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## Asymptotically log Fano varieties

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimensional strongly asymptotically log smooth log Fano varieties. Based on this classification we formulate an asymptotic logarithmic version of Calabi's conjecture for del Pezzo surfaces for the existence of Kähler-Einstein edge metrics in this regime. We make some initial progress towards its proof by demonstrating some existence and non-existence results, among them a generalization of Matsushima's result on the reductivity of the automorphism group of the pair, and results on log canonical thresholds of pairs. One by-product of this study is a new conjectural picture for the small angle regime and limit which reveals a rich structure in the asymptotic regime, of which a folklore conjecture concerning the case of a Fano manifold with an anticanonical divisor is a special case. © 2015 Elsevier Inc.

We consider Landau–Ginzburg models for smooth Fano threefolds of the principal series and prove that they can be represented by Laurent polynomials. We check that these models can be compactified to open Calabi–Yau varieties. In the spirit of Katzarkov's programme we prove that the numbers of irreducible components of the central fibres of compactifications of these pencils are equal to the dimensions of intermediate Jacobians of the corresponding Fano varieties plus 1. In particular, these numbers are independent of the choice of compactification. We state most of the known methods for finding Landau–Ginzburg models in terms of Laurent polynomials. We discuss the Laurent polynomial representation of the Landau–Ginzburg models of Fano varieties and state some related problems.

For any smooth quartic threefold in *P*4 we classify pencils on it whose general element is an irreducible surface birational to a surface of Kodaira dimension zero.

The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class associated to Euler's Gamma-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.

We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel–Mukai fourfold in more detail. Namely, we show this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational Gushel–Mukai fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel–Mukai varieties, which was one of the main motivations for this work.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.