# Tagged Questions

**6**

votes

**1**answer

144 views

### The moduli space of special Lagrangian submanifolds

Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures on the base space $B$. A theorem of Hitchin says that there ...

**0**

votes

**1**answer

313 views

### why quintics are Calabi-Yau?

Why quintics are Calabi-Yau? Is there a explicit formula of the holomorphic volume form?

**2**

votes

**1**answer

188 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, ...

**1**

vote

**0**answers

77 views

### Aysmptotic comparison of L^2 sections versus generating sections

Let $s_1,\ldots, s_k$ be linearly independent global holomorphic sections of a holomorphic line bundle $E$ over a compact algebraic manifold $X$, with volume form $\Omega$.
For $m$ large, let ...

**1**

vote

**2**answers

221 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

**1**answer

145 views

### Zeroes of a complex polynomial on a sphere as a manifold

Let $ f \in \mathbb{C}[z_1, \ldots, z_n]$ be a polynomial such that $f'(z) \neq 0$ if $z \neq 0$ ($f'$ means $\left( \frac{\partial f}{\partial z_1}, \ldots, \frac{\partial f}{\partial z_n}\right)$ ). ...

**7**

votes

**2**answers

260 views

### Necessary condition on Calabi-Yau manfiold to be a hypersurface in a Fano manifold

Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it
being a member of a base-point-free linear system in a nef-Fano fourfold?
What, in anything, is known ...

**8**

votes

**1**answer

237 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 ...

**2**

votes

**1**answer

124 views

### reference of extension of flat bundle

Does any one know where one can find a reference about the following fact?
Let $X$ be a smooth projective variety over an algebraically closed field $k$.
Fix two flat bundles $(L_i,\nabla_i)$ over ...

**6**

votes

**1**answer

352 views

### Explicit Kodaira-Spencer map of hyperelliptic curves

Let $g\geq 2$, and $$\mathcal T=\{(t_1,\cdots,t_{2g+2})~|~t_i\neq t_j,\forall i\neq j\}.$$
For any $t=(t_1,\cdots,t_{2g+2})\in \mathcal T$, let
$$Y_t=\left\{y^2=\prod_{i=1}^{2g+2}(x-t_i)\right\}.$$
...

**1**

vote

**2**answers

198 views

### When representation of two different coadjoint orbits are equivalent?

Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^*$ (where ...

**3**

votes

**0**answers

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 ...

**3**

votes

**3**answers

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

**0**answers

28 views

### Nonadjointable Equivariant Operators on Hermitian Vector Bundles

Let $V$ be an equivalent complex vector bundle over a homogeneous space $X$, and $D:\Gamma^\infty(V) \to \Gamma^\infty(V)$ an equivariant operator. If we put a Riemann structure on $X$, and an ...

**6**

votes

**2**answers

224 views

### Whitney stratification and affine grassmanian

Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by ...

**3**

votes

**0**answers

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 ...

**1**

vote

**0**answers

121 views

### Explicit formula for hermitian form on coadjoint orbit of $G$ on $\mathfrak{g}^*$

Let $G$ be a compact Lie group and $\mathfrak{g}$ be its Lie algebra and $\mathfrak{g}^*$ be its dual , then I am looking for explicit formula for hermitian form on coadjoint orbit of $G$ on ...

**0**

votes

**0**answers

93 views

### Dimension of the set of polarized sections

Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian, i.e. Maximal isotroic, dim$P_m=n$, ...

**6**

votes

**2**answers

466 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 ...

**5**

votes

**1**answer

172 views

### Holonomy group of Enriques surface

I expect that the holonomy group of an Enriques surface $S$ is $SU(2)\times C_2$. I think this can be proven by the fact that its double cover, which is a K3 surface, has the full $SU(2)$ holonomy, ...

**2**

votes

**0**answers

79 views

### Properties of algebraic vector fields which generates a $\mathbb{C^*}$ action

My question is rather vague and I apologize. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. I am interested in whether there are homological properties which distinguish algebraic ...

**-1**

votes

**4**answers

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 ...

**2**

votes

**2**answers

201 views

### Connections on the Hodge bundle?

Let $\mathcal{M}_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}_g$ whose fiber over a curve $C\in\mathcal{M}_g$ is the space of ...

**1**

vote

**2**answers

478 views

### Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?

**1**

vote

**1**answer

131 views

### on two definitions of irreducible connection

I have seen two kinds of definitions of irreducible connections on fibre bundls: A connection is said to be irreducible if
the holonomy group is precisely $G$ and not a proper subgroup.
or
2. ...

**0**

votes

**2**answers

140 views

### Segre class of smooth vector bundles over smooth manifolds?

Before you read the following question, please assume I have no knowledge in algebraic geometry.
Is it possible to define Segre class of a smooth complex vector bundle over a smooth manifold by using ...

**0**

votes

**2**answers

274 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 ...

**2**

votes

**1**answer

320 views

### looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*MÃ—\mathbb{R}$.i.e,
($J^1M=T^*MÃ—\mathbb{R}$)
Is there something like this identity for higher jet bundle $J^kM$?
I editted ...

**3**

votes

**1**answer

212 views

### on Brieskorn Manifolds

Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in
\mathbb{C}^{k+1}| -z_{0}^{3}+\sum_{i=1}^{k} z_{i}^{2}=0\}$(k odd, k>2) is a
topological manifold. Is it a smooth manifold?
In general, let $a_1, ...

