English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Let $X$ be a compact Riemann surface of genus $g\geq 2$ equipped with flatconical metric $|\Omega|$, where $\Omega$ be a holomorphic quadraticdifferential on $X$ with $4g-4$ simple zeroes. Let $K$ be the canonical linebundle on $X$. Introduce the Cauchy-Riemann operators $\bar \partial$ and$\partial$ acting on sections of holomorphic line bundles over $X$ ($K^2$ inthe definition of $\Delta^{(2)}_{|\Omega|}$ below) and, respectively,anti-holomorphic line bundles ($\bar { K}^{-1}$ below). Consider the Laplaceoperator $\Delta^{(2)}_{|\Omega|}:=|\Omega| \partial |\Omega|^{-2}\bar\partial$acting in the Hilbert space of square integrable sections of the bundle $K^2$equipped with inner product $<Q_1, Q_2>_{K^2}=\int_X\frac {Q_1\barQ_2}{|\Omega|}$.  We discuss two natural definitions of the determinant of the operator$\Delta^{(2)}_{|\Omega|}$. The first one uses the zeta-function of some specialself-adjoint extension of the operator (initially defined on smooth sections of$K^2$ vanishing near the zeroes of $\Omega$), the second one is an analog ofEskin-Kontsevich-Zorich (EKZ) regularization of the determinant of the conicalLaplacian acting in the trivial bundle. In contrast to the situation ofoperators acting in the trivial bundle, for operators acting in $K^2$ these tworegularizations turn out to be essentially different. Considering theregularized determinant of $\Delta^{(2)}_{|\Omega|}$ as a functional on themoduli space $Q_g(1, \dots, 1)$ of quadratic differentials with simple zeroeson compact Riemann surfaces of genus $g$, we derive explicit expressions forthis functional for the both regularizations. The expression for the EKZregularization is closely related to the well-known explicit expressions forthe Mumford measure on the moduli space of compact Riemann surfaces of genus$g$.

Flat conical Laplacian in the square of the canonical bundle and its regularized determinants