5
votes
0answers
41 views

Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
1
vote
0answers
34 views

on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, maybe semisimple); call its Cartan subalgebra $\mathbf t \subset \mathbf g$ and Weyl group $W$. Why does the construction $\mathcal ...
0
votes
0answers
29 views

If $[\omega]\in H^2(M,\mathbb Z)$ then $M$ is projective?

Let $(M,\omega)$ be a Kähler manifold with $[\omega]\in H^2(M,\mathbb Q)$ then why $M$ must be projective variety. As I know if $[\omega]\in H^2(M,\mathbb Z)$ then $M$ is projective by Kodaira theorem ...
0
votes
0answers
27 views

semi-positiveness of canonical line bundle under the condition Kodaira dimension be positive.

Let $M$ be a projective variety with positive Kodaira dimension, then why the canonical line bundle is semi-positive?. Is there any reference?
0
votes
2answers
43 views

Holomorphic Sphere $ S^2$ with (-1) self intersection number

What is the meaning of Holomorphic Sphere $ S^2$ with (-1)- self intersection number in intersection theory. Can we draw such sphere?
0
votes
1answer
43 views

Curve in an algebraic variety

Let $\lambda_1, \lambda_3, \lambda_3$ be distinct real numbers. Can it be that a curve of the form $$ t \mapsto \gamma(t) := (e^{\lambda_1 t}, e^{\lambda_2 t}, e^{\lambda_3 t}) $$ is contained for all ...
0
votes
0answers
32 views

$n$-connectivity of principal $G$-bundle

Page 202 of Switzer's Algebraic Topology: Let $\xi=(E,p,B)$ be a principal $G$-bundle such that $E$ is $n$-connected. Then for any pointed CW-complex $(X,x_0)$ of dimension at most $n$, the ...
3
votes
1answer
8 views

Containment of two varieties with a lot of intersection

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...
0
votes
2answers
30 views

Curve in union of hyperplanes

If a smooth curve $\gamma: [0,T] \to \mathbb R^n$ is contained in the union of hyperplanes $$ \bigcup_{i=1, \dots, N} H_i$$ does it then follow that one can always find time intervals $[t_0, t_1]$ ...
1
vote
3answers
76 views

Examples of smooth curves of genus $0$ and degree $d>2$.

Can we provide a source of explicit examples ? The degree assumption $d>2$ means that I would like to see examples which are not conics.
2
votes
0answers
59 views

cohomology ring of grassmannian

Let $G_k(\mathbb{R}^n)$ be the grassmannian consisting of all $k$-subspaces in $\mathbb{R}^n$. How to compute the cohomology ring $$H^*(G_k(\mathbb{R}^n);\mathbb{Z})$$ and what is the result?
0
votes
0answers
48 views

use Noether normalization theorem to integrate differential forms over singular subvarieties

Let $X \subset \mathbb C^n$ be an analytic subset. I would like to show that locally around any point $x \in X$ the regular part $X_\text{reg}$ has finite volume, perhaps using the theorem below. ...
8
votes
0answers
74 views

Why is the Leibniz rule a sufficient ingredient in the construction of the tangent space?

This is a very soft question, but I am wondering if anyone can shed light on why it is that the product rule (and linearity) provide exactly the right requirement for the space of derivations to be ...
1
vote
1answer
50 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
47 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
39 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
40 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
41 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
37 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
49 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
131 views

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

I have found somewhere that $T(S^n)$ is an algebraic variety in $\mathbb{C}^{n+1}$. But now I can not recall the explicit form of this variety and the source of this information. It will be helpful if ...
3
votes
0answers
63 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
65 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
98 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
47 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.
4
votes
1answer
79 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
55 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
99 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
61 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
104 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
352 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
198 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
40 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
126 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
100 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
63 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
51 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
96 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
58 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
21 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
17 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
47 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
69 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
14 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
88 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
46 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
77 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
63 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 $d$ ...