arXiv subjects
Publications and source records attributed to Mohamed El Alami.
We prove that under certain conditions, a normal crossings compactification of a Liouville domain determines a Maurer--Cartan element for the $L_\infty$ structure on its symplectic cohomology; and deforming by this element gives the quantum cohomology of the compactification.
We construct the $L_\infty$ structure on symplectic cohomology of a Liouville domain, together with an enhancement of the closed--open map to an $L_\infty$ homomorphism from symplectic cochains to Hochschild cochains on the wrapped Fukaya category. Features of our construction are that it respects a modified action filtration (in contrast to Pomerleano--Seidel's construction); it uses a compact telescope model (in contrast to Abouzaid--Groman--Varolgunes' construction); and it is adapted to the purposes of our follow-up work where we construct Maurer--Cartan elements in symplectic cochains which are associated to a normal-crossings compactification of the Liouville domain.
Given a Fano variety $Y$ and a simple normal crossings divisor $D\subseteq Y$ which is anti-canonical, we prove a formula relating counts of discs with boundary on a Lagrangian $L\subseteq Y\backslash D$ to counts of rational curves in $Y$, under suitable positivity assumptions on $L$. This formula seriously constrains the topology of $L$ in many examples. Our main application is a super-potential formula for Fano cyclic coverings $X$ of $Y$. As a corollary, we show that all the small components of the Fukaya category of a Fano hypersurface $X\subseteq \mathbb{P}^{n+1}$ are split-generated by monotone Lagrangian tori.
We study homological mirror symmetry of the singular hypersurface $X_0=V(t^{n+1}-x_0\dotsi x_n)\subseteq\mathbb{P}^{n+1}$. Following an SYZ type approach, we produce an LG-model, whose Fukaya-Seidel category recovers line bundles on $X_0$. As a byproduct of our approach, we answer a conjecture of N.Sheridan about generating the small component of the Fukaya category of the smooth index 1 Fano hypersurface in $\mathbb{P}^{n+1}$, without bounding co-chains.