3
votes
0answers
85 views

Moment map in the singular case

The moment map is defined on the symplectic manifold $(M,\omega)$, or particularly, $(M,\omega)$ is Kahler. While $\omega$ is smooth or differential enough, the definition is obvious to understand, in ...
2
votes
1answer
189 views

Cotangent bundle of coadjoint orbit is stein manifold?

Let me first define stein manifolds and coadjoint orbits. A complex manifold $X$ of complex dimension $n$ is called a Stein manifold if the following conditions hold: $X$ is holomorphically convex, ...
0
votes
1answer
134 views

Kahler structure on holomorphic principal bundles

Let $G$ be a compact complex Lie group and $M$ be a compact Kähler manifold. Does there exist any example of a holomorphic principal $G$-bundle over $M$ admitting Kähler structures?
2
votes
1answer
125 views

Atiyah classes of holomorphic vector bundles with trivial Chern classes

Let $X$ be compact Kahler and $E \to X$ a holomorphic vector bundle. Then $E$ has an Atiyah class, $At(E)$, valued in the sheaf cohomology $H^1(\Omega_X \otimes \operatorname{End} E)$. Suppose the ...
9
votes
0answers
194 views

What is the holomorphic sectional curvature?

Let $X$ be an $n$-dimensional complex manifold, let $\omega$ be a Kahler metric on $X$ and let $R$ be the $(4,0)$ curvature tensor of $\omega$. We can simplify the tensor $R$ in different ways, two of ...
1
vote
2answers
222 views

The Schottky group and the fundamental group of a compact Riemann surface

I am quoting the following description from a paper, "...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...
1
vote
1answer
137 views

A question about the first eigenvalue for two Kahler metrics

While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies ...
1
vote
1answer
231 views

A question about complex polarization

Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive dim$P\cap\bar P ...
2
votes
1answer
161 views

almost holomorphic line bundles

Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...
3
votes
1answer
176 views

Newlander-Nirenberg in dimension 2

What is the easiest (and what is the most elementary) way of proving Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce it to existence of non-trivial harmonic functions ...
1
vote
0answers
66 views

Boundary of fibers of submersions

Let $M$ be a smooth manifold with boundary (say of dimension $m$) and let $N$ be a smooth manifold with no boundary (say of dimension $n$ with $m\geq n$). We have the following classical result: ...
1
vote
1answer
105 views

Bakry-Emery Laplacian and Hodge Decomposition

I have a question about the Hodge Decomposition theorem. Let $(X,\omega)$ be a compact Fano Kaeler manifold, we know the Hodge theorem works very well with respect to the $\bar\partial-$Laplacian ...
8
votes
1answer
238 views

counterexample to the Chern number inequality on Fano manifold

We know that if an $n$-dimensional Fano manifold admits a Kahler-Einstein metric, it satisfies the following Chern number inequality $$nc_1^n\leq 2(n+1)c_2c_1^{n-2}.$$ My question is whether there ...
3
votes
0answers
234 views

holomorphic embeddings of the sphere into the quintic in degree 2

Is there an explicit way of classifying (with regard to their compatibiliy with $\Omega_+$ or $\Omega_-,$ see below) the various families of equivariant holomorphic embeddings from $\mathbb{CP}^1$ to ...
0
votes
1answer
76 views

geodesic convexity of regular sets for metrics with cone singularities

Consider $\mathbb{C}^2 = (r,\theta,\rho,\varphi)$ with the following cone metric in polar coordinates \begin{equation*} ds^2= (dr^2 + \alpha^2 r^2 d\theta^2) + (d\rho^2 + \beta^2\rho^2 d\varphi^2) ...
1
vote
1answer
190 views

Kahler-Einstein metrics on Toric manifolds are Torus-invariant?

let $(M,\omega)$ be a Kahler-Einstein toric manifold of complex dimension $m$. By toric manifold i mean a manifold that has an open dense subset $X$ biholomorphic to an algebraic torus ...
3
votes
3answers
290 views

Finiteness of De Rham cohomology of smooth quasi-projective varieties

Let $U$ be a smooth quasi-projective variety over $\mathbf{C}$. Let $U^{\infty}$ be $U$ but thought of as a smooth manifold. Q1: Is there a simple proof (so it should avoid Hironaka's ...
1
vote
1answer
132 views

