Rationality of varieties and $\mathbb{A}^1$-homotopy theory

In this article, we study the Bridgeland stability of the cotangent bundle on K3 surfaces and surfaces with ample canonical class. On a surface $X$ with ample canonical class $K$, we find values $t_0>0$ such that the cotangent bundle is semistable with respect to the Bridgeland stability $σ_{tK}$ for all $t > t_0$. We also study special cases such as the fake projective planes and the fake quadrics.

math.AG↗

arXiv2026

From wall structures to closed mirror symmetry. The case of $K_{\mathbb{P}^2}$: Renormalized periods over the positive real locus, closed Gromov-Witten invariants from wall functions, and tropical enumeration

We recover closed Gromov-Witten invariants and renormalized mirror periods for $K_{\mathbb{P}^2}$ by the same operation on a wall function. This gives a direct passage from the Gross-Siebert construction of intrinsic mirror pairs to classical enumerative mirror symmetry. The link is a polynomiality theorem for punctured invariants. Assuming the expected identification with the normalized slab function, we also obtain a finite tree sum for closed Gromov-Witten invariants in terms of types of plane tropical curves. The mechanism is expected to extend to more general Calabi-Yau mirror pairs.

math.AG↗

arXiv2026

The topology of Kähler manifolds and 1-forms without zeros

We study how the topology and geometry of a compact Kähler manifold relate to the zeros of its one-forms. We determine all implications among several closely related conditions. In particular, we construct a smooth projective variety for which the Aomoto complex is exact for every nonzero holomorphic one-form and every semisimple local system, although every closed real one-form, and hence every holomorphic one-form, has a zero.

math.AG↗