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We analyze the relationship between two compactifications of the moduli space of maps from curves to a Grassmannian: the Kontsevich moduli space of stable maps and the Marian–Oprea–Pandharipande moduli space of stable quotients. We construct a moduli space which dominates both the moduli space of stable maps to a Grassmannian and the moduli space of stable quotients, and equip our moduli space with a virtual fundamental class. We relate the virtual fundamental classes of all three moduli spaces using the virtual push-forward formula. This gives a new proof of a theorem of Marian–Oprea–Pandharipande: that enumerative invariants defined as intersection numbers in the stable quotient moduli space coincide with Gromov–Witten invariants.
Let
$\mathfrak{g}=\mbox{Lie}(G)$
be the Lie algebra of a simple algebraic group
$G$
over an algebraically closed field of characteristic
$0$
. Let
$e$
be a nilpotent element of
$\mathfrak{g}$
and let
$\mathfrak{g}_e=\mbox{Lie}(G_e)$
where
$G_e$
stands for the stabiliser of
$e$
in
$G$
. For
$\mathfrak{g}$
classical, we give an explicit combinatorial formula for the codimension of
$[\mathfrak{g}_e,\mathfrak{g}_e]$
in
$\mathfrak{g}_e$
and use it to determine those
$e\in \mathfrak{g}$
for which the largest commutative quotient
$U(\mathfrak{g},e)^{\mbox{ab}}$
of the finite
$W$
-algebra
$U(\mathfrak{g},e)$
is isomorphic to a polynomial algebra. It turns out that this happens if and only if
$e$
lies in a unique sheet of
$\mathfrak{g}$
. The nilpotent elements with this property are called non-singular in the paper. Confirming a recent conjecture of Izosimov, we prove that a nilpotent element
$e\in \mathfrak{g}$
is non-singular if and only if the maximal dimension of the geometric quotients
$\mathcal{S}/G$
, where
$\mathcal{S}$
is a sheet of
$\mathfrak{g}$
containing
$e$
, coincides with the codimension of
$[\mathfrak{g}_e,\mathfrak{g}_e]$
in
$\mathfrak{g}_e$
and describe all non-singular nilpotent elements in terms of partitions. We also show that for any nilpotent element
$e$
in a classical Lie algebra
$\mathfrak{g}$
the closed subset of Specm
$U(\mathfrak{g},e)^{\mbox{ab}}$
consisting of all points fixed by the natural action of the component group of
$G_e$
is isomorphic to an affine space. Analogues of these results for exceptional Lie algebras are also obtained and applications to the theory of primitive ideals are given.
In this paper we prove a version of curved Koszul duality for
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}$
-graded curved coalgebras and their cobar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations
$\mathsf{MF}(R,W)$
. We show how Dyckerhoff’s generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel–Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category
$\mathsf{MF}(R,W)$
. Similar results are also obtained in the orbifold case and in the graded case.
Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$
be a connected real semisimple Lie group,
$V$
be a finite-dimensional representation of
$G$
and
$\mu $
be a probability measure on
$G$
whose support spans a Zariski-dense subgroup. We prove that the set of ergodic
$\mu $
-stationary probability measures on the projective space
$\mathbb{P}(V)$
is in one-to-one correspondence with the set of compact
$G$
-orbits in
$\mathbb{P}(V)$
. When
$V$
is strongly irreducible, we prove the existence of limits for the empirical measures. We prove related results over local fields as the finiteness of the set of ergodic
$\mu $
-stationary measures on the flag variety of
$G$
.
We introduce a dual logarithmic residue map for hypersurface singularities and use it to answer a question of Kyoji Saito. Our result extends a theorem of Lê and Saito by an algebraic characterization of hypersurfaces that are normal crossing in codimension one. For free divisors, we relate the latter condition to other natural conditions involving the Jacobian ideal and the normalization. This leads to an algebraic characterization of normal crossing divisors. As a side result, we describe all free divisors with Gorenstein singular locus.