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Changjian Su

Publications and source records attributed to Changjian Su.

26 records · Page 2Linked to original sources
arXiv2021

Left Demazure-Lusztig operators on equivariant (quantum) cohomology and K theory

We study the Demazure-Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern-Schwartz-MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K theory), in any partial flag manifold. Along the way we advertise many properties of the left and right divided difference operators in cohomology and K theory, and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K theory, generating Schubert classes, and satisfying a Leibniz rule compatible with the quantum product.

math.AG↗
arXiv2020

Geometric properties of the Kazhdan-Lusztig Schubert basis

We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases produce dual bases, one of which can be interpreted as characteristic classes of the intersection homology mixed Hodge modules. In equivariant hyperbolic cohomology, we show that if the Schubert variety is smooth, then the class it determines coincides with the class of the Kazhdan-Lusztig basis; this was known as the Smoothness Conjecture. For Grassmannians, we prove that the classes of the Kazhdan-Lusztig basis coincide with the classes determined by Zelevinsky's small resolutions. These properties of the so-called KL-Schubert basis show that it is the closest existing analogue to the Schubert basis for hyperbolic cohomology; the latter is a very useful testbed for more general elliptic cohomologies.

math.AG↗
arXiv2020

Positivity of Segre-MacPherson classes

Let $X$ be a complex nonsingular variety with globally generated tangent bundle. We prove that the signed Segre-MacPherson (SM) class of a constructible function on $X$ with effective characteristic cycle is effective. This observation has a surprising number of applications to positivity questions in classical situations, unifying previous results in the literature and yielding several new results. We survey a selection of such results in this paper. For example, we prove general effectivity results for SM classes of subvarieties which admit proper (semi-)small resolutions and for regular or affine embeddings. Among these, we mention the effectivity of (signed) Segre-Milnor classes of complete intersections if $X$ is projective and an alternation property for SM classes of Schubert cells in flag manifolds; the latter result proves and generalizes a variant of a conjecture of Fehér and Rimányi. Among other applications we prove the positivity of Behrend's Donaldson-Thomas invariant for a closed subvariety of an abelian variety and the signed-effectivity of the intersection homology Chern class of the theta divisor of a non-hyperelliptic curve; and we extend the (known) non-negativity of the Euler characteristic of perverse sheaves on a semi-abelian variety to more general varieties dominating an abelian variety.

math.AG↗
arXiv2019

Structure constants for Chern classes of Schubert cells

A formula for the structure constants of the multiplication of Schubert classes is obtained in arXiv:1909.05283. In this note, we prove analogous formulae for the Chern--Schwartz--MacPherson (CSM) classes and Segre--Schwartz--MacPherson (SSM) classes of Schubert cells in the flag variety. By the equivalence between the CSM classes and the stable basis elements for the cotangent bundle of the flag variety, a formula for the structure constants for the latter is also deduced.

math.AG↗
arXiv2019

Stable bases of the Springer resolution and representation theory

In this note, we collect basic facts about Maulik and Okounkov's stable bases for the Springer resolution, focusing on their relations to representations of Lie algebras over complex numbers and algebraically closed positive characteristic fields, and of the Langlands dual group over non-Archimedean local fields.

math.RT↗
arXiv2017

On the K-theory stable bases of the Springer resolution

Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in computing the quantum connection of cotangent bundle of partial flag varieties. In this paper we study the K-theoretic stable bases of cotangent bundles of flag varieties. We describe these bases in terms of the action of the affine Hecke algebra and the twisted group algebra of Kostant-Kumar. Using this algebraic description and the method of root polynomials, we give a restriction formula of the stable bases. We apply it to obtain the restriction formula for partial flag varieties. We also build a relation between the stable basis and the Casselman basis in the principal series representations of the Langlands dual group. As an application, we give a closed formula for the transition matrix between Casselman basis and the characteristic functions.

math.AG↗
arXiv2015

Equivariant quantum cohomology of cotangent bundle of $G/P$

Let $G$ denote a complex semisimple linear algebraic group, $P$ a parabolic subgroup of $G$ and $\mathcal{P}=G/P$. We identify the quantum multiplication by divisors in $T^*\mathcal{P}$ in terms of stable basis, which is introduced by Maulik and Okounkov. Using this and the restriction formula for stable basis, we show that the $G\times\mathbb{C}^*$-equivariant quantum multiplication formula in $T^*\mathcal{P}$ is conjugate to the conjectured formula by Braverman.

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