Kähler Identities: from the untwisted to the twisted case

For any Kähler manifold $M$, we have the well known Kähler identities \begin{align*} [L,\partial^*] = i\overline{\partial}, & & [L,\overline{\partial}^*]=-i\partial, & & ...
3
votes
2answers
192 views

Integrable compatible complex structures

In reading Gauduchon's notes (cannot link them, anyway it is standard material) I ran into the following construction. Let $(M, \omega_0)$ be a compact symplectic manifold which is fixed. An almost ...
0
votes
0answers
105 views

When the leaves of a distribution are compact

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
3
votes
1answer
198 views

Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...
3
votes
0answers
134 views

Hilbert's syzygy theorem in the analytic setting

If $X$ is a projective variety then Hilbert's syzygy theorem says that any coherent sheaf of $\mathcal O_X$ modules has a finite global resolution by locally free modules. By GAGA, I believe this ...
3
votes
0answers
143 views

Vector bundle connection over complex manifold vs. over underlying real manifold

Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$ of rank $r$. Denote by $(X^{\mathbb{R}},g^{\mathbb{R}})$ the underlying Riemannian manifold of $(X,g)$. ...
6
votes
2answers
470 views

Trying to Understand Lefschetz Pencils

All: I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) below better, though I would appreciate insights on condition i), and in general. Please forgive if the presentation ...
2
votes
2answers
178 views

compute the automorphism of Iwasawa manifold

An Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup. We can also refer to Griffiths and Harris's Principles of Algebraic Geometry ...
-1
votes
4answers
284 views

An isomorphism on space of smooth sections

Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to $$A=\{f:L^{\times}\to ...
1
vote
1answer
97 views

Legal potentials on delzant polytopes

Let $P \subset \mathbb R^n$ be a Delzant polytope defined by inequalities $\ell_i(x) \geq 0, i=1, \ldots, d$. Of course, from the symplectic point of view, the inequalities $a_i \ell_i \geq 0$ still ...
0
votes
1answer
76 views

A subspace of the algebra of infinitesimal CR automorphisms

Let $(M^{2n+1}, D, J)$ be a strictly pseudoconvex CR sturcture on a compact $2n+1$-dimensional manifold, where $D$ is a nonintegrable distribution of codimension 1. The algebra of the infinitesimal CR ...
4
votes
1answer
163 views

Toric Fano Kahler manifolds and Delzant polytopes

Let $P$ be a Delzant polytope in $\mathbb R^n$, given by a set of inequalities $\ell_i(x) > 0$ where $\ell_i(x) = \sum_k \mu_k^i x_k - \lambda_i$. In his paper http://arxiv.org/abs/0803.0985 ...
2
votes
1answer
147 views

Properness of Ding-functional independent of the chosen Kahler metric

On Page 62 of "canonical metrics in Kahler geometry" written by Gang Tian, the author pointed out that properness of Ding-functional $F_\omega$ is independent of $\omega$, without proof. What I want ...
2
votes
1answer
160 views

A continuous version of Teichmuller uniqueness

By the Teichmuller uniqueness theorem, given a homeomorphism $f:X \rightarrow X$ where $X$ is the $n$-punctured sphere, there is a unique quasiconformal homeomorphism $g$ fixing $0$, $1$, and ...
1
vote
0answers
130 views

The Leibniz Rule for Vector Valued Holomorphic Forms

I am at the moment re-reading Voisin's book on complex geometry, from which I take the following: Let $M$ be a Kahler manifold, and let $E$ be a smooth vector bundle over $M$. Moreover, let us ...
5
votes
1answer
334 views

Are the Kahler Identities for a Holomorphic Vector Bundle Actually Interesting?

