1
vote
1answer
43 views

Twisting with a degree negative line bundle

Let $X$ be a Riemann surface. Let $M_1$ and $M_2$ be two holomorphic bundles on $X$. Does injectivity of $h^0(M_1)\to h^0(M_2)$ imply $h^0(M_1\otimes L)\to h^0(M_2\otimes L)$ is injective? Where $L$ ...
0
votes
0answers
39 views

Pushforward of a volume form

Let $X$ be a complex projective manifold with semi-ample line bundle $ K_X$ . Assume that $f: X\to X_{can}\subset \mathbb CP^N$ , and $f^{-1}(s)$ is nonsingular fibre, then I am looking for a proof ...
0
votes
1answer
33 views

Sign of first chern class with some conditions

Let $X$ be a compact Kahler algebraic variety which $K_M$ is big and nef, and $Kod(X)=dimX$ then why the first chern class $c_1(M)$ is negative or zero . I don't undrestand kawamata's theorem in this ...
0
votes
1answer
27 views

Rational first chern class of algebraic variety with zero Kodaira dimension.

Let $X$ be a compact Kahler algebraic variety which has zero Kodaira dimension. Then the integral first chern class vanishes? What about rational first chern class?
1
vote
0answers
38 views

differential of $f:X\to\Sigma$ as an elliptic surface,

Let $X$ be an algebraic surface surface and $\sum$ an algebraic curve, and assume, $f:X\to\Sigma$ be an elliptic surface, my question is Why the differential $df$ can be viewed as an injection of ...
1
vote
0answers
29 views

Definition - Logarithmic Zeros and Algebraic Vector Fields

In Algebraic Geometry, what is the difference between an algebraic vector field and a derivation (are they completely different or synonymous)? What does it mean to say an algebraic vector field has ...
4
votes
0answers
42 views

The Kähler form and the anticanonical line bundle

Let $M$ be a Kähler manifold. We say that $M$ is Fano if the anticanonical line bundle $K_M^*$ of $M$ is ample (or positive). On the other hand, I sometimes see the following definition (or ...
4
votes
2answers
80 views

is tangent bundle of $S^n$ an algebraic variety?

I have found somewhere that $T(S^n)$ is an algebraic variety in $C^n$. But now I can not recall the explicit form of this variety and the source of this information. It will be helpful if somebody ...
3
votes
0answers
57 views

Example for divisors, line bundles and meromorphic functions on $\mathbb{CP}^2$

I have been studying divisors using Griffiths/Harris (chapter 1.1) as well as Huybrechts (chapter 2.3). However, I cannot seem to find any very easy worked examples - i.e. $\mathbb{CP}^1$ or ...
0
votes
0answers
61 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
3
votes
2answers
95 views

geometric motivation for spaces with functions

Let $k$ be a field. A space with functions over $k$ is topological space X together with a family $O_X$ of k-subalgebras $O_X(U)\subseteq Map(U,k)$ for every open set $U$ that satisfy a) If ...
0
votes
0answers
46 views

What is $T^*(\mathbb{A}^1)$?

Let $\mathbb{A}^1$ be the affine line and $T^*(\mathbb{A}^1)$ the contangent space of $\mathbb{A}^1$. What is $T^*(\mathbb{A}^1)$? Is $T^*(\mathbb{A}^1) = \mathbb{A}^2$? Thank you very much.
3
votes
1answer
72 views

Application of Riemann Roch

I have read that thanks to Riemann Roch theorem, if get $\Sigma$ a compact Riemann Surface of genus $g$ there exists a conformal branch covering $\phi: \Sigma \rightarrow S^2$ of degree less than ...
2
votes
1answer
52 views

Orientability of algebraic manifolds

Is algebraic manifold always orientable? For example, unorientable Mobius strip $M$ can be represented as $$x(u,v)= \left(1+\frac{v}{2} \cos \frac{u}{2}\right)\cos u$$ $$y(u,v)= \left(1+\frac{v}{2} ...
3
votes
0answers
89 views

Developing intuition in algebraic geometry through differential geometry?

I'm interested in algebraic geometry (I am working through Ravi Vakil's notes and also have worked with curves and general varieties in the past), and have seen some basic definitions from ...
2
votes
0answers
55 views

Ample tangent bundle

I am looking for definition of ample tangent bundle or positive tangent bundle and why on a complex manifold , positive bisectional curvature means ample tangent bundle?
2
votes
0answers
82 views

Equation for the Maximum-Area Rotor in a Wankel Engine

I am attempting to construct a specialized Wankel Engine in Autodesk Inventor. For my particular project, the rotor must take up the maximum area, in order to separate the air spaces of the top and ...
8
votes
2answers
337 views

Trying to understand the use of the “word” pullback/pushforward.

