All Questions
Tagged with dg.differential-geometry cv.complex-variables
68 questions with no upvoted or accepted answers
8
votes
0
answers
315
views
Singularities of a morphism from a smooth projective variety to an abelian variety
Let $f: X\to A$ be a (flat) morphism from a smooth complex projective variety $X$ to an abelian variety $A$. Consider the following natural diagram:
$$T^*X\overset{df}{\longleftarrow}X\times H^0(A, \...
- 1,081
8
votes
0
answers
288
views
Are the Chern numbers of a hyperbolic-type compact complex manifold bounded in terms of the Euler number?
Let $X$ be an $n$-dimensional compact Kahler manifold with negative first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i
\geq 0$ with $\sum ik_i = n$, bounded in terms ...
7
votes
0
answers
168
views
The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space
Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
- 1,170
7
votes
0
answers
769
views
How much differs the category of real-analytic manifolds from $C^\infty$ ones?
I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (...
- 395
7
votes
0
answers
202
views
Biholomorphic neighborhoods of the boundary of Stein domains
Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
- 773
6
votes
0
answers
144
views
What does it mean for the torsion to blow up?
Consider the following theorem which is the main result of the Hermitian Curvature Flow paper by Jeffrey Streets and Gang Tian:
Theorem 1.1. Let $(M^{2n}, g_0, J)$ be a complex manifold with Hermitian ...
- 651
6
votes
0
answers
228
views
All complex surfaces embed into a common complex manifold
Is there a closed complex manifold into which every closed complex surface embeds?
user164740
6
votes
0
answers
163
views
Reference request: normal form of k-differentials and flat surfaces at a puncture
Let $f(z)$ be a holomorphic function defined on a punctured neighborhood of $z=0$ with non-essential singularity of degree $d$ at $0$ (namely, $f(z)=z^dh(z)$, where $h(z)$ is a holomorphic function ...
- 1,804
6
votes
0
answers
147
views
What is the meaning of complex values/multiplicities in dimension spectrum?
If we have a manifold $M$ (say smooth, closed) it can be equipped with the Laplace operator $\Delta$. One can consider the function $\textrm{trace}(\Delta^{-s})$ where $s$ is complex parameter and $\...
- 9,330
6
votes
0
answers
286
views
Is the space of holomorphic maps a manifold
To be more specific:
Let $Q\subset\mathbb{C}$ be a Lipschitz bounded domain, and $V$ is a compact complex manifold without boundary. Consider the set of holomorphic maps $f:Q\rightarrow V$, and $f\in ...
- 151
6
votes
0
answers
457
views
Jet differentials and hyperbolicity: possible mistake in the literature?
I was reading this note by Jingzhou Sun http://arxiv.org/abs/1109.1329
about Demailly's approach to hyperbolicity using jet differentials. The author seems to claim that there is a mistake in one of ...
- 61
6
votes
0
answers
1k
views
Computing the Chern class for a flat line bundle using the holonomy group?
Let $X$ be a Riemann surface of genus $g \geq 2$. I would like to consider flat line bundles on $X$. Flat line bundles can be identified with representations $\pi_1(X) \rightarrow U(1)$. From Chern-...
- 121
5
votes
0
answers
225
views
Energy bounds (or the lack thereof) for a functional between almost Hermitian manifolds
Suppose that $(M,g,J_M)$ and $(N,h,J_N)$ are two almost Hermitian manifolds. For a differentiable function $f:M\to N$ define its pseudoholomorphic energy to be
$E_+(f)=\frac{1}{4}\int_M |Df+J_N Df J_M|...
- 646
5
votes
0
answers
188
views
Proof of Tian's constant
Basically Tian's invariant relies fundamentally on a lemma of Homander which says that $e^{- \phi}$ is integrable for $\phi$ plurisubharmonic on a ball of radius $1$ under some extra assumption. I am ...
- 51
5
votes
0
answers
245
views
Dimension of highest discriminants of a morphism
Let $f: X\to Y$ be a flat morphism between smooth complex affine varieties. Let $Z$ be the closed set of most singular points of $f$ (in the sense: $p$ is a most singular point of $f$ if the tangent ...
- 1,081
5
votes
0
answers
104
views
On the embedding of manifolds into infinite-dimensional spaces
Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
- 5,529
5
votes
1
answer
752
views
Gaussian integral over a ball
How to compute the following integral?
$$\int_{\|x\|^2\leq R} \exp(-x^\ast G x+2\mathcal{Re}(x^\ast a)) \,dx,$$
where $x$ is an $M \times 1$ vector ($M\gg 1$), $G$ is a positive definite matrix, and $...
