arXiv subjects

Zhitong Su

Publications and source records attributed to Zhitong Su.

5 recordsLinked to original sources

arXiv2026

A Decomposition Lemma in Convex Integration via Classical Algebraic Geometry

In this paper, we prove a decomposition lemma for symmetric matrix fields on bounded domains: $D+\mathrm{Sym}\nablaΦ=\sum_i a_i^2ξ_i\otimesξ_i$ with uniform control on $Φ$ and $a_i^2$, using fewer than the usual $n(n+1)/2$ rank-one symmetric terms. Except possibly in dimensions $n=8,16$, the decomposition is shown to be optimal through algebraic arguments. This reduces the number of steps in convex integration for a nonlinear PDE system, improving Hölder regularity of flexible solutions in dimension $n\ge3$. This PDE is a partial linearization of the codimension-one local isometric embedding equation in the Nash--Kuiper theorem, and also yields improved regularity for very weak solutions of related 2D Monge--Ampére and $2$-Hessian systems. The improved Hölder exponent is any $α<(n^2+1)^{-1}$ for $n=2,4,8,16$ and any $α<(n^2+n-2ρ(n/2)-1)^{-1}$ otherwise, where $ρ$ is the Radon--Hurwitz number, related to Bott periodicity. The proof involves novel applications of algebraic geometry and topology that yield the optimality of decomposition, including Adams' theorem on vector fields on spheres, intersections of projective varieties, and projective duality, combined with an elliptic method that avoids loss of differentiability.

math.AP↗

arXiv2026

Conformal Embeddings via Heat Kernel

For any n-dimensional compact Riemannian Manifold $M$ with smooth metric $g$, by employing the heat kernel embedding introduced by Bérard-Besson-Gallot'94, we intrinsically construct a canonical family of conformal embeddings $C_{t,k}$: $M\rightarrow\mathbb{R}^{q(t)}$, with $t>0$ sufficiently small, $q(t)\gg t^{-\frac{n}{2}}$, and $k$ as a function of $O(t^l)$ in proper sense. Our approach involves finding all these canonical conformal embeddings, which shows the distinctions from the isometric embeddings introduced by Wang-Zhu'15.

math.DG↗

arXiv2026

Gamma conjecture II via global Gamma-I

For a Fano manifold $X$, Gamma conjecture II aims to use $\mathcal{D}_{\rm{coh}}^b(X)$ to describe the asymptotic behavior of its Dubrovin connection via $\widehatΓ$-integral structure. It was proposed by Galkin, Golyshev and Iritani, and can be regarded as a quantitative refinement of Dubrovin's conjecture on Fano manifolds with semisimple big quantum cohomology. As a step toward Gamma conjecture II, we define the Gamma-I property at points satisfying the (SR) condition, arising from the original Gamma conjecture I. We prove that the property holds globally in the following sense: if it holds at one such point, then it holds throughout the connected component of the (SR)-region containing that point. Based on this global Gamma-I property, we establish a strategy-type theorem relating Gamma conjecture II to the Gamma-I property at a possibly non-semisimple point, together with an analysis of small quantum cohomology. We further apply this theorem to prove Gamma conjecture II for del Pezzo surfaces; the proof combines Iritani's Galois action with addtional elementary operations on exceptional collections, and its most technically involved step consists in verifying the required global Gamma-I property.

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↗