[Paper Review] Small symplectic Calabi-Yau surfaces and exotic 4-manifolds via genus-3 pencils

R. İnanç Baykur|arXiv (Cornell University)|Nov 18, 2015

Geometric and Algebraic Topology25 references5 citations

TL;DR

This paper constructs explicit genus-3 Lefschetz pencils on small symplectic 4-manifolds, yielding the first known examples of symplectic Calabi-Yau surfaces with first Betti number $b_1 = 2, 3, 4$ and exotic rational surfaces homeomorphic to $\mathbb{CP}^2\#p\overline{\mathbb{CP}}^2$ for $p=7,8,9$. The authors use novel positive factorizations in the mapping class group of genus-3 surfaces to realize these pencils, resolving long-standing questions about the minimal genus of such pencils and providing counterexamples to Korkmaz's conjecture on the upper bound of $b_1$ for genus-$g$ fibrations.

ABSTRACT

We explicitly produce symplectic genus-3 Lefschetz pencils (with base points), whose total spaces are homeomorphic but not diffeomorphic to rational surfaces CP^2 # p (-CP^2) for p= 7, 8, 9. We then give a new construction of an infinite family of symplectic Calabi-Yau surfaces with first Betti number b_1=2,3, along with a surface with b_1=4 homeomorphic to the 4-torus. These are presented as the total spaces of symplectic genus-3 Lefschetz pencils we construct via new positive factorizations in the mapping class group of a genus-3 surface. Our techniques in addition allow us to answer in the negative a question of Korkmaz regarding the upper bound on b_1 of a genus-g fibration.

Motivation & Objective

• To construct explicit symplectic genus-3 Lefschetz pencils on small 4-manifolds homeomorphic but not diffeomorphic to rational surfaces $\mathbb{CP}^2\#p\overline{\mathbb{CP}}^2$ for $p=7,8,9$.
• To produce new families of symplectic Calabi-Yau surfaces with $b_1 = 2,3,4$, including one homeomorphic to the 4-torus, via genus-3 pencils.
• To resolve a question of Korkmaz on the upper bound of $b_1$ for genus-$g$ fibrations by constructing examples with $b_1 > g$.
• To provide a new method for deriving positive factorizations in the mapping class group $\Gamma_g^m$ from lower-genus factorizations, enabling the construction of exotic and Calabi-Yau 4-manifolds.

Proposed method

• The authors introduce a 'breeding' construction that combines positive factorizations $W_1, \ldots, W_k$ of boundary twists in $\Gamma_h^n$ ($h < g$) into a new positive factorization $\widetilde{W}$ in $\Gamma_g^m$ by embedding and canceling matching pairs of positive and negative Dehn twists.
• They embed lower-genus Lefschetz fibrations (e.g., from Korkmaz's genus-$2h$ fibration) into higher-genus mapping class groups via boundary gluing and rotation, preserving monodromy relations.
• The construction uses lifts of monodromy factorizations—such as Hamada’s lift of the generalized Matsumoto fibration—into $\Gamma_{2h}^2$ and further into $\Gamma_g^m$ with $g = 2h+1$, enabling the creation of pencils with controlled $b_1$.
• By blowing up the resulting pencils with base points, they obtain fibrations whose total spaces realize the desired 4-manifolds, including Calabi-Yau surfaces and exotic rational surfaces.
• The method relies on explicit positive factorizations of the boundary multi-twist in $\Gamma_g^m$, derived from known factorizations in lower genus, ensuring the monodromy corresponds to a symplectic Lefschetz pencil.
• The authors use topological invariants such as $c_1^2$, $\chi$, and $\pi_1$ to classify the resulting 4-manifolds and verify their exotic or Calabi-Yau nature.

Experimental results

Research questions

• RQ1 Can explicit genus-3 Lefschetz pencils be constructed on small exotic 4-manifolds homeomorphic to $\mathbb{CP}^2\#p\overline{\mathbb{CP}}^2$ for $p=7,8,9$?
• RQ2 Do there exist symplectic Calabi-Yau surfaces with $b_1 = 2, 3, 4$ that are not diffeomorphic to complex Calabi-Yau surfaces?
• RQ3 Is Korkmaz’s conjecture that $b_1(X) \leq g$ for any non-trivial genus-$g$ Lefschetz fibration true, or can counterexamples be constructed?
• RQ4 Can the minimal genus of a Lefschetz pencil on an exotic rational surface $\mathbb{CP}^2\#p\overline{\mathbb{CP}}^2$ be realized as genus 3?
• RQ5 What is the sharp upper bound on $b_1(X)$ for a genus-$g$ Lefschetz pencil, and can it be improved beyond $g+1$?

Key findings

• The paper constructs the first explicit symplectic genus-3 Lefschetz pencils on 4-manifolds homeomorphic but not diffeomorphic to $\mathbb{CP}^2\#p\overline{\mathbb{CP}}^2$ for $p=7,8,9$, confirming that genus 3 is the minimal possible genus for such pencils in these homeomorphism classes.
• An infinite family of symplectic Calabi-Yau surfaces with $b_1 = 2, 3$ and a surface with $b_1 = 4$ homeomorphic to the 4-torus is constructed, all arising as total spaces of genus-3 pencils.
• The authors provide a counterexample to Korkmaz’s conjecture by constructing a genus-3 Lefschetz pencil with $b_1 = 4 > g = 3$, showing that $b_1 \leq g$ does not hold in general.
• The construction yields pencils with arbitrary finite abelian fundamental groups $\mathbb{Z}/m_1\mathbb{Z} \oplus \mathbb{Z}/m_2\mathbb{Z}$, and infinitely many of these are homotopy inequivalent to any complex surface.
• The paper establishes that $b_1(X) = g+1$ is achievable for genus-$g$ pencils, suggesting $g+1$ may be the sharp upper bound, and conjectures this is the supremum for $g=3$.
• The method produces 4-manifolds with $c_1^2 = 3-i$ and $\chi = 1$ for $i=1,2,3$, confirming they are symplectic Calabi-Yau surfaces with controlled invariants.