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jmc
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Chow group of a (particular) motive [+ reference request]

I have two (not unrelated) questions. Let me first give a short introduction.

Introduction

For a general overview of the setup I refer to the introduction (§1) of [Zhang].

Let $k$ be a number field or a function field of a curve. Let $X/k$ be a curve, and assume that $X$ has a semistable model. Let $Y$ be the triple product $X^{3}$.

In [§5.1 (pp. 77v.), Zhang] he defines a motive $M$ as the kernel of the map $$ \bigwedge^{3} H^{1}(X)(2) \to H^{1}(X)(1) \colon a \wedge b \wedge c \mapsto a(b \smile c) + b(a \smile c) + c(a \smile b). $$ I think I have figured out what he means, and why this is a Chow motive.

A little further, he defines $\text{Ch}(M)$ as a subgroup of the kernel of the cycle map $\text{Ch}^{2}(Y) \to H^{4}(Y)$. It is the subgroup of elements $z$ satisfying

  • $z$ is stable under the action of the symmetric group $S_{3}$ on $Y$;
  • the push-forward $\text{pr}_{12,*} z$ equals $0$, where $\text{pr}_{12}$ is the projection on the first two coordinates;
  • if $i$ denotes the embedding $X^{2} \to Y \colon (x,y) \mapsto (x,x,y)$, and $\text{pr}_{2} \colon X^{2} \to X$ the projection on the second coordinate, then $\text{pr}_{2,*}i^{*} z = 0$.

Questions

Q1. Why this definition?

Q2.a. The notation suggests to me that there is a more general definition of $\text{Ch}(M)$, for Chow motives $M$. Is this true?

Q2.b. If so, can someone give me pointers (reference request) to a definition of $\text{Ch}(M)$, since all my Googles for "Chow group motive" give me results that use Chow groups to define motives (and I am actually not surprised to get those results.

I think that if $M$ is the motive corresponding to a smooth projective variety $X$, then $\text{Ch}(M)$ should equal $\text{Ch}(X)$. But what about the other motives?

References

[Zhang] Shou-Wu Zhang. “Gross–Schoen Cycles and Dualising Sheaves”. url: http://arxiv.org/0812.0371.pdf.

Chow group of a (particular) motive

I have two (not unrelated) questions. Let me first give a short introduction.

Introduction

For a general overview of the setup I refer to the introduction (§1) of [Zhang].

Let $k$ be a number field or a function field of a curve. Let $X/k$ be a curve, and assume that $X$ has a semistable model. Let $Y$ be the triple product $X^{3}$.

In [§5.1 (pp. 77v.), Zhang] he defines a motive $M$ as the kernel of the map $$ \bigwedge^{3} H^{1}(X)(2) \to H^{1}(X)(1) \colon a \wedge b \wedge c \mapsto a(b \smile c) + b(a \smile c) + c(a \smile b). $$ I think I have figured out what he means, and why this is a Chow motive.

A little further, he defines $\text{Ch}(M)$ as a subgroup of the kernel of the cycle map $\text{Ch}^{2}(Y) \to H^{4}(Y)$. It is the subgroup of elements $z$ satisfying

  • $z$ is stable under the action of the symmetric group $S_{3}$ on $Y$;
  • the push-forward $\text{pr}_{12,*} z$ equals $0$, where $\text{pr}_{12}$ is the projection on the first two coordinates;
  • if $i$ denotes the embedding $X^{2} \to Y \colon (x,y) \mapsto (x,x,y)$, and $\text{pr}_{2} \colon X^{2} \to X$ the projection on the second coordinate, then $\text{pr}_{2,*}i^{*} z = 0$.

Questions

Q1. Why this definition?

Q2.a. The notation suggests to me that there is a more general definition of $\text{Ch}(M)$, for Chow motives $M$. Is this true?

Q2.b. If so, can someone give me pointers to a definition of $\text{Ch}(M)$, since all my Googles for "Chow group motive" give me results that use Chow groups to define motives (and I am actually not surprised to get those results.

I think that if $M$ is the motive corresponding to a smooth projective variety $X$, then $\text{Ch}(M)$ should equal $\text{Ch}(X)$. But what about the other motives?

References

[Zhang] Shou-Wu Zhang. “Gross–Schoen Cycles and Dualising Sheaves”. url: http://arxiv.org/0812.0371.pdf.

