International Journal of Mathematics

We recover the wreath product X ≔ Sym²(ℂ²/± 1) as a transversal slice to a nilpotent orbit in 𝔰𝔭₆. By using deformations of Springer resolutions, we construct a symplectic deformation of symplectic resolutions of X.

Quantum integrable systems related with group U(1,3) are considered. The explicit expressions of waves functions, Harish–Chandra c-functions and S-matrices are given.

We argue that the existence of genus one fibrations with multisections of high degree on certain Calabi–Yau threefolds implies the existence of pairs of such varieties that are not birational, but are derived equivalent. It also (likely) implies the existence of non-birational counterexamples to the Torelli problem for Calabi–Yau threefolds.

We define and study a family of cubic surfaces in the projectivized tangent bundle over a four-dimensional projective space associated to the root system D₅. The 27 lines are rational over the base and we determine the classifying map to the moduli space of marked cubic surfaces. This map has degree two and we use it to get short proofs for some results on the Chow group of the moduli space of marked cubic surfaces.

Let M be a compact complex manifold with a positive holomorphic line bundle L, and K be its canonical line bundle. We give some sufficient conditions for the non-vanishing of H⁰(M, K + L). We also show that the criterion can be applied to interesting classes of examples including all compact locally hermitian symmetric spaces of non-compact type, Mostow–Siu [10] surfaces, Kähler threefolds given by Deraux [3] and examples of Zheng [17].

Let C be a curve of genus two. We denote by the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over C, and by J^d the variety of line bundles of degree d on C. In particular, J¹ has a canonical theta divisor Θ. The space is a double cover of ℙ⁸ = |3Θ| branched along a sextic hypersurface, the Coble sextic. In the dual , where J¹ is embedded, there is a unique cubic hypersurface singular along J¹, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre–Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.

We give sufficient conditions for a closed smooth hypersurface W in the n-dimensional Bergman ball to be interpolating or sampling. As in the recent work [5] of Ortega-Cerdà, Schuster and the second author on the Bargmann–Fock space, our sufficient conditions are expressed in terms of a geometric density of the hypersurface that, though less natural, is shown to be equivalent to Bergman ball analogs of the Beurling-type densities used in [5]. In the interpolation theorem we interpolate L² data from W to the ball using the method of Ohsawa–Takegoshi, extended to the present setting, rather than the Cousin I approach used in [5]. In the sampling theorem, our proof is completely different from [5]. We adapt the more natural method of Berndtsson and Ortega-Cerdà [1] to higher dimensions. This adaptation motivated the notion of density that we introduced. The approaches of [5] and the present paper both work in either the case of the Bergman ball or of the Bargmann–Fock space.

We consider the deformations of the two-dimensional complex analytic variety constructed from a genus 2 Riemann surface by attaching its self-product to its Jacobian in an elementary way. The deformations are shown to be unobstructed, the variety smooths to give complex projective manifolds whose invariants are computed and whose images under Albanese maps (re)verify an instance of the Hodge conjecture for certain abelian fourfolds.