Changzheng Li
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
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Changzheng Li has a total of 36 co-authors. Here are the top co-authors by number of shared publications:
- Mingzhi Yang
School of Mathematics, Sun Yat-Sen University, Guangzhou, China
7 shared publications - Jiayu Song
School of Mathematics, Sun Yat-Sen University, Guangzhou, China
6 shared publications - Hua-Zhong Ke
Sun Yat-sen University
5 shared publications - Leonardo Constantin Mihalcea
Department of Mathematics MC 225 Stanger Street Virginia Tech 0123 24061 Blacksburg VA
5 shared publications - Chi Zhang
Department of Mathematics, California Institute of Technology, Pasadena, Calif., U.S.A.
4 shared publications - Frank Sottile
Texas A&M University
4 shared publications
Latest Publications
- Open AccessQuantum Schubert calculus for smooth Schubert divisors of $F\ell_n$
We propose to study the quantum Schubert calculus for Schubert varieties, and investigate the smooth Schubert divisors X of the complete flag variety Fl_n. We provide a Borel-type ring presentation of the quantum cohomology of X. We derive the quantum Chevalley formula for X by geometric arguments. We also show that the quantum Schubert polynomials for X are the same as that for Fl_n introduced by Fomin, Gelfand and Postnikov.
Open AccessQuantum Schubert calculus for smooth Schubert divisors of $F\ell_n$We propose to study the quantum Schubert calculus for Schubert varieties, and investigate the smooth Schubert divisors X of the complete flag variety Fl_n. We provide a Borel-type ring presentation of the quantum cohomology of X. We derive the quantum Chevalley formula for X by geometric arguments. We also show that the quantum Schubert polynomials for X are the same as that for Fl_n introduced by Fomin, Gelfand and Postnikov.
- Open AccessToward quantum Pieri rule for $F\ell_n$ via Seidel representationBy using a ``quantum-to-classical" reduction formula on the Gromov-Witten invariants of flag vaireities $F\ell_n$, we provide a new proof of the Seidel operator on the quantum cohomology ring $QH^*(F\ell_n)$. Further, we reprove a quantum Pieri rule with respect to certain special Schubert class for $QH^*(F\ell_n)$. Finally, we propose a concrete conjecture on the corresponding quantum Pieri rule for the quantum $K$-theory of $F\ell_n$.Open AccessToward quantum Pieri rule for $F\ell_n$ via Seidel representationBy using a ``quantum-to-classical" reduction formula on the Gromov-Witten invariants of flag vaireities $F\ell_n$, we provide a new proof of the Seidel operator on the quantum cohomology ring $QH^*(F\ell_n)$. Further, we reprove a quantum Pieri rule with respect to certain special Schubert class for $QH^*(F\ell_n)$. Finally, we propose a concrete conjecture on the corresponding quantum Pieri rule for the quantum $K$-theory of $F\ell_n$.
- Open Access
Towards small quantum Chern character
We show a quantum version of Chern character homomorphism from the small quantum K-theory to the small quantum cohomology in the cases of projective spaces and incidence varieties, whose classical limit gives the classical Chern character homomorphism. We also provide a ring presentation of the small quantum K-theory of Milnor hypersurfaces.
Open AccessTowards small quantum Chern character
We show a quantum version of Chern character homomorphism from the small quantum K-theory to the small quantum cohomology in the cases of projective spaces and incidence varieties, whose classical limit gives the classical Chern character homomorphism. We also provide a ring presentation of the small quantum K-theory of Milnor hypersurfaces.
- Open Access
An anticanonical perspective on G/P Schubert varieties
We describe a natural basis of the Cartier class group of an arbitrary Schubert variety $X_{w,P}$ in a flag variety $G/P$ of general Lie type. We then characterise when the Schubert variety is factorial/Fano, along with an explicit formula for the anticanonical line bundle in these cases. We also prove that, for Schubert varieties in simply-laced types (only), being factorial is equivalent to being $Q$-factorial, and is equivalent to the equality of the Betti numbers $b_2(X_{w,P})=b_{2\ell(w)-2}(X_{w,P})$. Finally, we give a convenient characterisation of when a simply-laced Schubert variety is Gorenstein and when it is Gorenstein Fano.Open AccessAn anticanonical perspective on G/P Schubert varieties
We describe a natural basis of the Cartier class group of an arbitrary Schubert variety $X_{w,P}$ in a flag variety $G/P$ of general Lie type. We then characterise when the Schubert variety is factorial/Fano, along with an explicit formula for the anticanonical line bundle in these cases. We also prove that, for Schubert varieties in simply-laced types (only), being factorial is equivalent to being $Q$-factorial, and is equivalent to the equality of the Betti numbers $b_2(X_{w,P})=b_{2\ell(w)-2}(X_{w,P})$. Finally, we give a convenient characterisation of when a simply-laced Schubert variety is Gorenstein and when it is Gorenstein Fano. - Open Access
Unfolding of equivariant F-bundles and application to the mirror symmetry of flag varieties
We establish an unfolding theorem for equivariant F-bundles (a variant of Frobenius manifolds), generalizing Hertling-Manin's universal unfolding of meromorphic connections. As an application, we obtain the mirror symmetry theorem for the big quantum cohomology of flag varieties, from the recent works on the small quantum cohomology mirror symmetry, via the equivariant unfolding theorem.
