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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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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1 vote
0 answers

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 ...
  • 655
0 votes
1 answer

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 ...
  • 655
1 vote
0 answers

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 $...
2 votes
0 answers

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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7 votes
0 answers

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 ...
  • 4,184
1 vote
1 answer

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 ...
8 votes
1 answer

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^...
  • 2,671
4 votes
1 answer

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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6 votes
1 answer

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 ...
0 votes
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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 $f$ ...
3 votes
1 answer

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 ...
5 votes
0 answers

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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9 votes
1 answer

About Riemann-Roch without denominators

The Riemann-Roch without denominators can be expressed as follows: Let $f: X\rightarrow Y$ be a closed embedding of quasi-projective smooth $k$-varieties of codimension $d$ for some field $k$. Let $E$ ...
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1 vote
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Pushforward of tensor product of holomorphic vector bundles

Let $M$ be a complex manifold, $\Sigma$ a Riemann surface of genus $g$, and $E, F$ holomorphic vector bundles on $M\times\Sigma$. If $\pi: M\times \Sigma\to M$ is the projection map, is it possible to ...
  • 373
21 votes
1 answer

Günter Tamme's course "Arakelov theory and Grothendieck-Riemann-Roch"

On chapter III.4 ("Metrized $\mathcal{o}$-modules") of this book on algebraic number theory, Neukirch credits his treatment of the theory of finitely generated $\mathcal{o}$-modules to the course "...
  • 17.1k
3 votes
0 answers

Equivariant Riemann-Roch on DM stacks?

Does an equivariant version of (Toen)-Riemann-Roch theorem hold say over a smooth Deligne-Mumford stack $X$ over the complex numbers? Any references that state this explicitely? Are there formulas ...
  • 21.7k
7 votes
0 answers

Riemann-Roch for curves over Dedekind domains and base-change for modular forms

In p-adic properties of modular schemes and modular forms Katz formulates the following base change theorem as Theorem 1.7.1 Let $n\geq 3$ and $\overline{\mathcal{M}}_n$ be the compactified moduli ...
10 votes
2 answers

Faltings-Riemann-Roch Theorem

I found the famous Faltings book ``Lectures on arithmetic Riemann-Roch theorem". In the book, very analytic techniques such as Garding inequality or heat kernel are explained. I have no idea where ...
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8 votes
5 answers

$\lambda$-ring structure defined for a graded ring in Fulton–Lang's book

Given a commutative ring $A$ with unity, Grothendieck used universal polynomials to define a special $\lambda$-ring structure on $\Lambda(A):=1+t\:A[[t]]$. Suppose $A$ is graded, say $A=\bigoplus_{i=0}...
8 votes
1 answer

Serre duality and Hirzebruch-Riemann-Roch in the non-projective case

Serre duality and the Hirzebruch-Riemann-Roch formula are usually stated for $X$ a smooth projective algebraic variety. Do you know of a reference which proves these results for $X$ smooth and proper? ...