Does an Exotic CP2 With a Finite Order Symplectomorphism Exist?

The existence question becomes twelve filling questions

An exotic CP2 here means a symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP2. The paper asks a sharper version of the classical problem: does such a manifold exist if it must also carry a nontrivial finite order symplectomorphism? The answer is not settled by a construction or a nonexistence proof. Instead, the paper proves an equivalence that turns the question into a finite list of contact-geometry tests.

The reduction is exact. There are precisely 12 plumbed manifolds Y_Γ, one for each weighted graph Γ in the paper's Figure 1, and each carries a canonical contact structure ξ_LM called the Li-Mak contact structure. The paper's Theorem 1.1 states that an exotic CP2 with a nontrivial finite order symplectomorphism exists if and only if one of these 12 contact plumbed manifolds (Y_Γ, ξ_LM) admits a certain specific strong symplectic filling W_Γ. This is an if-and-only-if statement, not a probabilistic or partial result.

Why the extra symmetry assumption changes the method

Without symmetry, a hypothetical symplectic exotic CP2 is detected by having a positive canonical class, meaning c1(K)·[ω] > 0, a characterization the paper attributes to Taubes. The symmetry assumption gives a different route: the paper approaches existence through the minimal resolution of the symplectic quotient orbifold, an idea it credits to Keum in the algebraic setting for constructing a fake projective plane.

The symplectic implementation is deliberately different from the algebraic one. The paper relies on pseudoholomorphic curves together with results from symplectic geometry, contact geometry, and gauge theory. The 12 plumbed manifolds are the technical center of that route: each graph Γ determines a symplectic plumbing P_Γ and a symplectic divisor D_Γ with normal crossings, and Y_Γ is the oriented boundary −∂P_Γ. The symplectic structure on P_Γ depends on the areas of the surfaces in D_Γ, which are not part of the graph data.

The filling conditions that decide existence

The filling W_Γ is not an arbitrary strong symplectic filling. Theorem 1.1 imposes four conditions. Condition (i) requires a surjective homomorphism from π1(W_Γ) to Z_p extending the natural homomorphism ρ: π1(Y_Γ) → Z_p, with a p-fold covering W̃_Γ → W_Γ. Condition (ii) requires H1(W_Γ, Q) = 0 and that i*: H2(Y_Γ, Q) → H2(W_Γ, Q) is an isomorphism. Condition (iii) requires H1(W̃_Γ, Q) = 0. Condition (iv) is a pair of surjectivity statements relating the kernels of the maps from π1(Ỹ_Γ) to π1(P̃_Γ) and to π1(W̃_Γ).

These conditions separate three different conclusions. If W_Γ obeys (i) and (ii), the glued manifold M_Γ = W̃_Γ ∪ P̃_Γ has a pseudofree symplectic Z_p-action, positive canonical class, Euler characteristic χ(M_Γ) = 3, and signature Sign(M_Γ) = 1, placing it on the Bogomolov-Miyaoka-Yau line. It is a Q-homology CP2 if and only if (iii) also holds, and it is an exotic CP2 if and only if both (iii) and (iv) hold. The distinction matters: a Q-homology CP2 is not automatically an exotic CP2, and the paper keeps those thresholds separate.

The converse direction starts from a symplectic action

Theorem 1.1 also runs backward. Suppose there is a symplectic Q-homology CP2 with positive canonical class, denoted M, admitting a pseudofree symplectic Z_p-action for an odd prime p. Then there is a weighted graph Γ in Figure 1, with Γ = Γ_p or Γ_{p,i} for some i, such that the corresponding contact plumbed manifold (Y_Γ, ξ_LM) admits a strong symplectic filling W_Γ obeying conditions (i) through (iii), and M can be identified with the glued manifold M_Γ. If M is in fact an exotic CP2, then condition (iv) is also obeyed.

This converse is what makes the 12-graph list exhaustive rather than illustrative. Any pseudofree symplectic Z_p-action of odd prime order on a symplectic Q-homology CP2 with positive canonical class must fall into one of the 12 cases, and the minimal resolution of the quotient orbifold then contains a symplectic divisor whose plumbing graph is Γ. The paper notes that a symplectic Z_p-action of prime order on an exotic CP2 must be pseudofree with p > 2, so the odd-prime setting is not an arbitrary restriction.

The technical core fixes orders and local representations

The main technical result is a complete determination of the order and local representations of a pseudofree symplectic Z_p-action for an odd prime p on a rational homology CP2 with positive canonical class. The paper's classification work runs through the possible primes and eliminates cases one by one using lemmas about rotation numbers, δ-invariants, multiplicities, and characteristic sequences. For example, the text eliminates p = 41, p = 37, p = 31, p = 29, and p = 23 through case analyses of singularities of an invariant curve C, with the remaining possibilities recorded in Theorem 1.3.

The paper also draws a consequence about fake projective planes. Its result suggests that potentially there are symplectic Z_p-actions which are not finite automorphisms of a fake projective plane. That is stated as a suggestion, not as a completed classification of all such actions, and it should be read as a direction opened by the order-and-representation result rather than as a separate existence theorem.

A reverse-engineering construction on the BMY line

Beyond the equivalence, the paper introduces a construction that works under minimal assumptions on the symplectic fillings of the 12 contact plumbed manifolds. The construction produces a symplectic 4-manifold on the Bogomolov-Miyaoka-Yau line through a reverse-engineering process. In the forward direction of Theorem 1.1, the glued manifold M_Γ already has χ(M_Γ) = 3 and Sign(M_Γ) = 1, which is what places it on that line.

The construction is conditional. It does not by itself exhibit a filling of any of the 12 contact plumbed manifolds, and therefore does not by itself produce an exotic CP2. What it does is show what follows if the required fillings exist, and it gives the nonexistence side a concrete target: if none of the 12 contact plumbed manifolds admits a strong symplectic filling obeying (i) through (iv), then there is no symplectic exotic CP2 with a nontrivial finite order symplectomorphism.

What would settle the question

The paper's own priority is the finite list of 12 contact plumbed manifolds. A construction of an exotic CP2 with a nontrivial finite order symplectomorphism would follow from finding one graph Γ in Figure 1 whose (Y_Γ, ξ_LM) admits a strong symplectic filling W_Γ obeying (i) through (iv). A nonexistence proof would follow from showing that for every one of the 12 graphs, no strong symplectic filling obeys (i) through (iv). The paper states both directions explicitly.

The most decision-relevant distinction for a reader is therefore not whether the paper proves existence or nonexistence, but which of the four filling conditions a candidate filling satisfies. Conditions (i) and (ii) already give a positive canonical class and a manifold on the BMY line; adding (iii) upgrades it to a Q-homology CP2; adding (iv) is what upgrades it to an exotic CP2. Corollary 4.6 records a simpler sufficient route: if some Γ in Figure 1 has a strong symplectic filling W_Γ with i*: H2(Y_Γ, Q) → H2(W_Γ, Q) an isomorphism and an isomorphism ρ̄: π1(W_Γ) → Z_p satisfying ρ̄ ∘ i* = ρ, then a symplectic exotic CP2 with a nontrivial finite order symplectomorphism exists. That corollary, not the general equivalence, is the sharpest single threshold the paper offers for a positive answer.

Sources used in this answer