Open AccessUnfolding of equivariant F-bundles and application to the mirror symmetry of flag varieties
We establish an unfolding theorem for equivariant F-bundles (a variant of Frobenius manifolds), generalizing Hertling-Manin's universal unfolding of meromorphic connections. As an application, we obtain the mirror symmetry theorem for the big quantum cohomology of flag varieties, from the recent works on the small quantum cohomology mirror symmetry, via the equivariant unfolding theorem.
- 21 March 2025
- Published by Springer Nature in Science China Mathematics
- Vol. 68 (7) , 1523-1548
On Seidel representation in quantum K-theory of Grassmannians
- 21 March 2025
- Published by Springer Nature in Science China Mathematics
- Vol. 68 (7) , 1523-1548
On Seidel representation in quantum K-theory of Grassmannians
- Open Access
On the quantum cohomology of blow-ups of four-dimensional quadrics
We propose a conjecture relevant to Galkin's lower bound conjecture, and verify it for the blow-ups of a four-dimensional quadric at a point or along a projective plane. We also show that Conjecture $\mathcal{O}$ holds in these two cases.Open AccessOn the quantum cohomology of blow-ups of four-dimensional quadrics
We propose a conjecture relevant to Galkin's lower bound conjecture, and verify it for the blow-ups of a four-dimensional quadric at a point or along a projective plane. We also show that Conjecture $\mathcal{O}$ holds in these two cases. - Open Access
Mirror symmetry for certain blowups of Grassmannians
We classify when the blowup of a complex Grassmannian $G(k, n)$ along a smooth Schubert subvariety $Z$ is Fano. We compute almost all the two-point, genus zero Gromov-Witten invariants of the blowup when $Z=G(k, n-1)$. We further prove a mirror symmetry statement for the blowup $X_{2, n}$ of $G(2, n)$ along $G(2, n-1)$, by introducing a toric superpotential $f_{\rm tor}$ and showing the isomorphism between the Jacobi ring of $f_{\rm tor}$ and the small quantum cohomology ring $QH^*(X_{2, n})$.Open AccessMirror symmetry for certain blowups of Grassmannians
We classify when the blowup of a complex Grassmannian $G(k, n)$ along a smooth Schubert subvariety $Z$ is Fano. We compute almost all the two-point, genus zero Gromov-Witten invariants of the blowup when $Z=G(k, n-1)$. We further prove a mirror symmetry statement for the blowup $X_{2, n}$ of $G(2, n)$ along $G(2, n-1)$, by introducing a toric superpotential $f_{\rm tor}$ and showing the isomorphism between the Jacobi ring of $f_{\rm tor}$ and the small quantum cohomology ring $QH^*(X_{2, n})$. - Open Access
- 8 October 2024
- Published by Science China Press., Co. Ltd. in SCIENTIA SINICA Mathematica
- Vol. 54 (12) , 2009
Toward the quantum Pieri rule for
我们简要回顾梁乃聪与本文第一作者的关于旗簇量子上同调的函子性及由此获得的 Gromov-WItten 不变量的 ``量子 $\rightarraw$ 经典" 约化。利用该约化公式，我们重新证明了 Seidel 算子运算公式， 并由此证明旗簇 $F\ell_n$ 的量子上同调中关于一组特殊 Schubert 类的量子 Pieri 法则。我们也对 $F\ell_n$ 的量子 $K$ 理论相应的量子 Pieri 法则提出了具体猜想。- 8 October 2024
- Published by Science China Press., Co. Ltd. in SCIENTIA SINICA Mathematica
- Vol. 54 (12) , 2009
Open AccessToward the quantum Pieri rule for
我们简要回顾梁乃聪与本文第一作者的关于旗簇量子上同调的函子性及由此获得的 Gromov-WItten 不变量的 ``量子 $\rightarraw$ 经典" 约化。利用该约化公式，我们重新证明了 Seidel 算子运算公式， 并由此证明旗簇 $F\ell_n$ 的量子上同调中关于一组特殊 Schubert 类的量子 Pieri 法则。我们也对 $F\ell_n$ 的量子 $K$ 理论相应的量子 Pieri 法则提出了具体猜想。 - Open Access
A Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus
We construct a Plücker coordinate superpotential $\mathcal{F}_-$ that is mirror to a partial flag variety $\mathbb{ F}\ell(n_\bullet)$. Its Jacobi ring recovers the small quantum cohomology of $\mathbb{ F}\ell(n_\bullet)$ and we prove a folklore conjecture in mirror symmetry. Namely, we show that the eigenvalues for the action of the first Chern class $c_1(\mathbb{ F}\ell(n_\bullet))$ on quantum cohomology are equal to the critical values of $\mathcal{F}_-$. We achieve this by proving new identities in quantum Schubert calculus that are inspired by our formula for $\mathcal{F}_-$ and the mirror symmetry conjecture.Open AccessA Plücker coordinate mirror for partial flag varieties and quantum Schubert calculus
We construct a Plücker coordinate superpotential $\mathcal{F}_-$ that is mirror to a partial flag variety $\mathbb{ F}\ell(n_\bullet)$. Its Jacobi ring recovers the small quantum cohomology of $\mathbb{ F}\ell(n_\bullet)$ and we prove a folklore conjecture in mirror symmetry. Namely, we show that the eigenvalues for the action of the first Chern class $c_1(\mathbb{ F}\ell(n_\bullet))$ on quantum cohomology are equal to the critical values of $\mathcal{F}_-$. We achieve this by proving new identities in quantum Schubert calculus that are inspired by our formula for $\mathcal{F}_-$ and the mirror symmetry conjecture.