For a Kahler manifold $M$, and a smooth vector bundle $E$ over $M$, let us denote by $A^{(p,q)} := \Omega^{(p,q)} \otimes E$ the bundle of forms with values in $E$. Now with respect to a choice of ...
2
votes
1answer
160 views

Questions on the Hodge Dual of the Kähler Class

Let $M$ be a compact complex surface, i.e., a complex manifold with complex dimension two. Also, let us denote its Kähler metric by $g$ and its Kähler form by $K$. Let us denote the Kähler class by ...
0
votes
2answers
275 views

global sections of canonical line bundle of a projective variety

Given a smooth projective variety $X \subset \Bbb{CP}^k$ why is it true that global sections of $O(l)|_X, l >> 0 $ are just global sections of $O(l)$ on $\Bbb{CP}^k$ restricted to $X$? Here ...
6
votes
1answer
155 views

Intuition for holomorphic bisectional curvature

I would like to know the intuition behind the holomorphic bisectional curvature of Hermitian manifolds. I already know that the classical sectional curvature of a Riemannian (not necessarily complex) ...
2
votes
1answer
180 views

Complement of Donaldson's symplectic submanifold

I am just starting to learn more symplectic geometry on Stein manifolds. I understand that an important class of Stein manifolds arises as the complement of a Donaldson's codimension two symplectic ...
2
votes
1answer
108 views

Are harmonic maps quasiconformal at the boundary of hyperbolic spaces?

Let $M$ be a hyperbolic $3$-manifold and $X$ a negatively curved, simply connected space with geometric boundary $\partial X$. If $\rho:\pi_1M\rightarrow Isom(X)$ does not fix a point in $\partial ...
0
votes
1answer
147 views

Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me: Is there any open Ricci-flat ALE 4-manifold other than ...
4
votes
1answer
192 views

Is there extremal metric on toric Fano manifolds which futaki invariant is nonzero?

According the work by Wang & Zhu, on toric Fano manifolds there exists Kahler-Ricci solitons, if futaki=0 there exist CSCK metric, but if futaki invariant is not vanished, what about extremal ...
3
votes
2answers
218 views

The Szego projector, the dual disc bundle $\overline{D}$ and representation of $S^1$ on $H^2$($\overline{D}$)

This construction arises when constructing the Szego projector. Let's consider the dual disc bundle $\overline{D}$ of a positive Hermitian line bundle ($L$,$h$) over a compact Kahler manifold $M$, ...
5
votes
1answer
251 views

degrees of complex projective spaces and quadrics

A well-known result of Kobayashi and Ochiai says that an $n$-dimensional Fano maniofold $M$ is biholomorphic to $\mathbb{C}P^n$ or complex quadrics if its index is $n+1$ or $n$ respectively. In these ...
1
vote
0answers
145 views

Hyperkaehler Structures on the cotangent bundle

Let $M$ be a symplectic manifold (not Kaehler). Does there exists in a neighbourhood of the zero section in the cotangent bundle $T^{*}M$ a Hyperkaehler structure? I know that by the paper by Feix on ...
1
vote
1answer
294 views

Holomorphic vector field on Fano Kähler–Einstein manifold

Let $M$ be a compact Fano Kähler–Einstein manifold, and $V$ a holomorphic $(1,0)$ vector field on $M$. The Fano conditions say that $V = \nabla^{1,0} f$ for some smooth complex-valued function. By ...
2
votes
1answer
73 views

How a matrix of C^1 functions on a domain Ω in Cn generates a C^1 distribution in Ω

I am reading the paper Complex foliation generated by (1,1)-forms by M. Klimek. I have a problem understanding why is true a detail in one of the theorems: Let $\Omega$ be an open connected subset ...
1
vote
1answer
193 views

pull back of a Kähler form by a smooth circle group action on a compact Kähler manifold

Suppose a compact Kähler manifold $(M,\omega)$ admits a smooth circle action $g_t,~t\in S^1$. So the pull back of the Kähler form $g_t^{\ast}(\omega)$ is a nondegenerate two-form. Since the circle ...
3
votes
2answers
437 views

Is there a “by hand” proof on the symmetry of the Atiyah class of $TX$?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
4
votes
1answer
217 views

global section of local system from direct image

Deligne has a theorem in "Theorie de Hodge II" as follows: Let $S$ be a smooth separated scheme, and $f:X\to S$ be a smooth proper morphism. Let $\bar{X}$ be a non singular compactification of ...
5
votes
1answer
736 views

Hodge Theory (Voisin)

I have a strong understanding of Representation Theory but am interested in learning from Voisin, Hodge Theory and Complex Algebraic Geometry. What are the prerequisites to learning from this ...
3
votes
1answer
155 views

holomorphic sectional curvature and total scalar curvature

In a paper of Heier and Wong, It is written that from a pointwise argument due to Berger does follow that the scalar curvature (and thus also the total scalar curvature) of a Kaehler metric of ...