We prove that any compact Kahler 3-dimensional manifold which has no nontrivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of so-called simple manifolds, central in the bimeromorphic classication of compact Kahler manifolds. The proof follows from the Brunella pseudo-eectivity theorem, combined with fundamental results of Siu and of the second author on the Le- long numbers of closed positive (1;1)-currents, and with a version of the hard Lefschetz theorem for pseudo-eective line bundles, due to Takegoshi and Demailly-Peternell- Schneider. In a similar vein, we show that a normal compact and Kahler 3-dimensional analytic space with terminal singularities and nef canonical bundle is a cyclic quotient of a simple nonprojective torus if it carries no eective divisor. This is a crucial step towards completing the bimeromorphic classication of compact Kahler threefolds.
This paper performs a systematic study of Gushel–Mukai varieties —Fano mani- folds with Picard number 1, coindex 3, and degree 10 (higher-dimens ional analogues of prime Fano threefolds of genus 6). We introduce a new approach to the c lassification of these va- rieties which includes mildly singular varieties, gives a criterion for an iso morphism of such varieties, and describes their automorphisms groups. We carefully develop the relation between Gushel–Mukai varieties an d Eisenbud–Popescu– Walter sextics introduced earlier by Iliev–Manivel and O’Grady. We de scribe explicitly all Gushel–Mukai varieties whose associated EPW sextics are isomorph ic or dual (we call them period partners or dual varieties respectively). Finally, we show th at in dimension 3 and higher, period partners/dual varieties are always birationally isomo rphic.
Our goal in this paper is to discuss a conjectural correspondence between enumerative geometry of curves in Calabi-Yau 5-folds Z and 1-dimensional sheaves on 3-folds X that are embedded in Z as fixed points of certain C×-actions. In both cases, the enumerative information is taken in equivariant K-theory, where the equivariance is with respect to all automorphisms of the problem. In Donaldson-Thomas theories, one sums up over all Euler characteristics with a weight (−q)χ, where q is a parameter, informally referred to as the boxcounting parameter. The main feature of the correspondence is that the 3-dimensional boxcounting parameter q becomes in 5 dimensions the equivariant parameter for the C×-action that defines X inside Z. The 5-dimensional theory effectively sums up the q-expansion in the Donaldson-Thomas theory. In particular, it gives a natural explanation of the rationality (in q) of the DT partition functions. Other expected as well as unexpected symmetries of the DT counts follow naturally from the 5-dimensional perspective. These involve choosing different C×-actions on the same Z, and thus relating the same 5-dimensional theory to different DT problems. The important special case Z=X×C2 is considered in detail in Sections 7 and 8. If X is a toric Calabi-Yau threefold, we compute the theory in terms of a certain index vertex. We show the refined vertex found combinatorially by Iqbal, Kozcaz, and Vafa is a special case of the index vertex.
Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic 0. The natural Poisson structure on the zastava space Z can be described in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian reduction of the corresponding quotient of its universal enveloping algebra produces a quantization Y of the coordinate ring of Z. The same quantization was obtained in the finite (as opposed to the affine) case generically in arXiv:math/0409031 . We prove that Y is a quotient of the affine Borel Yangian. The analogous results for g=slN were obtained in our previous work arXiv:1009.0676 .
In a previous paper we established that for any del Pezzo surface Y of degree at least 4, the affine cone X over Y embedded via a pluri-anticanonical linear system admits an effective Ga-action. In particular, the group Aut(X) is infinite dimensional. In contrast, we show in this note that for a del Pezzo surface Y of degree at most 2 the generalized cones X as above do not admit any non-trivial action of a unipotent algebraic group.