For the de Rham mapping cone cochain complex induced by a smooth closed 2-form, we explicitly write down the associated mapping cone Thom form in the sense of Mathai–Quillen. Our construction uses the mapping cone covariant derivative, carrying the extra information brought by the 2-form. Our main tool is the Berezin integral. As the main result, we show that this Thom form is closed with respect to the mapping cone differentiation, its integration along the fiber is 1, and it satisfies the transgression formula.
The chromatic threshold δ _χ (H) of a graph H is the infimum of d>0 such that the chromatic number of every n-vertex H-free graph with minimum degree at least dn is bounded by a constant depending only on H and d. Allen, Böttcher, Griffiths, Kohayakawa, and Morris determined the chromatic threshold for every H; in particular, they showed that if χ (H)=r≥ 3 , then δ _χ (H) ∈{r-3/r-2, 2 r-5/2 r-3, r-2/r-1} . While the chromatic thresholds have been completely determined, rather surprisingly the structural behaviors of extremal graphs near the threshold remain unexplored. In this paper, we establish the stability theorems for chromatic threshold problems. We prove that every n-vertex H-free graph G with δ (G)≥ (δ _χ (H)-o(1))n and χ (G)=ω (1) must be structurally close to one of the extremal configurations. Furthermore, we give a stronger stability result when H is a clique, showing that G admits a partition into independent sets and a small subgraph on a sublinear number of vertices. We show that this small subgraph has fractional chromatic number 2+o(1) and is homomorphic to a Kneser graph defined by subsets of a logarithmic size set; both these two bounds are best possible. This is the first stability result that captures the lower-order structural features of extremal graphs. We also study two variations of chromatic thresholds. Replacing chromatic number by its fractional counterpart, we determine the fractional chromatic thresholds for all graphs. Another variation is the bounded-VC chromatic thresholds, which was introduced by Liu, Shangguan, Skokan, and Xu very recently. Extending work of Łuczak and Thomassé on the triangle case, we determine the bounded-VC chromatic thresholds for all cliques.
Let Q(z) be a holomorphic Hecke cusp newform of square-free level and u_j (z) traverse an orthonormal basis of Hecke–Maass cusp forms of full level. Let 1/4 + t_j^2 be the Laplace eigenvalue of u_j (z) . In this paper, we prove that there is a constant γ ( Q ) expressed as a certain Euler product associated to Q such that at least γ ( Q ) / 11 of the Rankin–Selberg special L-values L (1/2+it_j, Q ⊗ u_j) for t_j ⩽ T do not vanish as T →∞ . Further, we show that the non-vanishing proportion is at least γ ( Q ) · (4μ -3) / (4μ +7) on the short interval |t_j - T| ⩽ T^μ for any 3/4< μ < 1 .
We study the modulated Korteweg–de Vries equation (KdV) on the circle with a time non-homogeneous modulation acting on the linear dispersion term. By adapting the normal form approach to the modulated setting, we prove sharp unconditional uniqueness of solutions to the modulated KdV in L^2(𝕋) if a modulation is sufficiently irregular. For example, this result implies that if the modulation is given by a sample path of a fractional Brownian motion with Hurst index 0< H < 2/5 , the modulated KdV on the circle is unconditionally well posed in L^2(𝕋) . At a philosophical level, our normal form approach can be viewed as a controlled path approach to nonlinear Young integration, which allows for the construction of solutions to the modulated KdV (and the associated nonlinear Young integral) without assuming any positive (Hölder) regularity in time. As an interesting byproduct of our normal form approach, we obtain an improved Euler approximation scheme as compared to the classical sewing lemma approach. We also establish analogous sharp unconditional uniqueness results for the modulated Benjamin–Ono equation and the modulated derivative nonlinear Schrödinger equation (NLS) with a quadratic nonlinearity. In the appendix, we prove sharp unconditional uniqueness of the cubic modulated NLS on the circle in H^1/6(𝕋) .
We investigate in this paper the so-called pointed Shafarevich problem for families of primitive symplectic varieties. More precisely, for any fixed pointed curve (B, 0) and any fixed primitive symplectic variety X, among all locally trivial families of ℚ-factorial and terminal primitive symplectic varieties over B whose fiber over 0 is isomorphic to X, we show that there are only finitely many isomorphism classes of generic fibers. Moreover, assuming semi-ampleness of isotropic nef divisors, which holds true for all hyper-Kähler manifolds of known deformation types, we show that there are only finitely many such projective families up to isomorphism. These results are optimal since we can construct infinitely many pairwise non-isomorphic (not necessarily projective) families of smooth hyper-Kähler varieties over some pointed curve (B, 0) such that they are all isomorphic over the punctured curve B\{0} and have isomorphic fibers over the base point 0.