- 129
4
votes
0
answers
78
views
Higher-dimensional analogue of the relation between stable Higgs bundles and constant curvature metrics
In Hitchin's famous paper[1] on the self-dual Yang-Mills equations, he discussed the relation between the stable Higgs bundles and the Teichmüller space for a compact Riemann surface. Namely, through ...
- 353
4
votes
1
answer
848
views
Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory)
Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field ...
- 356
4
votes
0
answers
294
views
Holomorphic covers pulling back the volume form to any integer multiple
Let $M$ be a closed connected complex manifold with $\mathrm{dim}\:M=n$. Can there exist holomorphic covering maps $\phi_k:M\to M$ for all integers $k\geq 1$ such that $\phi_k^*:H^n(M, \mathbb{Z})\to ...
user145520
4
votes
0
answers
157
views
Analytic maps $\varphi: \mathbb C^n\to \mathbb C^n$ with degenerate differentials
Let $B^n\subset \mathbb C^n$ be a unit ball with center $p$ . Let $\varphi: B^n\to \mathbb C^n$ be a complex analytic map such that $d\varphi$ has rank at most $n-1$ at $p$. I would like to know if ...
- 14.3k
3
votes
0
answers
85
views
Can a punctured ball $(B\setminus\{0\})\subset\mathbb{C}^n$ be foliated by complete leaves?
Recently Antonio Alarcón proved that in the case of the unit ball $B\subset\mathbb{C}^n$ for $n\geq 2$ every smooth closed complex submanifold of dimension $q\leq n$, $V\subset\mathbb{C}^n$ defines a ...
- 166
3
votes
0
answers
185
views
Differentiable functions on $\mathbb{R}^n$ whose derivative is everywhere a scalar multiple of a special orthogonal matrix
The Cauchy–Riemann equations say that if $u : \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic then, regarded as a linear transformation of $\mathbb{R}^2$, its derivative is either zero or, up to a ...
- 11.2k
3
votes
0
answers
191
views
Holomorphic sectional curvature and Kobayashi hyperbolicity
Let $(M,g)$ be a compact Hermitian manifold. Let $\text{HSC}(g)$ denote the holomorphic sectional curvature of $g$. The implication $$\text{HSC}(g) < 0 \implies M \ \text{is Kobayashi hyperbolic}$$ ...
- 651
3
votes
0
answers
55
views
Infinitely many deformation equivalent Hodge diamonds II
Let $S$ be a connected smooth complex-analytic space. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
...
user164740
3
votes
0
answers
259
views
Stokes's Theorem with singularities on projective line
Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
- 789
3
votes
0
answers
148
views
Analytic Aspects of Rational Maps
I would like some help finding references for a analytic treatment of rational maps between compact complex manifolds (that is holomorphic maps defined away from a codimension at least 2 subvariety). ...
- 121
3
votes
0
answers
135
views
Asymptotic Expansion of Seiberg-Witten Differential?
Nekrasov & Okounkov proved (https://arxiv.org/pdf/hep-th/0306238.pdf) that the Seiberg-Witten prepotential can be given by
\begin{equation}
\mathcal{F}(\mathbf{a},\Lambda) = \lim_{\hbar\rightarrow ...
- 425
3
votes
0
answers
98
views
Euler characteristic of an exhaustion of compacts of a surface
Let $X$ be an open (connected) Riemann surface of finite Euler characteristic. And $K_1 \subset \cdots K_n \subset$ be an sequence of closures of bounded open subsets with smooth boundary of $X.$
...
- 750
3
votes
0
answers
84
views
Discrete set of critical points of a holomorphic map
I have originally posted this question on math.SE, but it received no attention, so I repost it here.
Let $U$ be an open domain in $\mathbb{C}^{n}$. Let $m\ge n$ and let $F:U\to C^{m}$ be a ...
- 5,529
3
votes
0
answers
165
views
Is a non vanishing holomorphic vector field necessarily a geodesible vector field?
Motivated by the "The obvious Fact" part of this answer,, we ask the following question:
First we recall a definition, which is used in the above link:
Definition: A non vanishing vector ...
- 356
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
2
votes
0
answers
108
views
The dual of the Lefschetz operator under a perturbation
Let $(X, \omega)$ be a compact Kähler, or more generally, Hermitian manifold. Let $L_{\omega} : \Omega^k(X) \to \Omega^{k+2}(X)$ denote the Lefschetz operator given by $$L_{\omega}(\alpha) : = \omega \...
