We study the monodromy action of the mixed braid group $B_{n,\mathcal{P}}$ on the first cohomology of cyclic branched covers of $\mathbb{P}^1$, which are mutually determined by a partition of branch points by equal ramification. The monodromy representation splits into irreducible representations on the $t$-eigenspaces of the deck transformation. For each, we construct an explicit spanning set using lifts of Pochhammer contours and figure-eight curves, and compute the Hermitian intersection form. The representation factors through a reduced mixed braid group by dropping $t$-invisible parts of the partition (those with trivial local monodromy). In this reduced representation, each generator acts by a complex reflection when the corresponding spanning class is non-isotropic, and by a unitary transvection when it is isotropic. Provided $\infty$ has non-trivial local monodromy, the factored representation is isomorphic to the reduced multivariate Burau representation evaluated at $t$.
We use a gluing procedure introduced by Ginzburg to describe the relative Langlands dual of the hyperspherical Hamiltonian $(\mathrm{GL}_n \times \mathrm{GL}_m)$-variety $T^*(\mathrm{Hom}(\mathbb{C}^m,\mathbb{C}^n))$, and in particular the Rankin-Selberg case $m=n$. We show that the dual is isomorphic to the triangle p...
Let $X$ carry a transversely holomorphic foliation, equivalently an elliptic involutive structure $V\subset T_{\mathbb C}X$, and let ${\mathcal O}_V$ be its sheaf of leafwise-constant, transversely holomorphic functions. We construct a finite superconnection model for the derived category of coherent ${\mathcal O}_V$-m...
Since the 1980s, the Bogomolov multiplier of a finite group has been known to obstruct rationality in complex algebraic geometry, and more recently it is understood to be responsible for any torsion in the oriented and stable unitary 2-dimensional $G$-equivariant bordism groups $\Omega_2^{SO,G}$ and $\Omega_2^{U,G}$. I...
We compare two mixed Hodge structures attached to a unipotent variation $\mathbb V$ of mixed Hodge--Tate structures on $X=\mathbb P^1\setminus{0,1,\infty}$: the cohomology of its extension by $*$ at $0$ and by $!$ at $\infty$, and the vanishing cycles at $1$. The semi-holonomy isomorphism constructed in [arXiv:2604.263...
Let $G$ be a torsion-free one-ended hyperbolic group and let $\phi\in\operatorname{Aut} (G)$. We prove that the mapping torus, or suspension, $$ M:=G_{\phi}=G\rtimes_\phi \mathbb{Z} $$ has solvable conjugacy problem. This builds on the pioneering work of Pr\'eaux who solved the conjugacy problem for all (geometrisable)...
Applying the BBDJS minimal model to the derived symplectification of a $-1$-shifted contact derived Artin stack and descending algebraically along the structural free $\mathbb{G}_m$-action, we construct an $\ell$-adic perverse sheaf on any oriented such stack, and use Verdier's specialization equivalence for monodromic...
Efe Izbudak· 1 citation
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