---
date: 2022-05-02 17:46
modification date: Monday 2nd May 2022 17:46:25
title: "hodge bundle"
aliases: [hodge bundle]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# hodge bundle

- Take the universal family $\mce_g \mapsvia{\pi} \Mg$
- Take the relate dualizing sheaf $\omega_{\mce_g/\Mg}$
- Define the Hodge bundle as $\Lambda_g \da \pi_* \omega_{\mce_g/\mcm_g} \in \QCoh \Mg$.
- The fibers $\Lambda_{g, p}$ for $p \da [C]$ a curve class are $H^{1, 0}(C) = H^0(C; \Omega_{C}^1)$.

## General construction

![](attachments/Pasted%20image%2020220502174627.png)

- Take $\mcx \to B$ a family of smooth curves
- Write $\omega_{\mcx/B}$ for the relative dualizing sheaf
- Take $\mcl\in \Pic(\mcx)$ and $M\in \Pic(\mcx)$ with $\deg M \gg 0$.
- Set $\det_\pi(\mcl) \da (\det \pi_* (\mcl \tensor M) ) \tensor (\det \pi_* ( \mcl\tensor \ro{M}{\sigma = 0}) )\inv$ where $\sigma\in H^0(M)$ such that $\ro{\sigma}{\mcx_p}\not\equiv 0$ for any fiber $\mcx_p$.
- Define $\lambda_n(B) \da \det_\pi(\omega^n_{\mcx/B})$.
- The **Hodge bundle** is $\lambda_1(B)$.