On the spectrum of the twisted Dolbeault Laplacian on line bundles over Kähler manifolds

Maximal hypoellipticity and Dolbeault cohomology representations for U(p,q)

Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormander condition. We prove here that this operator is not maximal hypoelliptic when G=U(p,q).

Bott-Chern Laplacian on almost Hermitian manifolds

Let $(M,J,g,ω)$ be a $2n$-dimensional almost Hermitian manifold. We extend the definition of the Bott-Chern Laplacian on $(M,J,g,ω)$, proving that it is still elliptic. On a compact Kähler manifold, the kernels of the Dolbeault Laplacian and of the Bott-Chern Laplacian coincide. We show that such a property does not hold when $(M,J,g,ω)$ is a compact almost Kähler manifold, providing an explicit almost Kähler structure on the Kodaira-Thurston manifold. Furthermore, if $(M,J,g,ω)$ is a connected compact almost Hermitian $4$-manifold, denoting by $h^{1,1}_{BC}$ the dimension of the space of Bott…

黎曼曲面上旋量丛的伪拉普拉斯算子行列式与$\zeta^{(\rm reg)}(1)$

Determinants of pseudo-laplacians and $ζ^{({\rm reg})}(1)$ for spinor bundles over Riemann surfaces

本文研究紧黎曼曲面上埃尔米特线丛中Dolbeault拉普拉斯算子的自伴扩张，引入Robin质量概念并推导其显式表达式，以此描述标量拉普拉斯正则化$\zeta(1)$的Ricci流演化，还给出Morpurgo零亏格曲面相关结论的新证法。

近埃尔米特流形上的更多微分算子

More Differential Operators on Almost Hermitian Manifolds

该研究放宽Michelsohn微分算子的Kähler流形条件，在紧近埃尔米特流形上引入两个推广的微分算子，证明其拉普拉斯核的类Kähler对称性及与近复几何中形式算子的对应关系。

某些非紧复流形上 Dirac-Dolbeault 算子的 $L^2$ 估计与 Bergman 核渐近

$L^2$-estimates for the Dirac-Dolbeault operator and Bergman kernel asymptotics on some classes of non-compact complex manifolds

本文推广经典方法，在混合曲率情形下获得了 Dirac-Dolbeault 算子的 $L^2$ 估计并界定了额外误差项，进而推导出 Kodaira Laplacian 的局部谱隙及 Bergman 核的完整渐近展开。

The inverse Penrose transform on Riemannian twistor spaces

With respect to the Dirac operator and the conformally invariant Laplacian, an explicit description of the inverse Penrose transform on Riemannian twistor spaces is given. A Dolbeault representative of cohomology on the twistor space is constructed from a solution of the field equation on the base manifold.

Hyperkaehler manifolds with torsion, supersymmetry and Hodge theory

Let M be a hypercomplex Hermitian manifold, (M,I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate (2,0)-form on (M,I). It is well known that M is hyperkaehler if and only if the form Ωis closed with respect to the de Rham differential. The M is called HKT (hyperkaehler with torsion) if this form is closed with respect to the Dolbeault differential (this condition is weaker. Conjecturally, all compact hypercomplex manifolds admit an HKT-metrics. We exploi…

全纯线丛高次幂的隧穿、Quillen 度量与解析挠率

Tunneling, the Quillen metric and analytic torsion for high powers of a holomorphic line bundle

本文研究紧复流形上全纯线丛高张量幂的 Dolbeault 拉普拉斯算子，给出指数小特征值平均分布公式，并由此获得 Quillen 度量渐近性的新证明。

On the Laplace-Beltrami operator on compact complex spaces

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2…

Calabi-Yau超曲面上线丛的Laplacian数值谱

Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces

本文首次数值计算了Calabi-Yau三重流形上取值于线丛的Dolbeault Laplacian谱，并通过三个例子验证了算法的准确性，为弦紧致化中Yukawa耦合计算提供了必要工具。

Twisted Adiabatic Limit for Complex Structures

Given a complex manifold $X$ and a smooth positive function $η$ thereon, we perturb the standard differential operator $d=\partial + \bar\partial$ acting on differential forms to a first-order differential operator $D_η$ whose principal part is $η\partial + \bar\partial$. The role of the zero-th order part is to force the integrability property $D_η^2=0$ that leads to a cohomology isomorphic to the de Rham cohomology of $X$, while the components of types $(0,\,1)$ and $(1,\,0)$ of $D_η$ induce cohomologies isomorphic to the Dolbeault and conjugate-Dolbeault cohomologies. We compute Bochner-Kod…

Scaling asymptotics of heat kernels of line bundles

We consider a general Hermitian holomorphic line bundle $L$ on a compact complex manifold $M$ and let ${\Box}^q_p$ be the Kodaira Laplacian on $(0,q)$ forms with values in $L^p$. The main result is a complete asymptotic expansion for the semi-classically scaled heat kernel $\exp(-u{\Box}^q_p/p)(x,x)$ along the diagonal. It is a generalization of the Bergman/Szegö kernel asymptotics in the case of a positive line bundle, but no positivity is assumed. We give two proofs, one based on the Hadamard parametrix for the heat kernel on a principal bundle and the second based on the analytic localizati…

Degenerating Hermitian metrics and spectral geometry of the canonical bundle

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $m$. In this paper we are interested in the Dolbeault operator acting on the space of $L^2$ sections of the canonical bundle of $reg(X)$, the regular part of $X$. More precisely let $\overline{\mathfrak{d}}_{m,0}:L^2Ω^{m,0}(reg(X),h)\rightarrow L^2Ω^{m,1}(reg(X),h)$ be an arbitrarily fixed closed extension of $\overline{\partial}_{m,0}:L^2Ω^{m,0}(reg(X),h)\rightarrow L^2Ω^{m,1}(reg(X),h)$ where the domain of the latter operator is $Ω_c^{m,0}(reg(X))$. We establish various properties such as closed range of $\…

Flat line bundles and the Cappell-Miller torsion in Arakelov geometry

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology side, the idea goes back to Fay's holomorphic extension of determinants of Dolbeault laplacians, and it is shown here to be equivalent to the holomorphic Cappell-Miller torsion. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in…

扭曲微分形式的狄利克雷格林核估计与索伯列夫型不等式

Dirichlet Green kernel estimates and Sobolev-type inequalities for twisted differential forms

该研究针对凯勒流形上的微分形式，证明了一致格林核估计，导出带边界项的索伯列夫型不等式，还得到无需伪凸性的\bar\nabla可解性及适用于扭曲德拉姆复形的类似结果。