Picard-Lefschetz theory is applied to path integrals of quantum mechanics, in order to compute real-time dynamics directly. After discussing basic properties of real-time path integrals on Lefschetz thimbles, we demonstrate its computational method in a concrete way by solving three simple examples of quantum mechanics. It is applied to quantum mechanics of a double-well potential, and quantum tunneling is discussed. We identify all of the complex saddle points of the classical action, and their properties are discussed in detail. However a big theoretical difficulty turns out to appear in rewriting the original path integral into a sum of path integrals on Lefschetz thimbles. We discuss generality of that problem and mention its importance. Real-time tunneling processes are shown to be described by those complex saddle points, and thus semi-classical description of real-time quantum tunneling becomes possible on solid ground if we could solve that problem.

Abstract. Let C be a compact complex curve included in a non-singular complex surface such that the normal bundle is topologically trivial. Ueda studied complex analytic properties of a neighborhood of C when C is non-singular or is a rational curve with a node. We propose an analogue of Ueda's theory for the case where C admits nodes. As an application, we study singular Hermitian metrics with semi-positive curvature on the anti-canonical bundle of the blow-up of the projective plane at nine points in arbitrary position.

We develop a new method for constructing K3 surfaces. We construct such a K3 surface X by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the K3 surface X is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction.

Our interest is a regularity of a minimal singular metric of a line bundle. One main conclusion of our general result in this paper is the existence of smooth Hermitian metrics with semi-positive curvatures on the so-called Zariski's example of a line bundle defined over the blow-up of P 2 at some twelve points. This is an example of a line bundle which is nef, big, not semi-ample, and whose section ring is not finitely generated. We generalize this result to the higher dimensional case when the stable base locus of a line bundle is a smooth hypersurface with a holomorphic tubular neighborhood.

We give a concrete expression of a minimal singular metric on a big line bundle on a compact Kähler manifold which is the total space of a toric bundle over a complex torus. In this class of manifolds, Nakayama constructed examples which have line bundles admitting no Zariski decomposition even after modifications. As an application, we discuss the Zariski closedness of non-nef loci.This expression enables us to compute the multiplier ideal sheaf J (h t min ) for each positive number t, whose stalk at x 0 ∈ X is defined bywhere ϕ min is the local weight function of h min around x 0 .Corollary 1.2. J (h t min ) is trivial at any point in X \ P(L 0 ). For a point x 0 ∈ P(L 0 ), the stalk J (h min ) x 0 of the multiplier ideal sheaf is the ideal of O X,x 0 which is generated by the polynomials {x p 1 x q 2 | (p + 1, q + 1) ∈ Int(S t ) ∩ Z 2 }, where we denote by S t the set {(tα, tβ) ∈ R 2 | α, β ≥ 0, a 2 (α + β) 2 ≥ (1 − α) 2 + (1 − β) 2 } ( For the shape of S t in this case, see Figure 1).