Questions tagged [grothendieck-riemann-roch]
The Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces.
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Is there Riemann-Roch without denominators for complex manifolds?
Let $X \subset Y$ be an inclusion of compact complex (possibly Kähler) manifolds. I'm wondering if "Riemann-Roch without Denominators" [1, Thm 15.3] holds in that situation. The statement is
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Simple Grothendieck-Riemann-Roch computation with relative Todd class
$\DeclareMathOperator\Tot{Tot}\DeclareMathOperator\ch{ch}\DeclareMathOperator\td{td}\DeclareMathOperator\ker{ker}\DeclareMathOperator\rk{rk}$I was wondering if the following is correct: Let $X=\Tot(L)$...
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What is the Todd class *really*?
My question is about how to think about the Todd class.
Usually this is presented via Grothendieck Riemann Roch (GRR): if $X$ is a smooth projective scheme over a field $\mathbf{C}$, the chern ...
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For a given $E\to X$ find $\xi\to \mathbb{P}E^*$ such that $\pi_*\operatorname{ch}(\xi)=\operatorname{ch}(E)$
$\DeclareMathOperator\ch{ch}$This is a specification of the question For given bundle $E\to X$ find $\xi\to Y$ such that $\pi_*ch(\xi)=ch(E)$.
Let $E\to X$ be a holomorphic vector bundle over a ...
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For given bundle $E\to X$ find $\xi\to Y$ such that $\pi_*ch(\xi)=ch(E)$
Let $\pi:X\to Y$ a projective morphism and $F\to X$ a vector bundle. The Grothendieck-Riemann-Roch theorem states that
$$\pi_*(ch(F)td(\pi))=ch(\pi_!F)$$
where $td(\pi)$ denotes the relative Todd ...
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Use of Porteus‘ formula in a paper of Beauville
In “Sur la cohomologie de certains espaces de modules de fibrés vectoriels”, Beauville calculates the Chern class of the diagonal $\Delta$ of the moduli space $M$ of certain stable bundles on a curve $...
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Generic rank of proper pushforward of the trivial line bundle
Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ ...
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Grothendieck Riemann Roch is abelian localisation on loop spaces
Abelian localisation says approximately that for a proper equivaraint map $f:X\to Y$ between schemes with a $\mathbf{G}_m$ action, the pushforward on cohomology $f_*\omega$ can be computed by the ...
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Can the dimension of Hom space between vector bundles on an algebraic curve predicted by Riemann-Roch type formula be the minimal possible?
Let us study vector bundles $E$ and $F$ on a smooth projective curve $C$. There is a Riemann-Roch type formula for the Euler characteristic $\chi(E,F)=dim\, Hom(E,F)-dim\, Ext^1(E,F)$ in terms of ...
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Original reference for Adams Riemann-Roch theorem
Let $f\colon Y\to X$ be a proper morphism between smooth quasiprojective $k$-algebraic varieties. Denote by $\psi^j$ the $j$-th Adams operation on the Grothendieck group of vector bundles and $\theta^...
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Can we move curves which are members of very ample systems?
Let us take the second degree Hirzebruch surface $\mathbb{F}_2$ which is a holomorphic $\mathbb{CP}^1$ bundle over $\mathbb{CP}^1$ having sections of self intersections $+2$ and $-2$. Let me denote ...
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Reflection-invariant monomial ideals and Alexander duality
First we give some definitions from Section 3 of the paper Monomials, Binomials, and Riemann-Roch by Manjunath and Sturmfels and then we restate a claim from that paper offered without proof. Finally ...
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$ch(L f^*\epsilon)$
I know the famous principle (or theorem depending on how you define) of Chern classes for locally free sheaves and a morphism $f: X'\rightarrow X$,
$ch(f^* \epsilon)=f^* ch(\epsilon)$.
But if $f$ ...
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A question on Grothendieck Riemann Roch
As an exercise for myself I wanted to check GRR in the following situation. Consider $P:X \rightarrow B$ to be an Weierstrass elliptic fibration with a section, and $X\times_B X$ be the fiber product ...
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Local family index theorem, but with Chern class?
Let $\pi:X\to B$ be a proper submersion with spin fibers, and $E\to X$ a Hermitian vector bundle with a unitary connection $\nabla$. Then the local family index theorem for spin Dirac operator twisted ...
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