In a classical-type flag variety, we consider a Schubert variety associated to a vexillary (signed) permutation, and establish a combinatorial formula for the Hilbert-Samuel multiplicity of a point on such a Schubert variety. The formula is expressed in terms of excited Young diagrams, and extends results for Grassmannians due to Krattenthaler, Lakshmibai-Raghavan-Sankaran, and for the maximal isotropic (symplectic and orthogonal) Grassmannians to Ghorpade-Raghavan, Raghavan-Upadhyay, Kreiman, and Ikeda-Naruse. We also provide a new proof of a theorem of Li-Yong in the type A vexillary case.The main ingredient is an isomorphism between certain neighborhoods of fixed points, known as Kazhdan-Lusztig varieties, which, in turn, relies on a direct sum embedding previously used by Anderson-Fulton to relate vexillary loci to Grassmannian loci.

In this paper we establish Lascoux's algorithm and Zelevinsky's inductive formula on Kazhdan-Lusztig polynomials for covexillary Schubert varieties in classical types. The main ingredients of the proof involve the intersection cohomology theory over an algebraic closure of the prime field and the method by the author's joint work with Anderson, Ikeda, and Kawago.

We use Jiang-Pareschi cohomological rank functions and techniques developed by Caucci and Ito to study syzygies of Kummer varieties, improving existing results by the second author.

In this article we propose formulas for the connected K-theory class of the pointed Brill-Noether loci in Prym varieties, which extends the result by Concini and Pragacz. Applying the formulas, we compute the holomorphic Euler Characteristics of the loci.

We express a Schubert expansion of the Chern-Mather class for Schubert varieties in the even orthogonal Grassmannian via integrals involving Pfaffians and pushforward of the small resolutions in the sense of Intersection Cohomology (IH) constructed by Sankaran and Vanchinathan, instead of the Nash blowup. The equivariant localization is employed to show the way of computing the integral. As a byproduct, we present the computations. For analogy and the completion of the method in ordinary Grassmannians, we also suggest Kazhdan-Lusztig classes associated to Schubert varieties in the Lagrangian and odd orthogonal Grassmannian.