Standard Embeddings of Smooth Schubert Varieties in Rational Homogeneous Manifolds of Picard Number 1

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomogeneous smooth Schubert varieties in symplectic Grassmannians and in the 20-dimensional $F_4$-homogeneous manifold associated to a short simple root.Comment: 22 page

On the anomaly of nonlocal symmetry in the chiral QED

We show that the anomaly of nonlocal symmetry can be canceled by the well-known Wess-Zumino acton which cancels the gauge anomaly in the two-dimensional chiral electrodynamics.Comment: 4 pages, latex, no figure

Diagrams for Legendrian VMRT

We describe varieties of minimal rational tangents on the wonderful symmetric varieties. An irreducible component of a variety of minimal rational tangents is a rational homogeneous space, and hence, we have a corresponding Dynkin diagram expression. On the other hand, we have diagrams from the (marked) Kac diagram of a symmetric space by marking adjacent nodes to the marked node, similar to the case of rational homogeneous spaces. We note that the above two diagrams coincide when the restricted root system is not of type A. This paper is a memo for "minimal rational curves on complete symmetric varieties".Comment: 14pag

Portfolio Optimization with Relative Tail Risk

This paper proposes analytic forms of portfolio CoVaR and CoCVaR on the normal tempered stable market model. Since CoCVaR captures the relative risk of the portfolio with respect to a benchmark return, we apply it to the relative portfolio optimization. Moreover, we derive analytic forms for the marginal contribution to CoVaR and the marginal contribution to CoCVaR. We discuss the Monte-Carlo simulation method to calculate CoCVaR and the marginal contributions of CoVaR and CoCVaR. As the empirical illustration, we show relative portfolio optimization with thirty stocks under the distress condition of the Dow Jones Industrial Average. Finally, we perform the risk budgeting method to reduce the CoVaR and CoCVaR of the portfolio based on the marginal contributions to CoVaR and CoCVaR