arXiv subjects

Pranav Haridas

Publications and source records attributed to Pranav Haridas.

5 recordsLinked to original sources

arXiv2026

Geometric Monodromy of Mixed Braid Groups and the Multivariate Burau Representation

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 $t$.

math.GT↗

arXiv2018

A $1$-point Quadrature domain of order $1$ not biholomorphic to a balanced domain

It is known that if $f: D_1 \to D_2$ is a polynomial biholomorphism with polynomial inverse and constant Jacobian then $D_1$ is a $1$-point Quadrature domain (the Bergman span contains all holomorphic polynomials) of order $1$ whenever $D_2$ is a balanced domain. Bell conjectured that all $1$-point Quadrature domains arise in this manner. In this note, we construct a $1$-point Quadrature domain of order $1$ that is not biholomorphic to any balanced domain.

math.CV↗

arXiv2018

A note on the smoothness of the Minkowski function

The Minkowski function is a crucial tool used in the study of balanced domains and, more generally, quasi-balanced domains in several complex variables. If a quasi-balanced domain is bounded and pseudoconvex then it is well-known that its Minkowski function is plurisubharmonic. In this short note, we prove that under the additional assumption of smoothness of the boundary, the Minkowski function of a quasi-balanced domain is in fact smooth away from the origin. This allows us to construct a smooth plurisubharmonic defining function for such domains. Our result is new even in the case of balanced domains.

math.CV↗

arXiv2016

Comments on the Green's function of a planar domain

We study several quantities associated to the Green's function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and $L^2$-cohomology of the capacity metric and critical points of the Green's function. The principal idea used is an affine scaling of the domain that furnishes quantitative boundary behaviour of the Green's function and related objects.

math.CV↗

arXiv2014

Quadrature domains in $\mathbb C^n$

We prove two density theorems for quadrature domains in $\mathbb{C}^n$, $n \geq 2$. It is shown that quadrature domains are dense in the class of all product domains of the form $D \times Ω$, where $D \subset \mathbb{C}^{n-1}$ is a smoothly bounded domain satisfying Bell's Condition R and $Ω\subset \mathbb{C}$ is a smoothly bounded domain and also in the class of all smoothly bounded complete Hartogs domains in $\mathbb{C}^2$.

math.CV↗