- 137
2
votes
0
answers
231
views
Does every non-compact hyperbolic manifold admit compact complex submanifolds?
Let $(X,\omega)$ be a complete Kähler manifold with a metric of negative holomorphic sectional curvature. Does $X$ admit a proper, positive-dimensional, compact complex submanifold?
In general, it is ...
- 1,379
2
votes
0
answers
358
views
Triangulating Riemann surfaces by using non-constant meromorphic functions
Let $X$ be a connected Riemann surface, i.e. $X$ is a one dimensional connected complex manifold (Hausdorff and second-countable as a topological space). The following is a classical result:
Theorem (...
- 395
2
votes
0
answers
158
views
Pull back of a Bounded form
Let $(X, \omega) $ be a complex manifold and let $\alpha $ be a $p$-form $\omega$-bounded on $X$.
Let $f:Y\to X$ be a holomorphic function.
Is $f^*\alpha$ $f^*(\omega)$-bounded on $Y$ ?
- 21
2
votes
0
answers
86
views
Estimates for tensors using local coordinates
Suppose that we have a Kähler manifold $(M, \omega_0)$ and another $(1,1)$ form $\eta$ on $M$. Let $\varphi$ be a smooth function such that $\omega_{\varphi} = \omega_0 + i \partial \bar \partial \...
- 121
2
votes
0
answers
117
views
Equivariant resolution of singularities with equivariant centres
From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (...
- 3,383
2
votes
0
answers
119
views
Covariant derivative of the Monge-Ampere equation on Kähler manifolds
I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
- 21
2
votes
0
answers
179
views
Questions about transformation or integral transformation
I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
- 3,094
2
votes
0
answers
221
views
why is this result about Gaussian analytic functions equivalent to the Crofton formula
I am reading Zeros of Gaussian Analytic Functions by Mikhail Sodin and he gives an much-too-easy proof of density of zeros of a Gaussian Analytic function.
Definition A Gaussian analytic function $...
- 22.8k
2
votes
0
answers
122
views
A parametrix for the $\bar\partial$ operator adapted to a holomorphic foliation
Let $X$ be a compact complex manifold with (regular) holomorphic foliation given by a holomorphic subbundle $\mathcal F$ of the tangent bundle. The foliation induces a filtration on differential forms....
- 51
2
votes
0
answers
238
views
Non-realizable CR structures?
Hill, Penrose, and Sparling have an example of a non-realizable CR structure, a 5-manifold $M^5$ that comes equipped with a "twisted version" of the Lewy operator for the quadric $Q^2$,
$v = \frac{1}{...
- 21
2
votes
0
answers
331
views
Is there asymptotic expansion of heat kernel of complex laplacian?
On real Riemannian manifold , the heat kernel of the laplacian have an asymptotic expansion . But on complex manifold , i haven't seen a result like this , i.e. the heat kernel of the Kodaira ...
- 399
2
votes
0
answers
390
views
Boundary behavior of Kähler cone with curvature restriction
Let $(M,\omega)$ be a compact Kähler manifold. The boundary behavior of Kähler cone is very interesting; however,it's hard to understand.
A fundamental result is due to Demailly and Paun: they ...
- 247
2
votes
0
answers
153
views
Holomorphic automorphism of strictly psudo-convex domain smooth on boundary
I am wondering if anything is known about this. I couldn't find anything in the literature.
In '74 C. Fefferman published a solution to the following problem.
Let $\sigma:D\rightarrow D$ be an ...
- 496
1
vote
0
answers
39
views
Currents with logarithmic poles compared with those with no poles
I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by
$$
'\...
- 161
1
vote
0
answers
156
views
Top cohomology of the canonical class of a compact non-Kähler manifold
Let $X$ be a complex compact manifold of complex dimension $n$. Let $K_X$ denote its canonical class. Is it true that the cohomology group
$$H^n(X,K_X)$$
is one dimensional?
Remark. If $X$ is Kähler ...
- 21.8k
1
vote
0
answers
88
views
Submersion function from a product space
Let $\Phi(x,y) \colon U_N \times U_M \to \mathbb{C}^n$ be a submersion, where $U_N \subset \mathbb{C}^N$ and $U_M \subset \mathbb{C}^M$.
Under which condition on $\Phi$ can I find some $s \in \...
1
vote
0
answers
70
views
Classification of principal monodromy elements
Let $(X,0)$ be a germ of normal analytic space with an isolated singularity at $0$, and let $Y:=X\backslash\{0\}$. Suppose $Y$ has a complex-hyperbolic metric which is complete at $0$. Burns-Mazzeo ...
- 1,121