# Elliptic curves in hyper-K\"ahler varieties

## TL;DR

This paper investigates the moduli space of elliptic curves in Fano and hyper-K"ahler varieties, revealing its structure, enumerating elliptic curves with fixed invariants, and connecting counts to Gromov--Witten invariants and modular forms.

## Contribution

It provides a detailed description of the moduli space of elliptic curves in Fano and hyper-K"ahler varieties, including explicit counts and formulas relating to Gromov--Witten invariants and modular forms.

## Key findings

- The moduli space of elliptic curves in a general Fano variety is a smooth curve of genus 631.
- A precise count of 3780 elliptic curves with fixed $j$-invariant in a general Fano.
- Explicit formulas for elliptic curve counts in hyper-K"ahler varieties of $K3[2]$-type in terms of modular forms.

## Abstract

We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic fourfold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely $3780$ elliptic curves of minimal degree with fixed (general) $j$-invariant. More generally, we express (modulo a transversality result) the enumerative count of elliptic curves of minimal degree in hyper-K\"ahler varieties with fixed $j$-invariant in terms of Gromov--Witten invariants. In $K3[2]$-type this leads to explicit formulas of these counts in terms of modular forms.