A cone conjecture for log Calabi-Yau surfaces

TL;DR

This work formulates and proves a Morrison‑style cone conjecture for log Calabi–Yau surfaces with singular boundary. By isolating a distinguished split representative $(Y_e,D_e)$ in each deformation type and analyzing the automorphism and monodromy actions on nef‑effective cones, the authors establish rational polyhedral fundamental domains for $Aut(Y_e,D_e)/K$ on $Nef^e(Y_e)$ and for $Adm$ on $Nef^e(Y_{gen})$. For boundary length $n\le 6$, they give explicit descriptions of the nef cone and construct new infinite families of Mori Dream Spaces. The results connect to cusp–singularity deformation theory and mirror symmetry, offering insight into both the Morrison cone conjecture and Looijenga’s geometric framework via the nef‑cone geometry of split log Calabi–Yau surfaces.

Abstract

We consider log Calabi-Yau surfaces $(Y, D)$ with singular boundary. In each deformation type, there is a distinguished surface $(Y_e,D_e)$ such that the mixed Hodge structure on $H_2(Y \setminus D)$ is split. We prove that (1) the action of the automorphism group of $(Y_e,D_e)$ on its nef effective cone admits a rational polyhedral fundamental domain; and (2) the action of the monodromy group on the nef effective cone of a very general surface in the deformation type admits a rational polyhedral fundamental domain. These statements can be viewed as versions of the Morrison cone conjecture for log Calabi--Yau surfaces. In addition, if the number of components of $D$ is $\le 6$, we show that the nef cone of $Y_e$ is rational polyhedral and describe it explicitly. This provides infinite series of new examples of Mori Dream Spaces.

Figures (12)

• Figure 5.1: This drawing shows arbitrarily many blowups of the surface $(\bar{Y}, \bar{D})$. The blowup at each point creates a chain of one $(-1)$-curve, intersecting the boundary component at one point, followed by some number $(-2)$-curves that lead to a common "central region". The center consists of additional curves.
• Figure 5.2: This drawing depicts a general "chain" that forms after arbitrarily many blowups. For a boundary of length $n$, the value of $i$ ranges from 1 to $n$.
• Figure 5.3: The drawing on the far left shows $(\bar{Y}, \bar{D})$ before any blowups. The middle drawing shows the first three blowups at the point $q$, and the figure on the right depicts $(Y, D)$ after arbitrarily many, say $p_{1}$, blowups at $q$.
• Figure 5.4: This drawing shows the curves $\bar{F}_{1}$ and $\bar{F}_{2}$ in the case $n=2$.
• Figure 5.5: The blowup of $(\bar{Y}, \bar{D})$ at the two points $q_{1}$ and $q_{2}$, a total of $p_{1}$ and $p_{2}$ times respectively, results in $(Y, D)$.

Theorems & Definitions (61)

• Theorem 1.1
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Definition 2.8
• Definition 2.9
• Definition 2.10
• Proposition 2.11
• Lemma 2.12