.wikipedia.org/wiki/Du_Val_singularity.
It is well known that they are classified by ADE, because the exceptional divisors arising in the ...
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1
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Hodge variation
I am reading Milne's online book of Shimura Varieties https://www.jmilne.org/math/xnotes/svi.pdf, I confused by a Definition of Hodge variation. On page 29, it was said something is called Hodge ...
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2
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Petries exotic circle action
In the paper "S^1-actions on homotopy complex projective spaces" by Petrie (Bulletin of the AMS, 1972), Petrie constructs a smooth circle action on $\mathbb{CP}^{3}$ (page 148). The fixed point set ...
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Compact embedding of the $\mathcal{C}^k$ norm on a compact Kahler manifold
Given a smooth complex valued function $f$ on a Kahler manifold $X$, we can define its $\mathcal{C}^k$ norm to be $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, ...
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A purely algebraic argument for existence of a section of a smooth projective morphism to the projective line
If I am reading this post correctly, any smooth projective $\mathbb{C}$-morphism of schemes $X\rightarrow \mathbb{P}^1$ admits a section. I am afraid of the topological argument presented there. Is ...
user142965
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A non-Kaehler manifold complex and symplectic in exactly one way
Does there exist a closed connected smooth manifold that admits exactly one (up to biholomorphism) integrable complex structure and exactly one (up to symplectomorphism and rescaling) symplectic ...
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Confusion about complex differential forms
I follow Kobayashi "Differential Geometry of Complex Vector Bundles", pages 11-12, prop. 4.9. Given a rank-$r$ Hermitian holomorphic vector bundle $(E,h)$ over a complex manifold $M$, there exists a ...
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$(-2)$-curves in complex $3$-folds
Let $X$ be a smooth complex $3$-fold,
and let $C \subset X$ be an embedded smooth rational curve whose
normal bundle $N_{C/X}$ is isomorphic to $\mathscr{O}(-1) \oplus \mathscr{O}(-1)$.
Is it true ...
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Automorphisms of a neighborhood of a negative curve
Let $X$ e a smooth complex surface and let $C\subset X$ be a smooth rational curve with negative self intersection.
Is there any known description of the automorphisms of a infinitesimal ...
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In which space are we solving the Kähler Ricci flow?
The Kähler Ricci flow on a compact Kähler manifold are formulated as $\frac{\partial}{\partial t}w(t) = -Ric(w)$, $w(0) = w_0$, where $w(t)$ is a family of Kähler metrics and $w_0$ is the initial ...
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A question on an extension of a holomorphic map
I encountered the following statement.
Let $X_1$ (resp. $X_2$) be a complex manifold, $D_1$ (resp. $D_2$) a divisor, and
$f:X_1 \rightarrow X_2$ a holomorphic mapping satisfying
1) $f$ is proper ...
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answers
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On the milnor number of analytic germ map
If $f:(\mathbb{C}^n,0) \to (\mathbb{C},0)$ is an analytic germ with an isolated singularity, then the Milnor number of $f$, denoted by $\mu(f)$ can be defined as $\dim_{\mathbb{C}} \mathcal{O}_n/\text{...
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How to define a "truncated solution complex" $RHom_{D_{X,x}}(M_x,\mathcal{O}_{X,x}/\mathfrak{m}_x^k)$?
Let $M$ be a regular holonomic $D_X$ module on a smooth complex variety $X$. The comparision theorem says that
$$RHom_{D_X}(M,\mathcal{O}_X)_x\cong RHom_{D_X,x}(M_x,\hat{\mathcal{O}}_{X,x}).$$
Now ...
- 2,768
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2
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Continuity of a differential of a Banach-valued holomorphic map
Originally posted on MSE.
Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...
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Algebra of meromorphic functions on a Riemann surface
Let $C$ be compact Riemann surface of high genus. Let $p$ be a general point on $C$ and $z$ a local coordinate around $p$.
Given a meromorphic function on $C$, regular outside $p$, we can look at its ...
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Suppose that two cohomologous forms agree on every restriction. Do they agree?
Let $\eta$, $\omega$ be two $(1,1)$-forms on $\mathbb{C}^m \times Y$, where $Y$ is a compact Kahler manifold with vanishing first Chern class, i.e., a Calabi-Yau manifold. Suppose that for all $z \in \...
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Submersion to $ T^{2}$
Let $ M$ be a $2n$-dimensional compact and connected manifold.
Suppose there is $\Omega\in\Omega^{1}(M,\mathbb{C}) $ a closed complex form whose real and imaginary parts represent linearly ...
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Show that these Kähler forms are cohomologous
Let $Y$ be a closed Kähler manifold with $c_1(Y)=0$ in $H^2(Y,\mathbb{R})$. Let $\omega$ be a Ricci-flat Kähler form on $\mathbb{C}^m \times Y$ such that $$A^{-1} (\omega_{\mathbb{C}^m} + \omega_Y) \...
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complexity of system of equations defining affine variety
Say you have an affine variety $X$ in $n$-dimensional affine space. (You can even assume we are over $\mathbb{C}$, but I believe the nature of my question is algebraic).
I want to bound from above ...
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A deformation of the second Hirzeburch surface $F_2$ over $\mathbb CP^1$
I would like to know if there exists a smooth complex projective $3$-fold $X$ that admits a fibration $\pi: X\to \mathbb CP^1$ such that all fibers are smooth, $\pi^{-1}(0)$ is the second Hirzebruch ...
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Rational functions on reduced complex varieties that extend to global holomorphic functions
Suppose $A$ is an integral domain and a finite type $\mathbb{C}$-algebra. Let $X := \text{Spec}(A)$ and $K := \text{Frac}(A)$ be the fraction field. Suppose $h \in K$ is a rational function that ...
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Coarse moduli space versus Kuranishi family
We will work over complex number field $\mathbb{C}$. Let $\mathscr{M}_h$ be the moduli functor for canonically polarized manifolds with $h$ the Hilbert polynomial. Let us denote by $M_h$ the coarse ...
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Closed subvariety that is unique in its small analytic neighborhood
Let $Y$ be some smooth projective variety over $\mathbb C$ with $\dim Y \geq 2$. For a closed sub-variety $X \hookrightarrow Y$, consider the following property:
There is some small open neighborhood ...
- 6,158
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Kähler metric of constant scalar curvature with positive bisectional curvature is Kähler-Einstein
Suppose $\omega$ is a Kähler metric of constant scalar curvature with positive bisectional curvature, how to prove $\omega$ is Kähler-Einstein?
I was told that we can use the following method:
Step ...
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Holomorphic versus algebraic $\mathbb C^*$-actions
I believe that the following is true:
Statement. A holomorphic $\mathbb C^*$-action on a complex projective manifold is algebraic if and only if it has a fixed point.
Where can I find a proof of ...
- 14k
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211
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Dense Stein subset in complex manifold
Let $X$ be a smooth proper algebraic variety. Then $X$ has a dense affine open subset. In particular, any smooth proper algebraic variety has a dense Stein open subset as the complement of a divisor.
...
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Projection of an invariant almost complex structure to a non-integrable one
My apologies in advance if my question is obvious or elementary.
We identify elements of $S^3$ with their quaternion representation $x_1 + x_2i + x_3j + x_4k$. We consider two independent vector ...
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Kaehler varieties
Let $X\rightarrow D$ be a proper holomorphic map of complex-analytic spaces that is a submersion away from the origin. Suppose that the central fiber is the analytification of a reduced scheme ...
user141225