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Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let Y denote the set of fixed points of H in X, and N(H) the normalizer of H in G. In this paper we study the natural map from the quotient of Y by N(H) to the quotient of X by G induced by the inclusion of Y in X. We show that, given G and H, this map is a finite morphism for all G-varieties X if and only if H is G-completely reducible (in the sense defined by J-P. Serre); this was proved in characteristic zero by Luna in the 1970s. We discuss some applications and give a criterion for the map of quotients to be an isomorphism. We show how to extend some other results in Lunas paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; then the double coset HgK is closed for generic g in G if and only if the intersection of generic conjugates of H and K is reductive.
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In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure $overline{B(1)}$ of type A, and study relations between an NCCR and crepant resolutions $Y$ and $Y^+$ of $overline{B(1)}$. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions $Y$ and $Y^+$ of $overline{B(1)}$ as moduli spaces of representations of the quiver. We also study the Kawamata-Namikawas derived equivalence between crepant resolutions $Y$ and $Y^+$ of $overline{B(1)}$ in terms of an NCCR. We also show that the P-twist on the derived category of $Y$ corresponds to a certain operation of the NCCR, which we call multi-mutation, and that a multi-mutation is a composition of Iyama-Wemysss mutations.
If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.
According to a well-known theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the nilpotent cone, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type $A_{2k-1}$. In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities which do not occur in the classical types. Three of these are unibranch non-normal singularities: an $SL_2(mathbb C)$-variety whose normalization is ${mathbb A}^2$, an $Sp_4(mathbb C)$-variety whose normalization is ${mathbb A}^4$, and a two-dimensional variety whose normalization is the simple surface singularity $A_3$. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, in analogy with Slodowys work for the regular nilpotent orbit.
The moduli space of stable pairs on a local surface $X=K_S$ is in general non-compact. The action of $mathbb{C}^*$ on the fibres of $X$ induces an action on the moduli space and the stable pair invariants of $X$ are defined by the virtual localization formula. We study the contribution to these invariants of stable pairs (scheme theoretically) supported in the zero section $S subset X$. Sometimes there are no other contributions, e.g. when the curve class $beta$ is irreducible. We relate these surface stable pair invariants to the Poincare invariants of Durr-Kabanov-Okonek. The latter are equal to the Seiberg-Witten invariants of $S$ by work of Durr-Kabanov-Okonek and Chang-Kiem. We give two applications of our result. (1) For irreducible curve classes the GW/PT correspondence for $X = K_S$ implies Taubes GW/SW correspondence for $S$. (2) When $p_g(S) = 0$, the difference of surface stable pair invariants in class $beta$ and $K_S - beta$ is a universal topological expression.
We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.
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