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

3
votes
1answer
247 views

Chow theorem in $\mathbb{C}^2$

I have the following question the answer to which I cannot find in the literature (but it must have been studied): Suppose that $M\subset\mathbb{C}^2$ is a real surface which may locally be written ...
0
votes
0answers
38 views

Differential operator of globally unbounded order on connected complex manifold?

Let $X$ be a connected complex manifold. Consider the module $\mathcal{D}_X$. I recall hearing somewhere that one has to be careful with regards to differential operators that are not globally of ...
2
votes
0answers
72 views

Universal cover of ladderly puntured complex plane

The twice punctured complex plane $\mathbb{C}-\{0,1\}$ has as its universal cover the upper half plane via elliptic modular function. I am looking for the constructions of the covering map from the ...
4
votes
0answers
272 views

Is Serre duality related to Pontryagin duality?

I am wondering if there is some relationship between Serre duality and Pontryagin duality for compact complex manifolds. In this case Serre duality reduces to the commutativity of Hodge-star operator ...
2
votes
0answers
35 views

Is singular Cauchy operator bounded in Morrey spaces?

The singular Cauchy operator is defined by $$S_\Gamma :f \to \int_\Gamma \frac{f(\xi)}{\xi-z} d\xi , z\in \Gamma.$$ Is this operator bounded in Morrey spaces and weighted Morrey spaces? i.e. is there ...
6
votes
0answers
86 views

Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?

For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$. If ...
5
votes
0answers
75 views

Good reduction of abelian varieties over valuation rings via coverings

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$, and let $A$ be an abelian variety over $K$. Suppose that there is a smooth proper scheme $\mathcal{X}$ over $\mathcal{O}_K$ whose ...
9
votes
2answers
297 views

Do abelian varieties have Neron models over arbitrary valuation rings?

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$. Let $A$ be an abelian variety over $K$. Does $A$ have a Neron model? If $\mathcal{O}_K$ is a discrete valuation ring, then this is ...
7
votes
0answers
149 views

Explicit descriptions of a flop

I want to know how to describe explicitly the flop of the following flop contraction. Because the construction is so natural and simple, I was wondering such descriptions should already exist in the ...
0
votes
0answers
65 views

What is the unitary $1$-parameter group generated by a vector field on a manifold?

If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...
1
vote
1answer
84 views

Chain rotation of a point

Let $n$ be a positive integer number and $P$ be a point in a plane. Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ points in the plane, we take modulo $m$ for $A_j$ (it is mean $A_{m+i}=A_{i}$ for $i=1, 2, \...
3
votes
1answer
147 views

Formal complex manifold without dd^c

Is there an example of compact complex manifold, which is formal, but does not admit complex structure satisfying $dd^c$-lemma?
4
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0answers
105 views

What is the Jarlskog invariant, conceptually?

Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity: $$J_{ij,k\ell} := \operatorname{...
7
votes
4answers
248 views

Lattices of PU(n,1) with large abelianization

I am interested in properly discontinuous cocompact subgroups of the group $PU(n,1)$ of automorphisms of the complex hyperbolic space $H^n_{\mathbb{C}}$, says for $n=2,3$. Is there such a lattice $G$ ...
5
votes
1answer
196 views

Is a smooth intersection of hypersurfaces equidimensional?

Let $X$ be a smooth projective complex algebraic variety. Let $V_i$, for $i=1,\dots, n$, be a collection of (smooth) connected hypersurfaces such that, for all $I\subseteq [n]$, the intersection $\...
2
votes
1answer
145 views

Why are modular curves non-trivial covers of the $j$-line

This is a very soft question. Let $n\geq 1$ and let $Y(n)$ be the (open) modular curve associated to $\Gamma(n)\subset SL_2(\mathbb{Z})$. Interpreted correctly, $Y(n)\to Y(d)$ is finite etale, ...
7
votes
1answer
240 views

How does one complexify a real $n$-dimensional Riemannian manifold $(M,g)$?

If $V$ is a real vector space, then the complexification of $V$ is formally defined as $V^{\mathbb{C}}=V\otimes_{\mathbb{R}}\mathbb{C}$. Is there an analogous complexification operation for a real $n$-...
5
votes
1answer
191 views

Fulton's deformation to the normal cone vs Verdier's

Let $X$ be a smooth variety over a field $k$, and let $Y$ be a smooth subvariety. In the literature, I've seen two versions of the deformation to the normal cone: Verdier's version: $\tilde{X}_Y^\...
6
votes
2answers
232 views

Harder Narasimhan filtration for the endomorphism bundle

Let $E$ be a vector bundle over a compact Riemann surface $X$, and let $$0=E_0\subsetneq E_1\subsetneq \ldots \subsetneq E_n=E$$ be its Harder-Narasimhan filtration: we have $V_i:=E_i/E_{i-1}$ ...
0
votes
1answer
123 views

Do line bundles with enough sections on surfaces have generic divisors which are irreducible?