**9**

votes

**1**answer

497 views

### moduli spaces are kahler?

I often heard from experts that "moduli spaces are Kahler". This sounds as a meta-theorem asserting that every time one defines reasonable moduli spaces, then there is a standard strategy to see ...

**2**

votes

**1**answer

121 views

### Split real form of $SL(2,\mathbb{C})$ description of the two sphere?

If we denote the parabolic subgroup of $SL(2,\mathbb{C})$ by $P$, then we have the well known isomorphism $SL(2,\mathbb{C})/P \simeq S^2$, where $S^2$ is the two sphere. Now the compact real form of ...

**0**

votes

**0**answers

166 views

### Fractional degree of a map?

Is there some natural notion of a fractional degree of a map?
The degree of a map is a generalization of the winding number,
and fractional winding numbers appear in the (mathematical physics)
...

**0**

votes

**0**answers

84 views

### Is equivariant homology class preserved in the limit?

Suppose $C \to (S,0)$ is a family of curves (with at most nodal singularities) parametrized by a pointed curve $(S,0)$, that is proper and flat. Let $u:C \to X/G$ be a family of maps to a quotient ...

**5**

votes

**1**answer

213 views

### Spin manifolds with one parallel spinor

Are there any examples of D-dimensional Ricci-flat Riemannian (spin) manifolds of dimension D= 2,3,4,5 with the dimension of the space of parallel spinors equal to 1? And the same question for the ...

**4**

votes

**1**answer

380 views

### Central extension of the algebraic loop group

I'm doing some constructions with the universal central extension $\widehat{\Omega G}$ of the loop group $\Omega G$ (here $G$ is a matrix group), where a priori the loops involved are just smooth, but ...

**4**

votes

**1**answer

206 views

### Normal form for a holomorphic Morse function

Similarly to Morse lemma, a holomorphic Morse function can be written, near a critical point, as $W_1^2+W_2^2+...+W_n^2+C$ , for well chosen coordinates $W_1,W_2,...,W_n$. I want to cite this result ...

**5**

votes

**1**answer

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

**1**answer

135 views

### Sections over projective bundles

I encounter a problem when I read compact complex surfaces.Let $X$ be a projective surface,$E$ is a vector bundle of rank 2 over $X$, $S \subset \mathbb{P}(E)$ is an irreducible subvariety such that ...

**5**

votes

**2**answers

280 views

### How to calculate the determinant bundle

Maybe, this is a problem of linear algebra. But I do not know how to calculate it. Let $E$ be a vector bundle of rank $2$ over an algebraic surface. If $H=S^{2n}E\bigotimes (\operatorname{det} ...

**1**

vote

**0**answers

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 ...

**9**

votes

**2**answers

410 views

### motivation for multiplier ideal sheaves

What is the origin of multiplier ideal sheaves?It was introduced ny Nadel.Yum Tong Siu,his advisor in his plenary lecture in 2002 icm mentions some thing that it arose in pde.Can anyone kindly ...

**3**

votes

**2**answers

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

**1**answer

216 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 ...

**0**

votes

**0**answers

109 views

### When a hyperplane of symmetric forms is determined by a quadric hypersurface?

Let $L$ be a 2D real vector space, $L^*$ its dual, and $\{V,\omega\}$ the symplectic space with $V=L\oplus L^*$ and $\omega$ unambiguously defined by $\omega(l,\lambda):=\lambda(l)$, for all $l\in L$ ...

**0**

votes

**1**answer

144 views

### QQ-Gorenstein smoothing of a surface and a finite group action

Let S be a singular complex algebraic surface. Let $\psi: \chi \rightarrow \Delta$ be a QQ-Gorenstein smoothing of S (central fibre is isomorphic to S). Let G be a finite group acting on the fibres of ...

**6**

votes

**2**answers

528 views

### Are there examples of Fano manifolds such that Tian's alpha invariant satisfies $\alpha_G(X)=\frac{n}{n+1}$ but without a Kähler-Einstein metric?

The alpha invariant $\alpha(X)$ of a Fano manifold $X$ of dimension $n$ is defined as the infimum of log canonical thresholds of (effective) $\mathbb{Q}$-divisors $D\sim_{\mathbb{Q}}-K_X$. Similarly, ...

**0**

votes

**1**answer

165 views

### Non-Symmetric Equivariant Riemannian Metrics on Homogeneous Spaces

For a homogeneous space $M = G/H$, the number of $H$-equivariant Riemannian metrics on $M$ is usually much smaller than the space of Riemannian metrics. I am wondering what happens when the symmetric ...

**8**

votes

**1**answer

1k views

### Yau's conjecture for positive Chern class

I heard in a conference that Yau's conjecture is open for positive Chern class. I read in an article that talked about some stability conditions necessary in this case. So I want to know if this ...

**4**

votes

**4**answers

384 views

### Submanifolds in the Grassmannian of n-dimensional subspaces determined by a submanifold in the Grassmannian of l-dimensional subspaces

Let $G_n(V)$ the Grassmannian of $n$-dimensional subspaces of a finite-dimensional $V$, and $l<n$. I've noticed that it is easy to associate a (possibly singular) submanifold $\tilde{M}\subseteq ...

**0**

votes

**0**answers

212 views

### Local System and Gauss-Manin connection

Fix a complex manifold $X$. Then if we have a line bundle $L=\mathcal{O}(D)$ together with Gauss-Manin connection $\nabla: L \rightarrow L \otimes \Omega^{1}_X$, we get the locally constant sheaf $F$ ...