论文标题
规范捆绑包的平方及其正则决定因素的平坦锥形拉普拉斯
Flat conical Laplacian in the square of the canonical bundle and its regularized determinants
论文作者
论文摘要
令$ x $为配备平面圆锥公制$ |ω| $的紧凑型Riemann表面，$ g \ geq 2 $，其中$ω$是$ x $的全态二次差异，$ x $，$ 4G-4 $简单的零。令$ k $为$ x $的规范线捆绑包。介绍Cauchy-Riemann运营商$ \ bar \ partial $和$ \ partial $作用于$ x $ $ x $（$ k^2 $的$Δ^{（2）} _ {|ω| |} $的定义，在$ x $上的套件（$ k^2 $），分别是抗holomormorphic line} $} - 以下）。考虑laplace操作员$δ^{（2）} _ {|ω|}：= |ω| \ partial |ω|^{ - 2} \ bar \ partial $作用于捆绑包的正方形空间$ k^2 $的正方形空间，配备了内部产品$ <q_1，q_2> _ {k^2} = \ int_x = \ int_x \ frac \ frac {q_1 \ bar q_1 \ bar q_1 \ bar q_1 \ bar q_1 \ bar q_2}} 我们讨论了操作员$δ^{（2）} _ {|ω|} $的两个自然定义。第一个使用了操作员的一些特殊自我预性扩展的Zeta功能（最初定义在$ k^2 $的平滑部分附近消失在$ω$的零附近），第二个是Eskin-Kontsevich-Zorich（EKZ）的类似物，对圆锥形Laplacian conial laplacian bun的确定性的定期化的类似物类似。与操作员在琐碎的捆绑中行动的情况相反，对于$ k^2 $行事的操作员，这两个正规化本质上是不同的。将$δ^{（2）} _ {|ω|} $的正则确定因素作为模量空间$ q_g（1，\ dots，1）$ quadratic差异的功能，在紧凑型riemann表面上具有简单的二次差差，我们对这两个定期的表达方式都定期表达。 EKZ正则化的表达与在$ g $属的紧凑型riemann表面的Mumford Measure上的众所周知的显式表达式密切相关。
Let $X$ be a compact Riemann surface of genus $g\geq 2$ equipped with flat conical metric $|Ω|$, where $Ω$ be a holomorphic quadratic differential on $X$ with $4g-4$ simple zeroes. Let $K$ be the canonical line bundle on $X$. Introduce the Cauchy-Riemann operators $\bar \partial$ and $\partial$ acting on sections of holomorphic line bundles over $X$ ($K^2$ in the definition of $Δ^{(2)}_{|Ω|}$ below) and, respectively, anti-holomorphic line bundles ($\bar { K}^{-1}$ below). Consider the Laplace operator $Δ^{(2)}_{|Ω|}:=|Ω| \partial |Ω|^{-2}\bar\partial$ acting in the Hilbert space of square integrable sections of the bundle $K^2$ equipped with inner product $<Q_1, Q_2>_{K^2}=\int_X\frac {Q_1\bar Q_2}{|Ω|}$. We discuss two natural definitions of the determinant of the operator $Δ^{(2)}_{|Ω|}$. The first one uses the zeta-function of some special self-adjoint extension of the operator (initially defined on smooth sections of $K^2$ vanishing near the zeroes of $Ω$), the second one is an analog of Eskin-Kontsevich-Zorich (EKZ) regularization of the determinant of the conical Laplacian acting in the trivial bundle. In contrast to the situation of operators acting in the trivial bundle, for operators acting in $K^2$ these two regularizations turn out to be essentially different. Considering the regularized determinant of $Δ^{(2)}_{|Ω|}$ as a functional on the moduli space $Q_g(1, \dots, 1)$ of quadratic differentials with simple zeroes on compact Riemann surfaces of genus $g$, we derive explicit expressions for this functional for the both regularizations. The expression for the EKZ regularization is closely related to the well-known explicit expressions for the Mumford measure on the moduli space of compact Riemann surfaces of genus $g$.
