Conjecture on finite-index geometric automorphism groups of symplectic 4-manifolds

Let $M$ be a smooth symplectic $4$-manifold, let $\Gamma_M$ denote its intersection form, let $A(\Gamma_M)$ be the group of automorphisms of $\Gamma_M$, and let $D(M)$ be the geometric automorphism group of $M$. A Kähler Calabi–Yau surface is a Kähler surface whose canonical class vanishes in real cohomology.

Finite-index geometric automorphism conjecture. Suppose $M$ has symplectic structures and $A(\Gamma_M)$ is infinite. Then $D(M)$ is of finite index if and only if $M$ is either a Kähler Calabi–Yau surface or

$$\mathbb C\mathbb P^2\# n\overline{\mathbb C\mathbb P}^2$$

with $2\leq n\leq 9$.

The conjecture characterizes when the geometric automorphism group has finite index in the full automorphism group of the intersection form. The paper notes that the assertion is known for Kähler Calabi–Yau surfaces and discusses Kodaira–Thurston manifolds as non-Kähler examples where the geometric automorphism group has infinite index; the complete classification remains open.