Homology supported in Lagrangian submanifolds in mirror quintic threefolds

In this note, we study homology classes in the mirror quintic Calabi–Yau threefold that can be realized by special Lagrangian submanifolds. We have used Picard–Lefschetz theory to establish the monodromy action and to study the orbit of Lagrangian vanishing cycles. For many prime numbers $p,$ we can compute the orbit modulo p. We conjecture that the orbit in homology with coefficients in $\mathbb {Z}$ can be determined by these orbits with coefficients in $\mathbb {Z}_p$ .

Quintic threefolds with triple points

We study the geometry of quintic threefolds [Formula: see text] with only ordinary triple points as singularities. In particular, we show that if a quintic threefold [Formula: see text] has a reducible hyperplane section then [Formula: see text] has at most [Formula: see text] ordinary triple points, and that this bound is sharp. We construct various examples of quintic threefolds with triple points and discuss their defect.

Quintic threefolds and Fano elevenfolds

The derived category of coherent sheaves on a general quintic threefold is a central object in mirror symmetry. We show that it can be embedded into the derived category of a certain Fano elevenfold. Our proof also generates related examples in different dimensions.

Thin monodromy in Sp(4)

We show that some hypergeometric monodromy groups in ${\rm Sp}(4,\mathbf{Z})$ split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank $2$. In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in $\mathbf{P}^{4}$ splits as $\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$. As a consequence, for a smooth quintic threefold $X$ we show that the group of autoequivalences $D^{b}(X)$ generated by the spherical twist along ${\mathcal{O}}_{X}$ and by tensoring with ${\mathcal{O}}_{X}(1)$ is an Artin group of dihedral type.

A Modular Quintic Calabi-Yau Threefold of Level 55

In this note we search the parameter space of Horrocks–Mumford quintic threefolds and locate a Calabi–Yau threefold that is modular, in the sense that the L-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55.