Essentially, my question is the following : Is everything we call "pullback" or "pushforward" an actual categorical pullback/pushout? I have seen tons of pullbacks in differential geometry but we ...
5
votes
1answer
156 views

Relations of Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, Wu class. [closed]

We have a quite some characteristic classes: Chern class, Stiefel-Whitney class, Pontryagin class, Euler class, Wu class, etc. I wonder whether some math experts can use simple words and basic ...
1
vote
1answer
46 views

projective homogeneous $G$-variety is equivariantly isomorphic to a partial flag variety $G/P$ where $P$ is projective variety.

I am looking for a proof(or refference) for this fact A projective homogeneous $G$-variety is equivariantly isomorphic to a partial flag variety $G/P$ where $P$ is parabolic subgroup.
-2
votes
1answer
39 views

reduced space of coadjoint orbit

Let $G$ be a compact Lie group and $\lambda\in \frak{g}^*$$=(Lie G)^*$ and $O_\lambda$ be the Coadjoint orbit through $\lambda\in \frak{g}^*$ and $\mu:O_\lambda\to\frak{g}^*$ be the moment map, ...
2
votes
1answer
119 views

What should a student (with algebraic-geometry minded) study in differential geometry?

One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a ...
2
votes
1answer
96 views

Lipschitz manifold and semi-algebraic is Lipschitz graph?

It is known that there are Lipschitz manifolds that are not Lipschitz graphs and $C^1$ manifold is locally $C^1$ graph. If $M\subset \mathbb{R}^m$ is a Lipschitz manifold (with the outer distance) ...
4
votes
0answers
62 views

Soft: Why does the existence of a singularity cause problems for deRham cohmology?

I've heard that if a variety has a singularity then the deRham theory has "problems". What exactly are these? Im guessing there is some sort of issue with the defintion of a differential form, but ...
1
vote
0answers
42 views

Lateral area of a cone with oblique base (elliptical base)

Can someone help me find the lateral area of a straight circular cone with oblique base ( inclined elliptical base) as seen at the following link. ...
1
vote
1answer
92 views

Complex line bundle at symplectic manifold

Let's say that there is a symplectic manifold $(M,\omega)$ with condition of $[\omega / 2\pi ]\in H^2(M;\mathbb{Z})$. Then in what condition can I get a complex line bundle $L\twoheadrightarrow M$ in ...
0
votes
0answers
54 views

Can Vanishing Cycles be Described as Fibers over Critical Points in a Lefschetz Fibrations?

I'm trying to see if it makes sense to see vanishing cycles in a Lefschetz fibration as the fibers over critical points. A Lefschetz fibration $f: M^4 \rightarrow X$ , where $M^4$ is a smooth ...
1
vote
0answers
20 views

Dual Translation plane

An affine plane $\mathcal{A}$ is called a translation plane if the translation group of $\mathcal{A}$ operates transitively on the point set of $\mathcal{A}$. So how do we define the dual translation ...
0
votes
0answers
15 views

Finding the Bessel function from its derivative

I have a situation: $A_k\frac{\partial J_m(k\rho)}{\partial \rho}=0$. where $k=k_1$ for $0\leq\rho\leq a$ and $k=k_2$ for $a \leq \rho \leq \Lambda-a$ with $a,\Lambda\leq \infty$. Can I proceed with ...
1
vote
2answers
45 views

space of sections of homogenious spaces

Let $G/H$ be a homogeneous space and then for homogeneous line bundle $L$ of $G/H$ the space of sections can be written as functions related to character of $H$. what about $\Gamma (G/H, L^2)$. then ...
0
votes
1answer
61 views

Definite integral of product of two bessel functions of different order and different argument

What is the solution of the integral: $\int_0^a J_m(k_2\rho)J_{m+1}(k_1\rho)d\rho$ where the integer $m\geq0$
4
votes
1answer
33 views

Derivations are determined by their values on linear functions

How are derivations of the $\mathbb R$ algebra of germs of differentiable real functions on a manifold completely determined by their values in germs of linear functions? Are derivations of more ...
0
votes
0answers
13 views

Orthogonal Array Planes

In an OA-plane (Orthogonal Array-Plane) how we define the touching of two lines? If we consider OA of strength of 3 with unit index , that is the plane OA-plane of rank 3 (we call this Laguerre ...
4
votes
1answer
79 views

