arXiv subjects

Huazhong Ke

Publications and source records attributed to Huazhong Ke.

4 recordsLinked to original sources

arXiv2025

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 $QH^*(X_{2, n})$.

math.AG↗

arXiv2025

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.

math.AG↗

arXiv2025

Revisiting Gamma conjecture I: counterexamples and modifications

We continue investigation of asymptotics of quantum differential equation for Fano manifolds, with a special regard to Gamma conjecture I and its underlying Conjecture $\mathcal{O}$. We introduce the A-model conifold value, a symplectic invariant of a Fano manifold, and propose modifications for Gamma conjecture I based on this new definition. We discuss an interplay of birational transformations with an extension of Gamma conjecture I over the Kähler moduli space. These heuristics are applied to rigorously identify the principal asymptotic class in the case of $\mathbb{P}^1$-bundles $X_n=\mathbb{P}_{\mathbb{P}^{n}}(\mathcal{O}\oplus\mathcal{O}(n))$. We observe, in particular, that for $X_n$ of dimension at least four, the Conjecture $\mathcal{O}$ holds just for even values of $n$, and in these cases we falsify the original non-modified Gamma conjecture I.

math.AG↗

arXiv2024

On Galkin's Lower Bound Conjecture

We estimate an upper bound of the spectral radius of a linear operator on the quantum cohomology of the toric Fano manifolds $\mathbb{P}_{\mathbb{P}^{n}}(\mathcal{O}\oplus\mathcal{O}(3))$. This provides a negative answer to Galkin's lower bound conjecture.

math.AG↗