Questions tagged [pushforward]

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4 votes
2 answers
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Change of coordinates for coends

I recall that there was a theorem mimicking the change of variables' integral formula. Surprisingly, I can't find it on the Fosco Loregian book. The change of variables formula states that, if $f: E \...
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-1 votes
0 answers
89 views

Reference about direct image sheaf

I was searching in Internet but I was not able to find a proper book/notes/etc. in which are carried out explicit computations with direct image sheaves. In particular I was interested in examples (...
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2 votes
1 answer
146 views

A question about pushforward measures and continuous Borel isomorphisms

It is fairly well known that if $\mu$ and $\nu$ are nonatomic measures on the standard Borel spaces $(X,B)$ and $(Y,C)$ such that $\mu(X)=\nu(Y)$. If $X$ and $Y$ are uncountable, then there exists a ...
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  • 33
1 vote
0 answers
102 views

Relation between push forward by diagonal morphism and higher direct image functors

Let $f : X \to Y$ be a morphism between two Noetherian schemes. Then $f_*$(respect to $R^1f_*$) sends coherent sheaves to coherent sheaves if and only if $f$ is universally closed (respect to ...
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0 votes
1 answer
166 views

Rewriting PDE as "push-forward"

Suppose that we have the following PDE $$\partial_t \mu_t = \nabla\cdot \left(\nabla \mu_t - (b*\mu_t)\mu_t\right), \tag{1}$$ with $\mu_0$ being a (smooth) probability measure/density on $\mathbb{R}^d$...
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  • 454
1 vote
1 answer
345 views

Push-forward of divisors and intersections

Let $f:X\rightarrow Y$ be a surjective finite morphism of varieties, with $X$ normal and $Y$ smooth. Let $D\subset X$ be a divisor and $C\subset Y$ a curve. Does the equality $$C\cdot f_{*}D = f^{*}C\...
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1 vote
1 answer
212 views

Pushforward of measure under surjective map

Let $X, Y, Z$ be measurable spaces with measures $\mu_X, \mu_Y, \mu_Z$ respectively. Let $\pi_Y : Y \times Z \rightarrow Y$ be the projection on $Y$ and $\pi_Z : Y \times Z \rightarrow Z$ the ...
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2 votes
1 answer
293 views

Pushforward of a very ample line bundle on a curve to $\mathbb{P}^1$

Let $p:C\to\mathbb{P}^1$ be a degree $k$ morphism from a smooth projective curve $C$ to the projective line and $L$ a very ample line bundle on $C$. We know that $p_*\mathcal{O}_C(L)$ is a rank $k$ ...
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  • 305
3 votes
0 answers
64 views

Relation between the orientation sheaves of the interior and the boundary of a topological manifold

Let $(M, \partial M)$ be an $n$-dimensional topological manifold with boundary. Let $\mathcal{O}_{M \setminus \partial M}$ and $\mathcal{O}_{\partial M}$ denote the orientation sheaves of $M \setminus ...
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  • 155
1 vote
1 answer
159 views

Gysin map for projective sub-bundles of exceptional divisors

Let $X$ be a smooth, projective variety, $Y \subset X$ a smooth, projective subvariety of codimension $3$. Denote by $\pi:\tilde{X} \to X$ the blow-up of $X$ along $Y$ and by $E$ the exceptional ...
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  • 2,056
3 votes
1 answer
448 views

Pushforward of curves

Let $Z$ be a subvariety of an irreducible projective variety $X$, and let $i:Z\rightarrow X$ be the inclusion. Let $N_1(X),N_1(Z)$ be the $\mathbb{Q}$-vector spaces of curves in $X$ and $Z$ ...
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1 vote
0 answers
518 views

Isomorphism of probability spaces

Consider a surjective map $f:(X, \sigma(f)) \to (Y, \mathcal{Y})$, if a measure $\nu$ is given in $(Y, \mathcal{Y})$ the pullback $\nu(f(\cdot))$ is a measure on $(X, \sigma(f))$, similarly if $\mu$ ...
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8 votes
1 answer
511 views

Pushforward of line bundle under "toric isogeny"

Let $(X,T)$ be a smooth complex toric variety of dimension $d$ with torus $T$ and toric boundary $D=X\setminus T$. Let $\phi : X\to X$ be a finite endomorphism of $X$ such that the restriction $$\phi|...
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  • 21.6k
1 vote
2 answers
687 views

Push-forward of a nef bundle

Let $f:X\rightarrow Y$ be a finite morphism between normal varieties. Let $E$ be a vector bundle on $X$ and let us consider its pushforwad $f_{*}E$. Does anyone know an example where $E$ is nef but $...
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  • 6,663
36 votes
3 answers
9k views

Pullback measures

Why do all measure theory textbooks present the concept of push-forward measure, but never the concept of pull-back measure? Doesn't the latter exist? It's true that the naive treatment of such a ...
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  • 4,740
2 votes
2 answers
508 views

sheaves for which the derived (compact or not) pushforward is zero

Conventions: sheaf = complex of constructible sheaves (in the l-adic setup with etale tplg or in the complex coefficients setup with analytical tplg). I would like to understand if there is an ...
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