Let $L$ be a line bundle on a smooth connected complete complex algebraic surface $X$. Assume that $L$ has enough sections i.e. that $H^0(L,X)$ has dimension $> 1$. A nonzero section $s$ of $L$ ...
3
votes
0answers
114 views

Algebrizing analytic quotients of algebraic spaces

Suppose that $Y$ is an algebraic space over $\mathbb{C}$ which is "nice" (say, separated and of finite type). Let $R\subset Y\times Y$ be a closed algebraic subspace which forms a categorical ...
0
votes
1answer
173 views

Exponential Sequence of Sheaves

Let $(X, \mathcal{O}_X)$ be a complex analytic space in the sense of Grauert, i.e., a $\mathbb{C}$-analytic ringed space which is locally isomorphic to a local model. We may assume that $X$ is a ...
5
votes
0answers
166 views

Asphericity of hypersurface complement in ${\mathbb C}^n$

How does one check that the following space is aspherical? $X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$. One way I can think of is to give ...
2
votes
0answers
98 views

Kähler manifold with a global potential

If $(X^{n},\omega)$ is a complete Kähler manifold with a global potential, i.e. $\omega=i\partial\bar{\partial}f$. There are many articles study the $L^{2}$-cohomology of $X$ under some conditions on $...
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vote
0answers
125 views

The comparison of certain modules arising from the Cauchy-Riemann differential operator

Let $\Gamma=C^{\infty}(\mathbb{R}^2)$ be the space of all smooth complex valued functions on the plane. We define the following Cauchy Riemann differential operator $D$ on $\Gamma$: $$D:\Gamma \...
2
votes
1answer
186 views

Admissible global residues on smooth variety with normal crossings divisor

Let $X$ be a smooth projective complex variety, and $D=\cup_{j=1}^m D_j$ a simple normal crossings divisor on $X$. Then we have an exact sequence $$0\to \Omega_X^1\to \Omega_X^1(\log D)\to \oplus_{j=1}...
2
votes
1answer
126 views

Movable divisor with base locus on a hyperkahler variety

I'm looking for an example of the following: $X$ is a hyperkahler fourfold (deformation equivalent to $K3^{[2]}$); $D$ is a movable divisor on $X$ with $D^4=0$; and the base locus of $D$ is a ...
5
votes
0answers
433 views

a question on Hodge and Atiyah's paper “integrals of the second kind on an algebraic variety”

I have a question on Hodge and Atiyah's paper "Integrals of the second kind on an algebraic variety". It is about the exact sequence below formula (14) and above formula (15) on page 71: $$H_{2n-q}(S)...
1
vote
0answers
62 views

Numerical equivalent positive non-degenerate divisor induced projective embedding involves Veronese map?

This is a part of material I do not understand from "Analytic Theory of Abelian Varieties" by Swinnerton-Dyer. Let $A=\mathbb{C}^n/\Lambda$ be an abelian variety with positive-definite Hermitian form ...
5
votes
0answers
135 views

Dimension of linear complex-symplectic reduction

Let $(V,\omega)$ be a finite-dimensional complex-symplectic vector space and $G$ be a complex reductive group acting linearly on $V$ by preserving $\omega$. Then, there is a moment map $$\mu:V\to\...
12
votes
2answers
738 views

Algebraic vs analytic normality

Let $X$ be a complex algebraic variety. We can ask if $X$ is normal as an algebraic variety, but also, if its analytification is normal as a complex analytic space. Is there a relationship between the ...
0
votes
0answers
73 views

Kelly's theorem for quadratic polynomials

Let $f_1, \ldots, f_m$ be homogeneous irreducible quadratic polynomials in $\mathbb{C}[x_1, \ldots, x_n]$. Assume that these polynomials are pairwise coprime. Denote $P:= f_1 \cdot f_2 \ldots \...
4
votes
0answers
80 views

Continuous function on a complex space that is holomorphic on the complement of a closed subspace

Let $X$ be a complex analytic space and $Y\subseteq X$ a closed complex subspace. Suppose that $f:X\to\mathbb{C}$ is a continuous function that is holomorphic on $X\setminus Y$. Is $f$ holomorphic on $...
-3
votes
1answer
138 views

Holomorphic line bundles with torsion Chern class [closed]

Suppose you have a holomorphic line bundle $L$ such that $L^{n}$ is a trivial holomorphic line bundle and the base complex manifold $M$ has no torsion cohomology classes in second degree (i.e. $H^{2}...
6
votes
1answer
218 views

Rationality of the moduli space of genus g curves

I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions ...
2
votes
0answers
50 views

Status of global spherical shell conjecture for minimal complex surfaces?

