Questions tagged [pushforward]
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24 questions
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Surjectivity of pushforward on image
Let $\mathcal X\subseteq\mathbb R^m$ be a Borel measurable set. $\Phi:\mathcal X\to\mathbb R^n$ be a continuous mapping and $\mathcal Y = \Phi(\mathcal X)\subseteq\mathbb R^n$ its image. Let $\mathcal ...
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Is the pushforward of a closed immersion ever fully-faithful at the level of Derived Categories?
Let $i: Z \rightarrow X$ be a closed immersion of schemes. Then, for any $\mathcal{O}_{Z}$-module $\mathcal{G}$, the counit of adjunction $i^{*}i_{*}\mathcal{G} \rightarrow \mathcal{G}$ is an ...
- 127
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Distribution of this integral of Fourier multiplier
In Barashkov and Gubinelli (2019) section 2, the authors make the claim that the distribution of $$Y_t = \int_0^t \langle D \rangle^{-1}\sigma_s(D)dX_s$$ is given by the pushforward $(\rho_t(D))_*\...
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Vanishing of all higher direct images for a non-flat morphism
Let $f: X \to Y$ be a proper morphism of algebraic varieties which is not flat. Then for a line bundle $L$ on $X$, is the vanishing of all higher direct images ($R^i f_* L = 0$ for all $i \geq 0$) ...
- 2,964
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243
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Behavior of divisors under push forward and pull back
Consider a birational morphism between smooth projective varieties $f:X\to Y$. I would like to understand the behavior of push-pull/pull-push of effective divisors under $f$. I know that if $D$ is an ...
- 109
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242
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Pull and push formula for degree for non-flat morphism
Let $\varphi\colon X_1\to X_2$ be dominant proper morphism of finite degree (in particular $\dim X_1=\dim X_2$) between varieties.
Let $D \subset X_2$ be a Cartier divisor.
Is it true that $$\varphi_*...
- 219
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338
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Derived pushforward of a projection
Given two smooth projective varieties, $X,Y$, consider their derived categories $D^b(X), D^b(Y)$. Let $\mathcal{F}$ a complex of coherent sheaves in $D^b(X \times Y)$, why the derived pushforward of ...
- 61
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Convex ordering of measures that are obtained by different push-forwards of a same measure
Suppose that we have a probability measure $\rho$ which is supported on $\mathbb{R}^d$ and absolutely continuous w.r.t. the Lebesgue measure. Take two vector fields $F, G : \mathbb{R}^d \rightarrow \...
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Approximation of two densities with a single transformation
Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, ...
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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 \...
- 2,152
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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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189
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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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434
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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}$$
with $\mu_0$ being a (smooth) probability measure/density on $\mathbb{R}^d$...
- 730
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Push-forward of divisors and intersections
Let $f:X\rightarrow Y$ be a surjective finite morphism of varieties, with $X$ normal and $Y$ smooth. Let $D\subset X$ be a divisor and $C\subset Y$ a curve. Does the equality
$$C\cdot f_{*}D = f^{*}C\...
user168627
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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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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 $k$ ...
- 439
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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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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 ...
- 2,126
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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 $Z$ ...
user114666
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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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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|...
- 23.3k
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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 $...
- 8,998
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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 ...
- 5,407
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579
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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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