11 problems

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 $(X,\omega)$ 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…

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…

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

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…

Let $\EuScript A$ be the Fukaya $A_\infty$-algebra of the Lefschetz fibration and $\EuScript B_{q,\bar b|M}$ the category associated to the fibrewise compactification. Suppose that…

Let a Lefschetz fibration arise from an anticanonical Lefschetz pencil, and choose the bulk term specified by the corresponding theorem. The Fukaya category has a canonical natural…

Slope inequality conjecture. For every genus-$g$ Lefschetz fibration $f:X\rightarrow S^2$, the slope inequality

Let $M$ be a $4$-manifold admitting a genus-$1$ simplified broken Lefschetz fibration (SBLF) structure with non-empty round singular locus. Write $\sharp$ for connected sum, and le…