A class VII surface is a compact complex surface $M$ such that $b_1(M)=1$ and $kd(M)=-\infty$. Class VII surfaces with vanishing second Betti number have been classified by Bogomolov (and are either ...
0
votes
1answer
179 views

Kähler form on complex projective algebraic variety [closed]

I am not very familiar with the notion of projective algebraic varieties, I work mostly from an algebraic topology/differential geometry point of view, but I am trying to find a prove for the ...
2
votes
0answers
58 views

Does there exist a leaf of this holomorphic foliation with non trivial holonomy?

Let's $\mathcal{F}$ be the holomorphic foliation of $\mathbb{C}^2$ tangent to the kernel of $\alpha=(sin x) dx -(cos x)dy$. Are all leaves of $\mathcal{F}$ simply connected? If the answer is no, ...
5
votes
2answers
372 views

Embedding of a complex submanifold in projective space

Suppose you have a projective manifold $M$, a very ample bundle $\scr L$ and a transverse holomorphic section $s \in H^0(\scr L)$. Then the zero set of $s$ is a complex submanifold $S_M$. Can we ...
2
votes
1answer
51 views

Set of sections whose zeroes avoid a given divisor is (Zariski) dense?

Let $X$ be a smooth complex projective variety of dimension $n$, and let $\mathcal{F}$ be a globally generated rank $n$ vector bundle on $X$. Let $D$ be a smooth divisor on $X$. Is it true that ...
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vote
0answers
22 views

What separates a cyclic polytope from a projective polytope?

I am having trouble understanding the difference between a cyclic polytope and a convex projective polytope as positive geometries. The link https://arxiv.org/pdf/1703.04541.pdf is the source of ...
6
votes
1answer
278 views

Are there non-projective, but algebraic, hyperkahler varieties?

Let $k$ be an algebraically closed field of characteristic zero. I am not sure what the right definition of a hyperkahler variety over $k$ is, but I think the following might be close enough. ...
6
votes
1answer
255 views

The Ricci Form and the First Chern Class

Let $(M, \omega)$ denote a compact Kähler manifold. Since $d\omega =0$, $\omega$ represents a cohomology class in $H^2(M, \mathbb{R})$. Let $\rho$ denote the Ricci form of $M$, in local coordinates, ...
4
votes
0answers
112 views

Structure of the Kähler cone

In Calabi's Extremal kahler metrics paper, MR0645743, on page 262, the author mentioned that "It is conjectured that the structure of Kähler cone is determined by a finite number of real analytic ...
2
votes
0answers
97 views

Degeneration of a metric

I want to understand how the metric degenerate on a family of projective varieties (mainly for abelian varieties.). Let $X$ be a smooth projective variety over $\mathbf{C}$. Let $B$ be a smooth ...
2
votes
2answers
177 views

When is the space of holomorphic sections of the tensor product of two line bundles given by the span of the tensor product of the basis?

Let $S$ be a compact complex manifold and $L_1, L_2 \longrightarrow S$ be two holomorphic line bundles. Under what conditions (hopefully something that is easy to check) on $L_1$ and $L_2$ is the ...
1
vote
0answers
118 views

(Real) holomorphic vector fields on compact Kähler manifolds

I am trying to prove Proposition 2.1.1 of Gauduchon's note on Kähler extremal metrics (page 67). In order to show that, for compact Kähler manifolds, the complex Lie algebra of real holomorphic vector ...
3
votes
2answers
413 views

Criteria for a coherent sheaf pushing forward from the universal cover

I would like to prove (or find a counterexample to) the following statement: Let $X$ be a complex analytic scheme and let $\pi: Y \to X$ be its universal cover. Let $F$ be a coherent sheaf on $X$ and ...
4
votes
1answer
126 views

Can an algebraic variety over a field $k$ be the union of proper closed subsets $(S_i)_{i\in I}$ with $I < k$

Let $k$ be an algebraically closed field (of characteristic zero, if it helps). Let $X$ be an algebraic variety over $k$. Let $I$ be an index set such that the cardinality of $I$ is smaller than the ...
4
votes
1answer
83 views

Examples of surfaces with negative Kahler curvature operator

Compact ball quotients are examples of compact Kahler surfaces with negative curvature operator. Are there any other examples ? What about nonpositive (other than the product of two Riemann surfaces ...