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The risk of inducing hypoglycaemia (low blood glucose) constitutes the main challenge associated with insulin therapy for diabetes. Insulin doses must be adjusted to ensure blood glucose values within normal range, but matching insulin doses to fluctuating glucose levels is difficult since even a slightly higher insulin dose than needed can lead to a hypoglycaemic incidence which can be anything from uncomfortable to life-threatening. It has thus been a long-standing goal to engineer a glucose-sensitive insulin that can auto-adjust its bioactivity in a reversible manner according to ambient glucose levels to ultimately achieve better glycaemic control while lowering the risk of hypoglycaemia. Here we report the design and properties of NNC2215, an insulin conjugate with bioactivity that is reversibly responsive to glucose over the physiological range as demonstrated in vitro and in vivo. NNC2215 was engineered by conjugating a glucose-binding macrocycle and a glucoside to insulin, thereby introducing a “switch” that can open and close in response to glucose and thereby equilibrate insulin between active and less active conformations. The insulin receptor affinity of NNC2215 increased 12-fold when glucose concentration was increased from 0 to 20 mM. In animal studies, the glucose-sensitive bioactivity of NNC2215 was demonstrated to lead to protection against hypoglycaemia while partially covering glucose excursions.

In this article, we investigate Muirhead's classical system of differential operators for the hypergeometric function 1F1 of a matrix argument. We formulate a conjecture for the combinatorial structure of the characteristic variety of its Weyl closure which is both supported by computational evidence as well as theoretical considerations. In particular, we determine the singular locus of this system., Comment: 26 pages. v2 changes: included Appendix discussing the singular locus for degenerate parameters

We study the deformation theory of rational curves on primitive symplectic varieties and show that if the rational curves cover a divisor, then, as in the smooth case, they deform along their Hodge locus in the universal locally trivial deformation. As applications, we extend Markman's deformation invariance of prime exceptional divisors along their Hodge locus to this singular framework and provide existence results for uniruled ample divisors on primitive symplectic varieties which are locally trivial deformations of any moduli space of semistable objects on a projective $K3$ or fibers of the Albanese map of those on an abelian surface. We also present an application to the existence of prime exceptional divisors.

We show that the Hilbert scheme compactification of the total space of Starr’s fibration on the space of twisted cubics on a cubic hypersurface in $${\mathbb P}^5$$ not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight which has a crepant resolution deformation equivalent to the symplectic eightfold constructed from twisted cubics on a smooth cubic fourfold. This yields another proof that the symplectic eightfold and the Hilbert scheme of four points on a K3 surface are deformation equivalent. As a byproduct we obtain similar results for the variety of lines on a singular cubic fourfold.

This article is a survey about or introduction to certain aspects of the complex geometry of a hypothetical complex structure on the six-sphere. We discuss a result of Peternell–Campana–Demailly on the algebraic dimension of a hypothetical complex six-sphere and give some examples. We also give an overview over an application of Huckleberry–Kebekus–Peternell on the group of biholomorphisms of such a complex six-sphere.

We develop the global moduli theory of symplectic varieties in the sense of Beauville. We prove a number of analogs of classical results from the smooth case, including a global Torelli theorem. In particular, this yields a new proof of Verbitsky's global Torelli theorem in the smooth case (assuming $b_2\geq 5$) which does not use the existence of a hyperk\"ahler metric or twistor deformations., Comment: Minor corrections. Final version, to appear in J. Reine Angew. Math

We construct a new twenty-dimensional family of projective eight-dimensional irreducible holomorphic symplectic manifolds: the compactified moduli space M_3(Y) of twisted cubics on a smooth cubic fourfold Y that does not contain a plane is shown to be smooth and to admit a contraction M_3(Y) -> Z(Y) to a projective eight-dimensional symplectic manifold Z(Y). The construction is based on results on linear determinantal representations of singular cubic surfaces., A proof that Z(Y) is simply-connected and irreducible hyperk\"ahler has been added. A couple of typos corrected

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak c$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak c \oplus {\mathfrak c}^*/W \to V\oplus V^*/\!\!/\!\!/ G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if $(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and certain further examples., Comment: 18 pages, final version

Let X be a compact hyperk\"ahler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L. We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair (X, L), which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian fibration there exists a smooth good minimal model, i.e., a hyperk\"ahler manifold birational to X on which the fibration is holomorphic., Comment: 28 pages; v2: minor corrections, added Thm 6.7; v3: implemented referees' comments, added slightly more detailed argument in Sect. 6.3.3, accepted for publication by Annales scientifiques de l'\'Ecole normale sup\'erieure

International audience; We generalize Huybrechts' theorem on deformation equivalence of birational irreducible symplectic manifolds to the singular setting. More precisely, under suitable natural hypotheses, we show that two birational symplectic varieties are locally trivial deformations of one another. As an application we show the termination of any log-minimal model program for a pair (X, ∆) of a projective irreducible symplectic manifold X and an effective R-divisor ∆. To prove this result we follow Shokurov's strategy and show that LSC and ACC for mlds hold for all the models appearing along any log-MMP of the initial pair.

