---
created: 2023-03-26T11:58
updated: 2024-05-03T23:22
aliases:
  - log CY pair
  - Q Fano
  - log Fano
  - log pair
  - log CY
---
# log pairs

Defined for normal varieties.

- Log pair: $(X, D)$ where $K_X + D \in \Cart_\QQ$.
- Log Fano: $X$ projective, $-K_X - D \in \Cart_\QQ^\amp$
- Log CY: $X$ projective and $K_X + D \sim_\QQ 0$
- $\QQ\dash$Fano: $(X,0)$ a [klt](log%20discrepancy.md) log Fano pair.

# Log CYs

A **log Calabi-Yau (CY) pair** is a pair $(X, D)$ with $X$ a proper variety and $D$ an effective $\QQ\dash$Cartier divisor such that the pair is log canonical and $K_X + D \sim_\QQ 0$.

## CY degenerations

A degeneration $\pi: \mcx \to \Delta$ is a **CY degeneration** if $\pi$ is proper, $K_{\mcx}\sim_\QQ 0$, and $(\mcx, \mcx_0)$ is dlt. This implies that $\mcx_t$ is a Calabi-Yau variety for $t\neq 0$ and $\mcx_0$ is a union of log CY pairs $(V_i, D_i)$. If $\mcx_t$ is a strict CY of dimension $n$, so $\pi_1 \mcx_t = 0$ and $h^i(\mcx_t, \OO_{\mcx_t}) = 0$ for $1 \leq i \leq n-1$, and $\dim \Gamma(\mcx_0) = n$, we say $\mcx$ is a [large complex structure limit](unipotent.md) or equivalently a [maximally unipotent](unipotent.md) or MUM degeneration.

If $n = 2$, Kulikov shows that $\Gamma(\mcx_0)$ is always isomorphic to a 2-sphere $S^2$. Whether $\Gamma(\mcx_0) \cong S^n$ or a quotient thereof for $n\geq 3$ more generally is an open question, posed by Kontsevich-Soibelman. It has recently been shown by Kollár-Xu that in the case of degenerations of Calabi-Yau or hyperkähler manifolds, the dual complex is always a rational homology sphere.

Why this is useful to us: [SYZ mirror symmetry](SYZ%20torus%20fibration.md).