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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Separating orthogonal vectors in $\mathbb{C}^2$

Is it possible to partition $\mathbb{C}^2$ into two sets $S$ and $S'$ such that, given any two nonzero orthogonal vectors $\mathbf{v}$ and $\mathbf{w}$ of $\mathbb{C}^2$, one of them lies in $S$ and ...
1 vote
0 answers
65 views

Exponentiate additive transition functions for $\mathbb{A}^1$-bundles

Consider an smooth complex elliptic curve $E$ glued from two affine curves ($p\in\mathbb{C}\setminus0$) $$C_{(x,y)}: y^2=x^3+px\\C_{(s,t)}: t^2=ps^3+s$$ via the coordinate change $s=1/x,t=y/x^2$. It ...
12 votes
2 answers
2k views

Negative holomorphic sectional curvature

Let X be a complex hermitian manifold with hermitian form $\omega$. How can you prove that if $\omega$ has negative holomorphic sectional curvature, then its scalar curvature is negative, too?
3 votes
1 answer
213 views

Hyperelliptic integrals

I am learning about hyperelliptic curves and hyperelliptic integrals. I encountered some problems when reading the book by Gesztesy and Holden (F. Gesztesy, H. Holden, Soliton Equations and Their ...
1 vote
0 answers
104 views

Monomorphism/Isomorphism of $C_4$-tangent cones for complex varieties

Suppose that $(M,\mathcal{O}_M)$ is a reduced complex analytic space (or complex algebraic variety if you prefer). The tangent linear fiber space $TM$ associated to $M$ is defined as the analytic ...
0 votes
0 answers
269 views

Space which is diffeomorphic to CP^2 # -CP^2

The manifold $\mathbb{CP}^2 \# -\mathbb{CP}^2$, the non-trivial $\mathbb{S}^2$ bundle over $\mathbb{S}^2$, is known to be diffeomorphic to the space that we will now describe. Represent $\mathbb{S}^3 ...
2 votes
0 answers
51 views

Lefschetz operator on bundle-valued forms

For a holomorphic vector bundle $V \rightarrow X$ endowed with a Hermitian structure, one may define the corresponding Dolbeault-like operators $\bar{\partial}_V: \Omega^{p,q}(V) \rightarrow \Omega^{p,...
12 votes
2 answers
416 views

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 ...
1 vote
0 answers
88 views

Effective Torelli theorem for K3 surfaces

The proof of the Torelli theorem I've seen goes something like: Put $M$ the moduli space of marked $K3$ surfaces, and $D$ the period domain s.t there is a natural map $$P: M \to D$$. Up to lies, here ...
2 votes
0 answers
80 views

Does Kobayashi isometry map preserve complex geodesics?

Let $\gamma_1, \gamma_2$ are real geodesics in a domain $D$ and these two real geodesics are lying in the same complex geodesics, the question is, are $f\left(\gamma_1\right)$ and $f\left(\gamma_2\...
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1 answer
102 views

An optimization problem with variables on the exponential of a complex number

$$\min_t \quad\operatorname{Re} \sum\limits_{i = 1}^N {\left( {{e^{ - j2\pi {f_i}t}}{r_i}} \right)}, $$where $\operatorname{Re}$ refers to get the real part of a complex number, $\{f_i\}$ is an ...
3 votes
0 answers
184 views

An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$

It is known that the only spheres that admit an almost complex structures are $\Bbb S^2$ and $\Bbb S^6$ (Borel and Serre, 1953). In particular, $\Bbb S^4$ cannot be given an almost complex structure (...
2 votes
1 answer
197 views

Homotopy classes of homeomorphism vs. Homotopy classes of a biholomorphism

This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand: $$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\...
2 votes
1 answer
174 views

Parabolic Schwarz lemma

Trying to follow the computation in Song and Tian - The Kähler–Ricci flow on surfaces of positive Kodaira dimension, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they ...
6 votes
1 answer
282 views

Is this $\mathbb C$-fibration over compact Riemann surface trivial?

I have a question about a complex manifold $M$ and a holomorphic submersion $p : M \to S$ to a compact complex curve satisfying the following conditions: $p^{-1}(x)$ is biholomorphic to $\mathbb{C}$ ...
-1 votes
1 answer
116 views

A complex complex integral operation [closed]

This question mainly asks about the integral of complex numbers. This question originates from the optical properties of the axicon angle.$\lambda, f,R,ρ_{0},k$ are a constant. $$m(r,\varphi)=\frac{1}{...
1 vote
0 answers
117 views

Is every curve on a projective three-fold a homology-theoretic complete intersection of sorts?

