Loewner’s equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function of its boundary. The first main result of the present paper is that the Dirichlet energy of this driving function (also known as the Loewner energy) is equal to the Dirichlet energy of the log-derivative of the (appropriately defined) uniformizing conformal map. This description of the Loewner energy then enables to tie direct links with regularized determinants and Teichmüller theory: We show that for smooth simple loops, the Loewner energy can be expressed in terms of the zeta-regularized determinants of a certain Neumann jump operator. We also show that the family of finite Loewner energy loops coincides with the Weil–Petersson class of quasicircles, and that the Loewner energy equals to a multiple of the universal Liouville action introduced by Takhtajan and Teo, which is a Kähler potential for the Weil–Petersson metric on the Weil–Petersson Teichmüller space.

We study singular Hermitian metrics on vector bundles. There are two main results in this paper. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections. As an application, we show the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose standard approximations are not Nakano semipositive. The aim of the second main result is to determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for example, certain rank 2 bundles on elliptic curves.

The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model. The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rødland. A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.

Let M and N be two compact complex manifolds. We show that if the tautological line bundle O_{T^∗}M(1) is not pseudo-effective and O_{T^∗}N(1) is nef, then there is no non-constant holomorphic map from M to N. In particular, we prove that any holomorphic map from a compact complex manifold M with RC-positive tangent bundle to a compact complex manifold N with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

The case of dimension two was proved by Andreotti-Frankel [-7] and the case of dimension three by Mabuchi [18] using the result of Kobayashi-Ochiai [12]. Our method of proof uses harmonic maps and the characterization of the complex projective space obtained by Kobayashi-Ochiai [15]. According to the result of Kobayashi-Ochiai the complex projective space of dimension n is characterized by the fact that its first Chern class equals 2c1 (F) for some 2> n+ 1 and some positive holomorphic line bundle F over it. Since by the result of Bishop-Goldberg [-2] the second Betti number of a compact Kiihler manifold M of positive bisectional curvature is 1, for the Main Theorem it suffices to show that ca (M) is 2 times a generator of H2 (M, 7Z.) for some 2> 1+ dim M. This can be done by proving that a generator of the free part of H2 (M, Z) can be represented by a rational curve, because the tangent bundle of M restricted to the rational curve splits into a direct sum of holomorphic line bundles over the rational curve according to the result of Grothendieck [-11]. The existence of the rational curve is obtained in the following way. According to the result of Sacks-Uhlenbeck [22] and its improved formulation by Meeks-Yau [-19], the infimum of the energies of maps from S 2 to M representing the generator of rcz (M) can be achieved by a sum of stable harmonic maps f~ from S 2 to M (1< i< m). The key step in our proof is to show that each f~ is either holomorphic or conjugate holomorphic. The known methods of proving the complex-analyticity of a harmonic map use the formula for the Laplacian of the energy function [23, 25, 26] or a variation of it [24]. Here we use