Chow group of a (particular) motive [+ reference request]

I have two (not unrelated) questions. Let me first give a short introduction.

Introduction

For a general overview of the setup I refer to the introduction (§1) of [Zhang].

Let $k$ be a number field or a function field of a curve. Let $X/k$ be a curve, and assume that $X$ has a semistable model. Let $Y$ be the triple product $X^{3}$.

In [§5.1 (pp. 77v.), Zhang] he defines a motive $M$ as the kernel of the map $$ \bigwedge^{3} H^{1}(X)(2) \to H^{1}(X)(1) \colon a \wedge b \wedge c \mapsto a(b \smile c) + b(a \smile c) + c(a \smile b). $$ I think I have figured out what he means, and why this is a Chow motive.

A little further, he defines $\text{Ch}(M)$ as a subgroup of the kernel of the cycle map $\text{Ch}^{2}(Y) \to H^{4}(Y)$. It is the subgroup of elements $z$ satisfying

  • $z$ is stable under the action of the symmetric group $S_{3}$ on $Y$;
  • the push-forward $\text{pr}_{12,*} z$ equals $0$, where $\text{pr}_{12}$ is the projection on the first two coordinates;
  • if $i$ denotes the embedding $X^{2} \to Y \colon (x,y) \mapsto (x,x,y)$, and $\text{pr}_{2} \colon X^{2} \to X$ the projection on the second coordinate, then $\text{pr}_{2,*}i^{*} z = 0$.

Questions

Q1. Why this definition?

Q2.a. The notation suggests to me that there is a more general definition of $\text{Ch}(M)$, for Chow motives $M$. Is this true?

Q2.b. If so, can someone give me pointers (reference request) to a definition of $\text{Ch}(M)$, since all my Googles for "Chow group motive" give me results that use Chow groups to define motives (and I am actually not surprised to get those results.

I think that if $M$ is the motive corresponding to a smooth projective variety $X$, then $\text{Ch}(M)$ should equal $\text{Ch}(X)$. But what about the other motives?

References

[Zhang] Shou-Wu Zhang. “Gross–Schoen Cycles and Dualising Sheaves”. url: http://arxiv.org/0812.0371.pdf.

Source Link
jmc
jmc
  • 5.5k
  • 27
  • 60

Chow group of a (particular) motive

I have two (not unrelated) questions. Let me first give a short introduction.

Introduction

For a general overview of the setup I refer to the introduction (§1) of [Zhang].

Let $k$ be a number field or a function field of a curve. Let $X/k$ be a curve, and assume that $X$ has a semistable model. Let $Y$ be the triple product $X^{3}$.

In [§5.1 (pp. 77v.), Zhang] he defines a motive $M$ as the kernel of the map $$ \bigwedge^{3} H^{1}(X)(2) \to H^{1}(X)(1) \colon a \wedge b \wedge c \mapsto a(b \smile c) + b(a \smile c) + c(a \smile b). $$ I think I have figured out what he means, and why this is a Chow motive.

A little further, he defines $\text{Ch}(M)$ as a subgroup of the kernel of the cycle map $\text{Ch}^{2}(Y) \to H^{4}(Y)$. It is the subgroup of elements $z$ satisfying

  • $z$ is stable under the action of the symmetric group $S_{3}$ on $Y$;
  • the push-forward $\text{pr}_{12,*} z$ equals $0$, where $\text{pr}_{12}$ is the projection on the first two coordinates;
  • if $i$ denotes the embedding $X^{2} \to Y \colon (x,y) \mapsto (x,x,y)$, and $\text{pr}_{2} \colon X^{2} \to X$ the projection on the second coordinate, then $\text{pr}_{2,*}i^{*} z = 0$.

Questions

Q1. Why this definition?

Q2.a. The notation suggests to me that there is a more general definition of $\text{Ch}(M)$, for Chow motives $M$. Is this true?

Q2.b. If so, can someone give me pointers to a definition of $\text{Ch}(M)$, since all my Googles for "Chow group motive" give me results that use Chow groups to define motives (and I am actually not surprised to get those results.

I think that if $M$ is the motive corresponding to a smooth projective variety $X$, then $\text{Ch}(M)$ should equal $\text{Ch}(X)$. But what about the other motives?

References

[Zhang] Shou-Wu Zhang. “Gross–Schoen Cycles and Dualising Sheaves”. url: http://arxiv.org/0812.0371.pdf.