Return to Question

Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties.

We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where the fiber over any $(A,\theta)$ is the corresponding principally polarized abelian variety.

If $(A,\theta)\in\mathcal{A}_g$. Let $V$ be a rank $r$ vector bundle on $A$ with chern classes $c_i$.

1. I was told that there is a coarse moduli space $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ parametrizing basically pairs $(A,F)$ where $A\in \mathcal{A}_g$ and $F$ is a coherent sheaf on $A$ with rank $r$ and chern classes $c_i$. That is $F\in\mathcal{M}_A(r,c_i)$. Are there any conditions for this scheme to exist?

2. Consider the fiber product of $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)\times_{\mathcal{A}_{g}}\chi_g$. Can we find a universal sheaf $\mathcal{E}$ on this product, which when restricted to the fiber over any $(A,F)\in\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ gives $F$? When can we find such a sheaf?

How do I make these notions precise?

It would be great if someone can also direct me to references of the above especially of (1) and (2). Thanks in advance!

Edits: I have made edits based on Prof. Jason Starr's comment

Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties.

We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where the fiber over any $(A,\theta)$ is the corresponding principally polarized abelian variety. Let $L$ be the universal line bundle corresponding to the universal theta divisor.

If $(A,\theta)\in\mathcal{A}_g$. Let $V$ be a rank $r$ vector bundle on $A$ with chern classes $c_i$.

1. I was told that there is a family of coarse moduli spaces $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ parametrizing basically pairs $(A,F)$ where $A\in \mathcal{A}_g$ and $F$ is a coherent $\mu_{L|_{A}}$ -semistable sheaf on $A$ with rank $r$ and chern classes $c_i$. That is $F\in\mathcal{M}_A(r,c_i)$. Are there any conditions for this space to exist?

2. Consider the fiber product of $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)\times_{\mathcal{A}_{g}}\chi_g$. Can we find a universal sheaf (or a quasi-universal sheaf) $\mathcal{E}$ on this product, which when restricted to the fiber over any $(A,F)\in\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ gives $F$? When can we find such a sheaf?

How do I make these notions precise?

It would be great if someone can also direct me to references of the above especially of (1) and (2). Thanks in advance!

Edits: I have made edits based on Prof. Jason Starr's comment

Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties.

We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where the fiber over any $(A,\theta)$ is the corresponding principally polarized abelian variety.

If $(A,\theta)\in\mathcal{A}_g$. Let $V$ be a rank $r$ vector bundle on $A$ with chern classes $c_i$.

1. I was told that there is a scheme(?) $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ parametrizing basically pairs $(A,F)$ where $A\in \mathcal{A}_g$ and $F$ is a coherent sheaf on $A$ with rank $r$ and chern classes $c_i$. That is $F\in\mathcal{M}_A(r,c_i)$. Are there any conditions for this scheme to exist?

2. Consider the fiber product of $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)\times_{\mathcal{A}_{g}}\chi_g$. Can we find a universal sheaf $\mathcal{E}$ on this product, which when restricted to the fiber over any $(A,F)\in\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ gives $F$? When can we find such a sheaf?

How do I make these notions precise?

It would be great if someone can also direct me to references of the above especially of (1) and (2). Thanks in advance!

Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties.

We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where the fiber over any $(A,\theta)$ is the corresponding principally polarized abelian variety.

If $(A,\theta)\in\mathcal{A}_g$. Let $V$ be a rank $r$ vector bundle on $A$ with chern classes $c_i$.

1. I was told that there is a coarse moduli space $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ parametrizing basically pairs $(A,F)$ where $A\in \mathcal{A}_g$ and $F$ is a coherent sheaf on $A$ with rank $r$ and chern classes $c_i$. That is $F\in\mathcal{M}_A(r,c_i)$. Are there any conditions for this scheme to exist?

2. Consider the fiber product of $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)\times_{\mathcal{A}_{g}}\chi_g$. Can we find a universal sheaf $\mathcal{E}$ on this product, which when restricted to the fiber over any $(A,F)\in\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ gives $F$? When can we find such a sheaf?

How do I make these notions precise?

It would be great if someone can also direct me to references of the above especially of (1) and (2). Thanks in advance!

Edits: I have made edits based on Prof. Jason Starr's comment

Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties.

We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where the fiber over any $(A,\theta)$ is the corresponding principally polarized abelian variety.

If $(A,\theta)\in\mathcal{A}_g$. Let $V$ be a rank $r$ vector bundle on $A$ with chern classes $c_i$.

1. I was told that there is a scheme(?) $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ parametrizing basically pairs $(A,F)$ where $A\in \mathcal{A}_g$ and $F$ is a coherent sheaf on $A$ with rank $r$ and chern classes $c_i$. That is $F\in\mathcal{M}_A(r,c_i)$. Are there any conditions for this scheme to exist?

2. Consider the fiber product of $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)\times_{\mathcal{A}_{g}}\chi_g$. Can we find a Poincare line bundle $\mathcal{E}$ on this product, which when restricted to the fiber over any $(A,F)\in\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ gives $F$? When can we find such a line bundle?

How do I make these notions precise?

It would be great if someone can also direct me to references of the above especially of (1) and (2). Thanks in advance!

Let $\mathcal{A}_g$ be the moduli space (stack?) of $g$-dimensional principally polarized abelian varieties.

We have the universal family of abelian varieties $\chi_g\rightarrow \mathcal{A}_g$, where the fiber over any $(A,\theta)$ is the corresponding principally polarized abelian variety.

If $(A,\theta)\in\mathcal{A}_g$. Let $V$ be a rank $r$ vector bundle on $A$ with chern classes $c_i$.

1. I was told that there is a scheme(?) $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ parametrizing basically pairs $(A,F)$ where $A\in \mathcal{A}_g$ and $F$ is a coherent sheaf on $A$ with rank $r$ and chern classes $c_i$. That is $F\in\mathcal{M}_A(r,c_i)$. Are there any conditions for this scheme to exist?

2. Consider the fiber product of $\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)\times_{\mathcal{A}_{g}}\chi_g$. Can we find a universal sheaf $\mathcal{E}$ on this product, which when restricted to the fiber over any $(A,F)\in\mathcal{M}_{\mathcal{A}_{g}}(r,c_i)$ gives $F$? When can we find such a sheaf?

How do I make these notions precise?

It would be great if someone can also direct me to references of the above especially of (1) and (2). Thanks in advance!