Problems understanding the construction of Hitchin moduli space in his paper “The self-duality equations on a riemann surface”

First, if this post must be broken up in separate questions, please tell me so. I thought it would be better if I simply posed my questions in one thread, as they are directly related to each other. ...
1
vote
1answer
41 views

Parabolic Cusp of an Action on the Upper Half Plane

This is a basic definition question. Parabolic bundles are used in certain counting arguments in my research area. I asked my advisor for a reference on these, and he directed me to the paper of Mehta ...
5
votes
1answer
68 views

Problem about parallel curve- differential geometry

Let $\alpha (s)$ , $s\in [0,L]$, be a smooth positively oriented regular Jodan curve which is arc-length parametrized. The curve $\beta(s)=\alpha (s) +\lambda n(s)$, where $\lambda$ is a positive ...
6
votes
1answer
61 views

Relation between different definitions of degree in complex geometry

Consider a holomoprhic map from a Riemann surface $$ f: \Sigma_g \to \mathbb{CP}^n. $$ This is given by some homogeneous polynomials in some variables. How can we show that the homogeneous degree $d$ ...
4
votes
1answer
66 views

Connection(gauge field) in Fubini-Study metric is pull back of a connection A of line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^{N-1}$

One can describe a $\mathbb{CP}^{N-1}$ manifold with a Fubini-Study metric $g^{FS}$, and there is a connection one form $v$ on it. A is connection one form(gauge field) of a line ...
1
vote
0answers
124 views

Top current research topics in Complex Algebraic and Differential Geometry

I am currently a first year PhD candidate and I need some help figuring out the top current research topics in Complex Algebraic and Differential Geometry (can be in both, or just one of the two). I ...
0
votes
0answers
38 views

Central subgroups …

Have a question about the central subgroup. E is an elation group. Let $E_{x,y}$ be a stabilizer of points $x$ and $y$. Then the group $E_{x,y}$ is a central subgroup both $E_{x}$ and $E_{y}$, ...
0
votes
2answers
45 views

Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
1
vote
1answer
35 views

How do I convert OS coordinates (X and Y) to longitude and latitude coordinates?

How do I convert OS coordinates (X and Y) - Eastings and Northings to longitude and latitude coordinates? For example X and Y below X (Eastings): 347904 Y (Northings): 287484
1
vote
0answers
19 views

How do I figure out the grid references necessary to create a circular 1 mile radius around one location?

How do I figure out the grid references necessary to create a circular 1 mile radius around one location? For instance: Grid Reference: SO 47904 87484 X (Eastings): 347904 Y (Northings): 287484 ...
11
votes
4answers
229 views

Topics in Differential geometry $\cap$ Algebraic geometry

I find (both: differential and algebraic) geometry fascinating. I'm just beginning my graduate studies, but I'd like to know some topics/theorems in the intersection of these two (since I don't really ...
7
votes
2answers
117 views

When does a SES of vector bundles split?

Given a short exact sequence of smooth vector bundles, $$0\to A \to B \to C \to 0$$ on a manifold $M$, it is an easy exercise that sequence splits. One approach is to pick a Riemannian metric on ...
1
vote
1answer
78 views

How do I figure out the longitude and latitude coordinates necessary to create a circular 1 mile radius around one location?

How do I figure out the longitude and latitude coordinates necessary to create a circular 1 mile radius around one location? I don't mind how many coordinates that takes. For instance: Latitude = ...
0
votes
1answer
70 views

Parametrization of a Curve

I am working through my script of Algebraic Geometry and have some questions to Parametrization: What is the exact definition of Parametrization? We wrote in our script: $k \to C$ with $t\mapsto ...
1
vote
1answer
101 views

Book Recommend Differential Geometry of Algebraic Manifolds

I just want to study Differential Geometry of Algebraic Manifolds. but I can`t find a book about that. Is there any good book for studying Differential Geometry of Algebraic Manifolds??
0
votes
1answer
67 views

Fano manifolds and positive Ricci curvature.

I already know that if a Kahler manifold, $M$, is Fano, then $M$ has Kahler metrics with positive Ricci curvature. What about the converse? If $Ric(M)$ is positive, does this mean that $-K_M$ is ...
2
votes
1answer
39 views

Intersection of two open sets in the projective plane

I want to compute the cohomology groups of the real projective plane, $P^2$, using Mayer Vietoris exact sequence. Now $H^0(P^2)=\mathbb{R}$, $H^2(P^2)=0$ being $P$ not orientable, so my problem really ...