34 problems

Stipsicz's conjecture. Every fiber-sum indecomposable Lefschetz fibration admits an exceptional section. Equivalently, every Lefschetz fibration should be a fiber-sum of blown-up p…

Let $f:X\to\mathbb{S}^2$ be a relatively minimal genus-$g$ Lefschetz fibration, where $g\geq 2$, and let its slope be … Here $K_f^2(X)=c_1^2(X)+8(g-1)$ and…

Let $f:X\to\mathbb{S}^2$ be a relatively minimal Lefschetz fibration, meaning that no fiber contains a $(-1)$-sphere. It is decomposable if it can be expressed as a fiber sum of tw…

For integers $p,q,p',q'$ defining nontrivial torus knots with $g(p,q)=g(p',q')$, let $F(p,q;p',q')$ be the fiber sum along a regular fiber of the rational surfaces $Z(p,q)$ and…

Let a hyperelliptic SLF be a hyperelliptic smooth Lefschetz fibration of a four-manifold. Assume that all its singular fibers are irreducible. Holomorphicity conjecture. Every such…

A smooth Lefschetz fibration is a smooth fibration of an oriented 4-manifold over an oriented 2-manifold whose singular points have the local holomorphic form of two complex lines…

BLF achirality conjecture. Not all closed, smooth, oriented $4$-manifolds admit BLFs; in particular, it is possible that $\mathbb{C}P^2$ is necessarily achiral as a fibration, even…

Let $Y$ be a closed oriented $3$-manifold equipped with a fibration $f:Y\to S^1$ of fiber genus at least $2$, let $h\in H_1(Y;\mathbb{Z})$, let $c\in H^2(Y;\mathbb{R})$ evaluate po…

Let $(X,\omega)$ be a symplectic 4-manifold and let $\alpha\in H^2(X;\mathbb{Z})$. Let $f\colon X'\to S^2$ be a Lefschetz fibration obtained from a sufficiently high-degree Lefsche…

Signature conjecture. The signature of this Lefschetz fibration is

Let $S^{[3]}$ and $S^{[4]}$ be the strata appearing in the semistable degeneration, and let $[N]$ denote the cohomology class of the monodromy operator. The relevant restriction ma…

Siebert–Tian's stabilization conjecture. Any genus $2$ Lefschetz fibration should become holomorphic after fiber sum with sufficiently many copies of the rational genus $2$ Lefsche…

Let $f$ be a genus-$g$ Lefschetz fibration with slope $\lambda_f$, and call $f$ hyperelliptic when its monodromy lies in the hyperelliptic mapping class group. Xiao's hyperelliptic…

Giroux–Pardon conjecture. There is a Weinstein Lefschetz fibration

Let $f:X\to S^2$ be a nontrivial genus-$g$ Lefschetz fibration, where $X$ is a smooth, closed, compact $4$-manifold, and let $b_1(X)$ be its first Betti number. Baykur's improved B…

Let $f:X\to S^2$ be a genus-$2$ Lefschetz fibration, and let $\pi_1(X)$ denote the fundamental group of its total space. Abelian fundamental-group conjecture. The group $\pi_1(X)$…

Let $f:X\to S^2$ be a genus-$2$ Lefschetz fibration, and let $\pi_1(X)$ denote the fundamental group of its total space. Two-generator conjecture. The group $\pi_1(X)$ has at most…

Veronese projection conjecture. The Lefschetz bipencil constructed in this section is isomorphic to the composition

Let $X=(\mathbb{C}^*)^n\subset\mathbb{C}^{n+1}$. A normal crossings divisor in $\mathbb{P}^n$ is the union of $n+1$ hyperplanes in general position. The categorical intersection co…

A hyperelliptic Lefschetz fibration is a Lefschetz fibration whose regular fiber is a hyperelliptic curve. Consider such a fibration over the 2-sphere $S^2$ with no reducible fiber…

The fillings under consideration are the explicit Stein fillings associated with the rational cuspidal curves of types $\mathcal{A}_p$ and $\mathcal{B}_p$, and Figures … give diagr…

For each $g\geq 2$, let … where $\lambda_f$ denotes the slope of $f$. Infimum-slope conjecture. … for all $g\geq 2$. The paper establishes that the infimum is positive and that, fo…

Let $f\colon X\to S^2$ be a symplectic Lefschetz fibration, with total space $X$ a closed symplectic $4$-manifold. Its signature is denoted by $\sigma(X)$. Stipsicz's conjecture. E…

Consider the Lefschetz fibration … constructed from a pencil of quadrics, and let $\varphi\in\operatorname{Symp}_{ct}(T^*\mathbb{R}\mathbb{P}^3)$ be the compactly supported symplec…

A Lefschetz fibration … would be used to construct a compactly supported symplectomorphism, called a model $\mathbb{H}\mathbb{P}^2$-twist. Model quaternionic projective-plane twist…