0
votes
0answers
109 views

Hermitian metric on $f_*(\Omega_{X/X_{can}}^{n-k})$

Let $X$ be an n- dimensional algebraic manifold . Suppose that its canonical line bundle $K_X$ is semi-positive and $0<k=Kod(X)<n $ . Let $f: X\to X_{can}\subset \mathbb CP^N$ Here $X_{can}$ ...
0
votes
1answer
183 views

First chern class of fibers of compact Kaehler algebraic variety

Let $M$ be an compact Kähler algebraic variety and suppose $K_M$ is semi-ample. Consider the holomorphic map $\pi:X\to \Sigma \subset \mathbb CP^N$ with $Kod(M)=dim_\mathbb C\Sigma$ (here $Kod$ means ...
3
votes
0answers
105 views

On the cohomology of Kontsevich graph complex

Kontsevich's formality theorem asserts that a certain quasi-isomorphism of chain complexes between the graded Lie algebra of polyvector fields on $\mathbb{R}^n$ and the dg Lie algebra of ...
0
votes
1answer
254 views

Yang-Mills equations are not elliptic [closed]

How does one prove that the Yang-Mills equations (from classical Yang-Mills theory) are not elliptic? Alternatively, how does one calculate the principal symbol of the Yang-Mills equations? Can ...
3
votes
1answer
159 views

Regular singularities and the infinitesimal site

Suppose I have a smooth non-proper algebraic variety $X/\mathbb{C}$. A vector bundle with flat connection (``differential equation'') on $X$ extends, as was noted by Grothendieck, to a coherent ...
2
votes
1answer
180 views

The stability of vector bundle with trivial Chern classes is independent of ample divisor, a direct proof?

Let $X$ be a smooth projective variety over $\mathbb{C}$. For an ample divisor H, we can define the slop of vector bundle with respect to $H$, then we can define stablilty of vector bundle with ...
5
votes
1answer
250 views

A geometric characterization of smooth points of a complex algebraic variety

Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric. Fix an arbitrary point $p\in X$. Let ...
5
votes
2answers
188 views

Local maxima and minima of the trace of a product of $SL_2^\pm(\mathbb{R})$-matrices

I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my ...
2
votes
0answers
90 views

Chern-Weil theory for degenerated metric

If $\omega$ is a Kähler metric on a compact complex manifold $X$, the standard Chern-Weil theory says that the Chern classes $c_{i}(M)$ can be represented by forms involving the curvature of ...
7
votes
1answer
428 views

Geometric interpretation for fourth coeficient of the polynomial for Hirzebruch–Riemann–Roch theorem

Hey guys :) I have a question on a theorem which is known as Tian-Yau-Zelditch theorem now. If there is some reference for following question, please let me know Let $(M,\omega)$ be a compact ...
1
vote
1answer
156 views

When $\frac{\text{Aut}(G/P,L)}{S^1}$ is discrete?

Let $(M,\omega)$ be a Kähler manifold with a pre-quantum Line bundle $L$ and $\text {Aut}(M,L)$ means the group biholomorphisms of $M$ which lift to holomorphic bundles maps $L\to L$. My question is ...
1
vote
1answer
111 views

characterization of structure group

Somebody tell me that: For a bundle(maybe polystable) over algebraic manifold, take a symmetric power of the bundle and tensor with its determinant line bundle to some power. Assume that the ...
1
vote
0answers
169 views

Orbital integral by using symplectic quotient

Let $(M,\omega)$ be a compact symplectic Hamiltonian $G$-manifold and $\Gamma_{\text hol}(M,L)$ be the space of holomorphic sections of the line bundle $L\to M$ I am looking for a proof for ...
1
vote
1answer
175 views

examples of Kähler manifolds with trivial Hodge numbers and first Chern classes

Yesterday I asked the following question to which abx has given a positive answer. examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes But I suddenly realized ...
0
votes
1answer
291 views

A question about Quantized closed Kaehler manifolds

Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding $$\phi: (M,\omega)\to  (\mathbb CP^N, ...
2
votes
1answer
246 views

examples of Kähler manifolds with trivial odd Betti numbers and first Chern classes

To my limited knowledge, many compact Kähler manifolds have trivial odd Betti numbers. For instance, flag manifolds $G/P$,where $G$ is a semisimple complex Lie group and $P$ a parabolic subgroup, and ...
3
votes
1answer
147 views

Deformation of Hitchin-Simpson correspondence

Let $X$ be a compact Riemann surface, Hitchin-Simpson correspondence over $X$ says that irreducible representations of $\pi_1(X)$ one-to-one correspondend to stable Higgs bundles with vanishing Chern ...
2
votes
0answers
295 views

The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)

(Migrated from MSE) While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...
0
votes
0answers
72 views

Comparison of support of a divisor and its class in de Rham cohomology

Let $X$ be a smooth surface over $\mathbb{C}$ and $D$ be an effective divisor. Suppose that the linear system corresponding to $D$ is zero dimensional. Denote by $[D] \in H^2(X,\mathbb{Z})$ the ...
0
votes
1answer
127 views

extension of Riemannian metric on real affine variety

Given a Riemannian metric $g$ on the real part $X_R$ of a real affine variety $X$, is there a "natural" way to extend $g$ to be a Riemannian metric on $X$?
1
vote
1answer
131 views

Kaehler form on weighted projective space

The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by: $$\log(\sum_{i=0}^n |z_i|^2)).$$ What is the analogous formula for a Kaehler ...
9
votes
2answers
387 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 (symplectic and complex) on the base space $B$. A theorem of ...
3
votes
1answer
515 views

Why quintics are Calabi-Yau?

Why quintics are Calabi-Yau? Is there a explicit formula of the holomorphic volume form?
2
votes
1answer
216 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
1answer
182 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
2answers
239 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
151 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
2answers
282 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
1answer
254 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
1answer
129 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
1answer
364 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
2answers
212 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
0answers
237 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
3answers
302 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
0answers
33 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
2answers
243 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
0answers
144 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
0answers
126 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
0answers
94 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
2answers
555 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
1answer
175 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
0answers
83 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
4answers
291 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
2answers
228 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
2answers
495 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
1answer
147 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
2answers
147 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
2answers
317 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
1answer
339 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 $T^*M×\mathbb{R}$.i.e, ($J^1M=T^*M×\mathbb{R}$) Is there something like this identity for higher jet bundle $J^kM$? I editted ...
3
votes
1answer
220 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, ...