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A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.

2 votes
1 answer
1k views

Fubini-Study metric induced by submersion

The Fubini-Study metric $g:=g_{FS}$ is the unique $U(n+1)$-invariant Riemannian metric on the complex projective space $\mathbb{CP}^{n}$ the complex projective space which by $U(n+1)$-invariance can …
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3 votes
0 answers
233 views

Zero section of quasi-coherent bundle

Let $S$ be a scheme and let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module. Then we can construct a graded quasi-coherent $\mathcal{O}_S$-algebra $\mathscr{A}:= Sym(\mathcal{E})$ and define …
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2 votes

Sheaf of relative Kähler differentials intuitively

supplement/ an "almost" answer: I noticed that OP's of several related questions (1, 2) asked about similar issue. The best explanation I found there was that for a smooth manifold $X$ the tangent spa …
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0 votes
1 answer
322 views

Self-intersection of zero section of line bundle over elliptic base curve

Let $C$ be an elliptic curve over $k=\mathbb{C}$ and $\mathcal{L}$ a line bundle of degree $d$. It induces naturally a $\mathbb{A}^1$-fibration $L \to C$ where $L=\underline{\operatorname{Spec}} (\ope …
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13 votes
2 answers
3k views

Sheaf of relative Kähler differentials intuitively

Let $f: X \to Y$ be a separated morphism between $k$-varieties or more general schemes of finite type. The most common way in standard literature on algebraic geometry to define the sheaf of relative …
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-1 votes
1 answer
216 views

Almost Complex Structure extending to Complex Structure, aka "Integrable"

Let $M$ be a smooth manifold of (real) dimension $2n$. An almost complex structure $J$ on $M$ is a linear vector bundle isomorphism $J \colon TM\to TM$ on the tangent bundle $TM$ such that $J^2 = − 1 …
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1 vote
0 answers
214 views

Find torsion classes using flat bundles

My question refers to a discussion from this older thread on Neron-Severi group of a Kähler manifold. In the comments below Ted Shifrin's answer there arose a discussion when the map $H^2(X,\mathbb{Z …
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5 votes
1 answer
271 views

Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf

Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) consist …
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