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Changzheng Li

Publications and source records attributed to Changzheng Li.

At least 37 records · Page 2Linked to original sources
arXiv2020

On the fundamental group of open Richardson varieties

We compute the fundamental group of an open Richardson variety in the manifold of complete flags that corresponds to a partial flag manifold. Rietsch showed that these log Calabi-Yau varieties underlie a Landau-Ginzburg mirror for the Langlands dual partial flag manifold, and our computation verifies a prediction of Hori for this mirror. It is log Calabi-Yau as it isomorphic to the complement of the Knutson-Lam-Speyer anti-canonical divisor for the partial flag manifold. We also determine explicit defining equations for this divisor.

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arXiv2019

Pseudotoric structures and special Lagrangian torus fibrations on certain flag varieties

We construct pseudotoric structures (à la Tyurin) on the two-step flag variety $F\ell_{1, n-1; n}$, and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important in the study of SYZ mirror symmetry. As an application, we speculate how our constructions can explain the number of terms in the superpotential of Rietsch's Landau-Ginzburg mirror.

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arXiv2019

Euler characteristics in the quantum $K$-theory of flag varieties

We prove that the sheaf Euler characteristic of the product of a Schubert class and an opposite Schubert class in the quantum $K$-theory ring of a (generalized) flag variety $G/P$ is equal to $q^d$, where $d$ is the smallest degree of a rational curve joining the two Schubert varieties. This implies that the sum of the structure constants of any product of Schubert classes is equal to 1. Along the way, we provide a description of the smallest degree $d$ in terms of its projections to flag varieties defined by maximal parabolic subgroups.

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arXiv2019

Gamma conjecture I for del Pezzo surfaces

Gamma conjecture I and the underlying Conjecture $\mathcal{O}$ for Fano manifolds were proposed by Galkin, Golyshev and Iritani recently. We show that both conjectures hold for all two-dimensional Fano manifolds. We prove Conjecture $\mathcal{O}$ by deriving a generalized Perron-Frobenius theorem on eigenvalues of real matrices and a vanishing result of certain Gromov-Witten invariants for del Pezzo surfaces. We prove Gamma conjecture I by applying mirror techniques proposed by Galkin-Iritani together with the study of Gamma conjecture I for weighted projective spaces.

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arXiv2017

Conjecture $\mathcal{O}$ holds for the odd symplectic Grassmannian

Let $\mathrm{IG}(k, 2n+1)$ be the odd-symplectic Grassmannian. Property $\mathcal{O}$, introduced by Galkin, Golyshev and Iritani for arbitrary complex, Fano manifolds $X$, is a statement about the eigenvalues of the linear operator obtained by the quantum multiplication by the anticanonical class of $X$. We prove that property $\mathcal{O}$ holds in the case when $X= \mathrm{IG}(k, 2n+1)$ is an odd-symplectic Grassmannian. The proof uses the combinatorics of the recently found quantum Chevalley formula for $\mathrm{IG}(k, 2n+1)$, together with the Perron-Frobenius theory of nonnegative matrices.

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arXiv2017

A conjectural Peterson isomorphism in K-theory

We state a precise conjectural isomorphism between localizations of the equivariant quantum K-theory ring of a flag variety and the equivariant K-homology ring of the affine Grassmannian, in particular relating their Schubert bases and structure constants. This generalizes Peterson's isomorphism in (co)homology. We prove a formula for the Pontryagin structure constants in the K-homology ring, and we use it to check our conjecture in few situations.

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arXiv2016

On the Conjecture $\mathcal{O}$ of GGI for $G/P$

In this paper, we show that general homogeneous manifolds $G/P$ satisfy Conjecture $\mathcal{O}$ of Galkin, Golyshev and Iritani which `underlies' Gamma conjectures I and II of them. Our main tools are the quantum Chevalley formula for $G/P$ and a theory on nonnegative matrices including Perron-Frobenius theorem.

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arXiv2015

Equivariant Pieri Rules For Isotropic Grassmannians

We give a Pieri rule for the torus-equivariant cohomology of (submaximal) Grassmannians of Lie types B, C, and D. To the authors' best knowledge, our rule is the first manifestly positive formula, beyond the equivariant Chevalley formula. We also give a simple proof of the equivariant Pieri rule for the ordinary (type A) Grassmannian.

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arXiv2015

On equivariant quantum Schubert calculus for G/P

We show a Z^2-filtered algebraic structure and a "quantum to classical" principle on the torus-equivariant quantum cohomology of a complete flag variety of general Lie type, generalizing earlier works of Leung and the second author. We also provide various applications on equivariant quantum Schubert calculus, including an equivariant quantum Pieri rule for any partial flag variety of Lie type A.

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arXiv2014

Mirror symmetry for exceptional unimodular singularities

In this paper, we prove the mirror symmetry conjecture between the Saito-Givental theory of exceptional unimodular singularities on Landau-Ginzburg B-side and the Fan-Jarvis-Ruan-Witten theory of their mirror partners on Landau-Ginzburg A-side. On the B-side, we compute the genus-zero generating function from a perturbative formula of primitive forms introduced by the first three authors recently. This computation matches the orbifold-Grothendieck-Riemann-Roch and WDVV calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction techniques. Our result establishes the first examples of LG-LG mirror symmetry of all genera for weighted homogeneous polynomials of central charge greater than one (i.e. which contain negative degree deformation parameters).

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arXiv2014

Primitive forms via polyvector fields

We develop a complex differential geometric approach to the theory of higher residues and primitive forms from the viewpoint of Kodaira-Spencer gauge theory, unifying the semi-infinite period maps for Calabi-Yau models and Landau-Ginzburg models. We give an explicit perturbative construction of primitive forms with respect to opposite filtrations and primitive elements. This leads to a concrete algorithm to compute the Taylor expansions of primitive forms as well as the description of their moduli space for all weighted homogenous cases. As an example, we present unknown perturbative expressions for the primitive form of E_12 singularity and illustrate its application to Landau-Ginzburg mirror symmetry with FJRW-theory.

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arXiv2013

Quantum Pieri rules for tautological subbundles

We give quantum Pieri rules for quantum cohomology of Grassmannians of classical types, expressing the quantum product of Chern classes of the tautological subbundles with general cohomology classes. We derive them by showing the relevant genus zero, three-pointed Gromov-Witten invariants coincide with certain classical intersection numbers.

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arXiv2013

K-theoretic Gromov-Witten invariants of lines in homogeneous spaces

Let X=G/P be a homogeneous space and e_k be the class of a simple coroot in H_2(X). A theorem of Strickland shows that for almost all X, the variety of pointed lines of degree e_k, denoted Z_k(X), is again a homogeneous space. For these X we show that the 3-point, genus 0, equivariant K-theoretic Gromov-Witten invariants of lines of degree e_k are equal to quantities obtained in the (ordinary) equivariant K-theory of Z_k(X). We apply this to compute the structure constants N_{u,v}^{w, e_k} for degree e_k from the multiplication of two Schubert classes in the equivariant quantum K-theory ring of X. Using geometry of spaces of lines through Schubert or Richardson varieties we prove vanishing and positivity properties of N_{u,v}^{w,e_k}. This generalizes many results about K-theory and quantum cohomology of X, and also gives new identities among the structure constants in equivariant K-theory of X.

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