We systematically study the moduli theory of symplectic varieties (in the sense of Beauville) which admit a resolution by an irreducible symplectic manifold. In particular, we prove an analog of Verbitsky's global Torelli theorem for the locally trivial deformations of such varieties. Verbitsky's work on ergodic complex structures replaces twistor lines as the essential global input. In so doing we extend many of the local deformation-theoretic results known in the smooth case to such (not-necessarily-projective) symplectic varieties. We deduce a number of applications to the birational geometry of symplectic manifolds, including some results on the classification of birational contractions of $K3^{[n]}$-type varieties., Comment: Final version, to appear in J. Eur. Math. Soc

We develop an invariant deformation theory, in a form accessible to practice, for affine schemes $W$ equipped with an action of a reductive algebraic group $G$. Given the defining equations of a $G$-invariant subscheme $X \subset W$, we device an algorithm to compute the universal deformation of $X$ in terms of generators and relations up to a given order. In many situations, our algorithm even computes an algebraization of the universal deformation. As an application, we determine new families of examples of the invariant Hilbert scheme of Alexeev and Brion, where $G$ is a classical group acting on a classical representation, and describe their singularities., Comment: 43 pages, final version, to appear in J. Pure Appl. Algebra

We investigate the stability of fibers of coisotropic fibrations on holomorphic symplectic manifolds and generalize Voisin's result on Lagrangian subvarieties to this framework. We present applications to the moduli space of holomorphic symplectic manifolds which are deformations equivalent to Hilbert schemes of points on a $K3$ surface or to generalized Kummer manifolds., Comment: Minor changes in the Introduction and in the body of the paper to improve the presentation of the results

We show that the base manifold of a Lagrangian fibration on a hyperk\"ahler manifold is isomorphic to complex projective space. This generalises a theorem of J.-M. Hwang to the K\"ahler case., Comment: 4 pages; v2: minor changes, final version, to appear in IMRN

We develop a Hodge theory for relative simple normal crossing varieties over an Artinian base scheme. We introduce the notion of a mixed Hodge structure over an Artin ring, which axiomatizes the structure that is found on the cohomology of such a variety. As an application we prove that the maps between the graded pieces of the Hodge bundles have constant rank., Comment: 26 pages. arXiv admin note: substantial text overlap with arXiv:1112.1887

We generalize Voisin's theorem on deformations of pairs of a symplectic manifold and a Lagrangian submanifold to the case of Lagrangian normal crossing subvarieties. Partial results are obtained for arbitrary Lagrangian subvarieties. We apply our results to the study of singular fibers of Lagrangian fibrations., Comment: 21 pages, rewritten according to the referee's suggestions, minor errors and typos corrected, main results improved, applications added

Animal studies indicate some persistent organic pollutants (POPs) may be endocrine disruptors and suggest similar health risks for humans. Recent studies have further suggested an association between exposure to POPs and the prevalence of type 2 diabetes. Now an experimental animal study reports evidence of a causal link between POP exposure and insulin resistance syndrome, a cluster of metabolic disorders—including type 2 diabetes—that are marked by sustained high blood sugar [EHP 118:465–471; Ruzzin et al.]. POPs accumulate in fatty tissue, where they can remain for years because they are not easily broken down. Fatty fish are a potential source of POP exposure in many human populations. However, n-3 polyunsaturated fatty acids in fish oil may have beneficial health effects, possibly including protective effects on insulin resistance, that could counterbalance any adverse effects of POPs in fatty fish. In the current study, rats were exposed for 28 days to high-fat diets that contained either crude fish oil (from farmed Atlantic salmon) or fish oil that was refined to remove POPs. As expected, the crude fish oil contained much higher levels of POPs than the refined oil. Gene expression profile comparisons of the livers of the 2 treatment groups showed that POP exposure disrupted lipid homeostasis. Rats on the unrefined fish oil diet gained more weight overall and had increased triacylglycerol, diacylglycerol, and total cholesterol levels compared with rats fed refined fish oil. They also showed impaired insulin action in response to the high-fat diet, whereas the high-fat diet did not seem to cause insulin resistance in the rats fed refined fish oil. Further analysis revealed a reduction in the ability of insulin to stimulate glucose uptake in adipocytes treated with a mixture of POPs that was comparable to the mixture of chemicals in crude fish oil. Adipocyte responses to insulin varied with exposures to different mixtures of individual POPs. The authors conclude that dietary exposure to POPs may be a risk factor for insulin resistance and associated metabolic disorders. Furthermore, the metabolic effects of POP exposures exacerbated deleterious effects of a high-fat diet on rats and appeared to negate protective effects of n-3 polyunsaturated fatty acids on insulin resistance.