Let $C$ be a curve on a smooth projective three-fold $M$ equipped with the restriction of the Fubini-Study metric $\omega$. I'd like to know if there exists a surface $S$ such that for every closed $(...
1 vote
2 answers
253 views

A simple question about a statement of Kähler Manifold and Moishezon Manifold

I saw a statement in a question Non-compact Kähler manifolds which admit a positive line bundle, which says that any compact complex manifold that admits a positive line bundle must be a Kähler ...
2 votes
1 answer
221 views

The indecomposable bundle on an elliptic curve

M. Atiyah (Theorem 5, p. 432 of "Vector bundles on an elliptic curve") defines an indecomposable bundle of degree $0$ that has a global section for each rank $r$ (I'm thinking on an elliptic ...
0 votes
1 answer
165 views

The intersection number $C\cdot D=\deg(D_{/C})$

Let $S$ be an algebraic complex surface, and $D=[(U_\alpha,f_{\alpha})]$ is a Cartier divisor over $S$, and let $\cal{O}_S(D)$ be the sheaf associated to $D$. And let $C$ be a complex compact curve in ...
5 votes
5 answers
888 views

Two arcs in the complement of a disc must intersect?

Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$. Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...
2 votes
0 answers
142 views

Do the nearby cycle and Beilinson's vanishing cycle functors commute?

Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...
18 votes
2 answers
4k views

What is a Futaki invariant, what is the intuition behind it, and why is it important?

As the question title suggests, what is a Futaki invariant, what is the intuition behind it, and why is it important?
1 vote
0 answers
146 views

Reference for application of local cohomology to complex manifolds with points removed

Reference request - I am looking at Dolbeault cohomology on compact complex manifolds (not Riemann surfaces) with points removed. I have been told that the key to doing this is to look at Local ...
1 vote
1 answer
288 views

Three-dimensional analogues of Hirzebruch surfaces

There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$....
0 votes
1 answer
78 views

Holomorphic automorphisms of analytifications of proper varieties

I have two questions of a similar nature: Is it true that any holomorphic automorphism of an analytification of a proper variety over $\mathbb{C}$ is algebraic? If the first is not the case, is ...
6 votes
0 answers
196 views

Obstruction to the existence of global resolution of coherent sheaf

It is well known that any coherent sheaf on a complex manifold (or more generally on some complex spaces) admits locally a resolution with locally free sheaves. It is also well known that for non-...
5 votes
1 answer
227 views

Rozansky-Witten invariants of hyperkahler manifolds and independence of complex structure

Recently I have been learning about Rozansky-Witten invariants, mainly through Hitchin-Sawon's paper "curvature and characteristic numbers of hyperkahler manifolds" and through Justin Sawon'...
1 vote
0 answers
44 views

Determining a class in Dolbeault cohomology that defines a principal $\mathbb{C}$-bundle over a compact torus

This is a cross-post from MSE Consider a standard complex torus $\mathbf{T}=\mathbb{C}/(\mathbb{Z}\oplus i\mathbb{Z}).$ It could be obtained in another way. $\mathbf{T}$ is a quotient $(\mathbb{C}^\...
3 votes
1 answer
148 views

$n$-th root of meromorphic functions of several complex variables

Let $f:\Omega\rightarrow\mathbb{C}$ be a "nice" meromorphic function of several complex variables on some domain $\Omega$. I wonder if the following claim is true. Claim. $f$ admits a global ...
5 votes
0 answers
195 views

$C^1$ manifold with complex structure

Let $M$ be a manifold. A complex structure on $M$ is an endomorphism $J \in \text{End}(TM)$ such that $J^2 = -\text{id}$ together with the vanishing of the Nijenhuis tensor. If $J$ is real-analytic, ...
3 votes
1 answer
366 views

On Simpson's motivicity conjecture

Simpson's motivicity conjecture says that for any rigid, flat irreducible connection $(V,\nabla)$ on a smooth complex variety $M$, there exists a proper smooth morphism $f:X \to M$ s.t. $(V,\nabla)$ ...
4 votes
0 answers
113 views

Non-Kähler complex structure on $S^2 \times T^4$

Consider $M = S^2 \times T^4$. Then we can construct a non-Kähler complex structure as follows. Let $L$ be a line bundle over $\mathbb{P}^1$ such that there are two sections $s_1, s_2 \in H^0 (L)$ ...
1 vote
1 answer
298 views

