A Luna étale slice theorem for algebraic stacks

Abstract

We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, isétale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.

The generalized Franchetta conjecture for some hyper-Kähler varieties, II

“…2.2] using [BM14b]. In our general setting where A is a 2-Calabi-Yau category, Li-Pertusi-Zhao [LPZ20, §3] showed that the singularity of M σ (A , v) has the same local model as in the classical case and that the construction of the crepant resolution in [LS06] can be adapted by using [AHR20]. For details we refer to [LPZ20, §3], where the proof was written for cubic fourfolds, but works in general.…”

Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation

“…, c 2 dim X ), which we call the type of the sheaves under consideration. This is an algebraic stack locally of finite type over C since it satisfies Artin's axioms [Alp15, Theorem 2.20]; see also [AHR15,Theorem 2.19]. See also the beginning of Section 3 for more information on the basic properties of X .…”

The moduli space of Fano manifolds with Kähler–Ricci solitons

“…then we obtain a test configuration of X ∞ with the central fiber X ∞ from the reductivity of the stabilizer Aut T (X ∞ ), which allows to apply theétale slice theorem [AHR,Theorem 2.1] and the Hilbert-Mumford theorem. Since the central fiber admits a Kähler-Ricci soliton, the modified Donaldson-Futaki invariant of this test configuration is zero.…”