In this article we give a vanishing result for the cohomology groups $H^{p,q}(X,S^{\nu }E\otimes L)$, where $E$ is a vector bundle generated by sections and $L$ is an ample line bundle on a smooth projective variety $X$. We also give an application related to a result of Barth-Lefschetz type. A general nonvanishing result under the same hypothesis is given to prove the optimality of the vanishing result for some parameter values.

Let $L=(1/2)\mathrm{\nabla }\cdot a\mathrm{\nabla }+b\cdot \mathrm{\nabla }+W$ be a critical elliptic operator on ${\mathbb{R}}^d$. For a certain class of potential functions, we consider the generalized principal eigenvalues $\lambda \left(t\right)$ of $L_t=L+tV$. We show that it is differentiable if and only if $L_0$ is null critical.

Malliavin calculus is applicable to functionals of stable processes by using subordination. We prepare Malliavin calculus for stochastic differential equations driven by Brownian motions with deterministic time change, and the conditions that the existence and the regularity of the densities inherit from those of the densities of conditional probabilities. By using these, we prove regularity properties of the solutions of equations driven by subordinated Brownian motions. In [4] a similar problem is considered. In this article we consider more general cases. We also consider equations driven by rotation-invariant stable processes. We prove that the ellipticity of the equations implies the existence of the density of the solution, and we also prove that the regularity of the coefficients implies the regularity of the densities in the case when the equations are driven by one rotation-invariant stable process.

In this article we use a continuous family of multisections of the moduli space of pseudoholomorphic discs to partially improve the construction of the Lagrangian Floer cohomology of [11] in the case of $\mathbb{R}$ coefficient. Namely, we associate a cyclically symmetric filtered $A_{\infty }$-algebra to every relatively spin Lagrangian submanifold. We use the same trick to construct a local rigid analytic family of filtered $A_{\infty }$-structures associated to a (family of) Lagrangian submanifolds. We include the study of homological algebra of pseudoisotopy of cyclic (filtered) $A_{\infty }$-algebras.

There are countably many equivalence classes of principal $Sp\left(2\right)$-bundles over $S^4$, classified by the integer value second Chern class. We show that the corresponding gauge groups ${\mathcal{G}}_k$ have the property that if there is a homotopy equivalence ${\mathcal{G}}_k\simeq {\mathcal{G}}_{k\prime }$, then $(40,k)=(40,k\prime )$, and we prove a partial converse by showing that if $(40,k)=(40,k\prime )$, then ${\mathcal{G}}_k$ and ${\mathcal{G}}_{k\prime }$ are homotopy equivalent when localized rationally or at any prime.

For a rational map $\varphi :X\to G$ from a normal algebraic variety $X$ to a commutative algebraic group $G$, we define the modulus of $\varphi$ as an effective divisor on $X$. We study the properties of the modulus. This work generalizes the known theories for curves $X$ to higher-dimensional varieties.

We discuss the spectral theory of second-order differential operators that describe the vibration of strings, diffusion processes, and others. M. G. Krein established a one-to-one correspondence between the spectral measure and the string in the case of regular left boundaries, and this correspondence was extended by S. Kotani to a certain class of strings with singular left boundary. In this article we study the relationship between the asymptotic behavior of the spectral measure and that of the corresponding string. Although the results are basically for Kotani’s strings, some are also applicable to Krein’s.