We introduce the notion of non-lc ideal sheaves. It is an analogue of the notion of multiplier ideal sheaves. We establish the restriction theorem, which seems to be the most important property of non-lc ideal sheaves.

In this article, we investigate the converging radius of the “generalized zeta function,” which is, roughly speaking, the generating function of the number of effective cycles. In Section 3, we give the explicit value of the converging radius when the codimension of the cycles is $1$. In Section 4, we deal with $1$-dimensional cycles on a projective space and give a lower bound of the convergent radius.

In this article we prove the meromorphic continuation to the entire complex plane and also the functional equation of the zeta functions of twisted $4$-dimensional quaternionic Shimura varieties in quite general cases.

In the articles [4] and [7], we completed the determination of group realizations ${\mathfrak{g}}_{ev}$ and ${\mathfrak{g}}_0$ of $2$-graded decompositions $\mathfrak{g}={\mathfrak{g}}_{-2}\oplus {\mathfrak{g}}_{-1}\oplus {\mathfrak{g}}_0\oplus {\mathfrak{g}}_1\oplus {\mathfrak{g}}_2$ of exceptional Lie algebras $\mathfrak{g}$ for the universal exceptional Lie groups. In the present article, which is a continuation of [5] and [8], we determine group realizations of subalgebras ${\mathfrak{g}}_{ev}$, ${\mathfrak{g}}_0$ and ${\mathfrak{g}}_{ed}$ of $3$-graded decompositions of exceptional Lie algebras $\mathfrak{g}$ for the universal exceptional Lie groups of type $E_8$.

In this article, we compute ${\mathit{\text{MU}}}^{*}\left(\mathit{\text{BSO}}\right(2m\left)\right)$ and show that it is generated as an ${\mathit{\text{MU}}}^{*}$-algebra by Conner-Floyd Chern classes $c_i$ and one $2m$-dimensional element $y_m$. The case $\mathit{\text{BO}}\left(n\right)$ was studied by W. S. Wilson, and the case $\mathit{\text{BSO}}(2m+1)$ is derived directly from the result. We obtain the result for $\mathit{\text{BSO}}\left(2m\right)$ by using (equivariant) stratification methods introduced to compute Chow rings by Guillot, Molina, Vezzosi, and Vistoli.

Let $G$ be a connected complex reductive linear algebraic group, and let $K\subset G$ be a maximal compact subgroup. The Lie algebra of $K$ is denoted by $\mathfrak{k}$. A holomorphic Hermitian principal $G$-bundle is a pair of the form $(E_G,E_K)$, where $E_G$ is a holomorphic principal $G$-bundle and $E_K\subset E_G$ is a $C^{\infty }$-reduction of structure group to $K$. Two holomorphic Hermitian principal $G$-bundles $(E_G,E_K)$ and $({E\prime }_G,{E\prime }_K)$ are called holomorphically isometric if there is a holomorphic isomorphism of the principal $G$-bundle $E_G$ with $E{\prime }_G$ which takes $E_K$ to $E{\prime }_K$. We consider all holomorphic Hermitian principal $G$-bundles $(E_G,E_K)$ over the upper half-plane $\mathbb{H}$ such that the pullback of $(E_G,E_K)$ by each holomorphic automorphism of $\mathbb{H}$ is holomorphically isometric to $(E_G,E_K)$ itself. We prove that the isomorphism classes of such pairs are parameterized by the equivalence classes of pairs of the form $(\chi ,A)$, where $\chi :\mathbb{R}\to K$ is a homomorphism, and $A\in \mathfrak{k}{\otimes }_{\mathbb{R}}\mathbb{C}$ such that $[A,d\chi (1\left)\right]=2\sqrt{-1}\cdot A$. (Here $d\chi :\mathbb{R}\to \mathfrak{k}$ is the homomorphism of Lie algebras associated to $\chi$.) Two such pairs $(\chi ,A)$ and $(\chi \prime ,A\prime )$ are called equivalent if there is an element $g_0\in K$ such that $\chi \prime =Ad\left(g_0\right)◦\chi$ and $A\prime =Ad\left(g_0\right)\left(A\right)$.

We consider linear, smooth, hyperbolic systems with time-dependent coefficients and size $N$. We give a condition sufficient for the well-posedness of the Cauchy Problem in some Gevrey classes. We present some Levi conditions to improve the Gevrey index of well-posedness for the scalar equation of order $N$, using the transformation in [DAS] and following the technique introduced in [CT]. By using this result and adding some assumptions on the form of the first-order term, we can improve the well-posedness for systems. A similar condition has been studied in [DAT] for systems with size $3$.

We completely determine the range of Sobolev regularity for the Dirac- Klein-Gordon system, the quadratic nonlinear Dirac equations, and the wave-map equation to be well posed locally in time on the real line. For the Dirac-Klein-Gordon system, we can continue those local solutions in nonnegative Sobolev spaces by the charge conservation. In particular, we obtain global well-posedness in the space where both the spinor and scalar fields are only in $L^2\left(\mathbb{R}\right)$. Outside the range for well-posedness, we show either that some solutions exit the Sobolev space instantly or that the solution map is not twice differentiable at zero.

In this article, we discuss the growth of solutions of the second-order nonhomogeneous linear differential equation

$f\prime \prime +A_1\left(z\right)e^{az}f\prime +A_0\left(z\right)e^{bz}f=F$,

where $a$, $b$ are complex constants and $A_j\left(z\right)\not\equiv 0$ $(j=0,1)$, and $F\not\equiv 0$ are entire functions such that $max\left\{\rho \right(A_j\left)\right(j=0,1),\rho (F\left)\right\}<1$. We also investigate the relationship between small functions and differential polynomials $g_f\left(z\right)=d_{2}f\prime \prime +d_{1}f\prime +d_{0}f$, where $d_0\left(z\right),d_1\left(z\right),d_2\left(z\right)$ are entire functions that are not all equal to zero with $\rho \left(d_j\right)<1$ $(j=0,1,2)$ generated by solutions of the above equation.