Motivation for the definition of $L^p$ norm for quadratic and Beltrami differentials

According to Riemann surfaces, dynamics and geometry by C. McMullen (Course notes), the definition for a quadratic differential $\phi$ on a Riemann surface $X$ is given by $$ \|\phi\|_p = \left(\...
2 votes
1 answer
226 views

Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample

Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
4 votes
1 answer
218 views

Homogeneous polynomials cutting out complex abelian varieties

This is an update to a previous question of mine. The more clarified questions, results and definitions make me feel like this warrants a separate post instead of a large edit of the original one. ...
4 votes
0 answers
240 views

Has anyone studied the derived category of Higgs sheaves?

Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A Higgs bundle on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-...
4 votes
0 answers
185 views

Elementary proof Hilbert-Mumford stability criterion for $\operatorname{GL}_n(\mathbb{C})$

In An elementary proof of the Hilbert-Mumford criterion, B. Sury gives an elementary proof of the Hilbert-Mumford semi-stability criterion for $G = \operatorname{GL}_n(\mathbb{C})$ (and $G = \...
1 vote
0 answers
60 views

Embedding toric varieties in other toric varieties as a real algebraic hypersurface

In the question On a Hirzebruch surface, I've seen that the $n$-th Hirzebruch surface is isomorphic to a surface of bidegree $(n,1)$ in $\mathbb{P}^1\times \mathbb{P}^2$. I am trying to answer the ...
6 votes
0 answers
165 views

How exactly does the Kreck-Stolz description of elliptic homology match the one by Totaro?

In Kreck, Matthias; Stolz, Stephan, $\mathbf H\mathbf P^2$-bundles and elliptic homology, Acta Math. 171, No. 2, 231-261 (1993). ZBL0851.55007. the $n$th elliptic homology group of a space $X$ is ...
6 votes
0 answers
140 views

Fourier transform and Hodge-$*$ operator

Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says $$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$ ...
7 votes
1 answer
114 views

Picture of the isotopy class of a degree $d$ smooth complex curve

All smooth complex curves of degree $d$ in $\mathbb{C}P^2$ are isotopic. Let $C$ be such a curve. I often picture $\mathbb{C}P^2$ as a 2-dimensional disk bundle over $S^2$ (of Euler class 1) which ...
4 votes
1 answer
685 views

Understanding Remmert-Stein extension theorem

I'm trying to study the Remmert-Stein theorem in analytic geometry. This is an important result which can be used to prove the Proper Mapping theorem. A preliminary result is stated in various books (...
1 vote
1 answer
292 views

Self-intersection of the diagonal on a surface

Let $X$ be a smooth projective curve over the complex numbers, and take $\Delta$ the diagonal divisor on $X\times X$. Using the adjunction formula, one computes $\Delta\cdot\Delta =2-2g$ for $g$ the ...
3 votes
1 answer
291 views

Example of a morphism of complex spaces or "nice schemes" that is not cohomologically flat in any point

Suppose that $f:X\rightarrow S$ is a proper, separated morphism of complex spaces (with $S$ reduced) and $\mathcal{F}$ a is $f$-flat coherent sheaf on $X$. From (well-)known results it is known that ...
0 votes
1 answer
128 views

Comparative between Kobayashi hyperbolicity and hyperbolicity in the sense of Koszul

In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to ...
7 votes
1 answer
319 views

A non-Kähler compact complex manifold with negative sectional curvature

I am looking for an example of a compact complex manifold with negative sectional (not holomorphic) curvature which is not Kählerian. Can such an example exist?
2 votes
0 answers
150 views

Product subvariety of a simple abelian variety

Terminology: A subvariety $V$ of an abelian variety $A/\mathbb{C}$ is called a product if there are integral closed subvarieties $U,W\subset A$ such that $\dim U,\dim W>0$ and the sum morphism $U\...
12 votes
3 answers
1k views

Foliations by holomorphic curves on complex surfaces

On a complex surface, does there exist a non-singular foliation by holomorphic curves that is NOT a holomorphic foliation, i.e. a transversally holomorphic foliation? The surface should be compact ...
0 votes
1 answer
258 views

Self-intersection of zero section of line bundle over elliptic base curve

Let $C$ be an elliptic curve over $k=\mathbb{C}$ and $\mathcal{L}$ a line bundle of degree $d$. It induces naturally a $\mathbb{A}^1$-fibration $L \to C$ where $L=\underline{\operatorname{Spec}} (\...

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