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We show that the category of vector fields on a geometric stack has the structure of a Lie 2-algebra. This proves a conjecture of R.~Hepworth. The construction uses a Lie groupoid that presents the geometric stack. We show that the category of vector fields on the Lie groupoid is equivalent to the category of vector fields on the stack. The category of vector fields on the Lie groupoid has a Lie 2-algebra structure built from known (ordinary) Lie brackets on multiplicative vector fields of Mackenzie and Xu and the global sections of the Lie algebroid of the Lie groupoid. After giving a precise formulation of Morita invariance of the construction, we verify that the Lie 2-algebra structure defined in this way is well-defined on the underlying stack.
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A Lie 2-group $G$ is a category internal to the category of Lie groups. Consequently it is a monoidal category and a Lie groupoid. The Lie groupoid structure on $G$ gives rise to the Lie 2-algebra $mathbb{X}(G)$ of multiplicative vector fields, see (Berwick-Evans -- Lerman). The monoidal structure on $G$ gives rise to a left action of the 2-group $G$ on the Lie groupoid $G$, hence to an action of $G$ on the Lie 2-algebra $mathbb{X}(G)$. As a result we get the Lie 2-algebra $mathbb{X}(G)^G$ of left-invariant multiplicative vector fields. On the other hand there is a well-known construction that associates a Lie 2-algebra $mathfrak{g}$ to a Lie 2-group $G$: apply the functor $mathsf{Lie}: mathsf{Lie Groups} to mathsf{Lie Algebras}$ to the structure maps of the category $G$. We show that the Lie 2-algebra $mathfrak{g}$ is isomorphic to the Lie 2-algebra $mathbb{X}(G)^G$ of left invariant multiplicative vector fields.
Assume M is a 3-dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X is an element of A. The following result is proved: If K is a locally maximal compact set of zeroes of X and the Poincare-Hopf index of X at K is nonzero, there is a point in K at which all the elements of A vanish.
We prove that the space of coinvariants of functions on an affine variety by a Lie algebra of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem include Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves under Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving groups under the incompressible vector fields, and arbitrary varieties with isolated singularities under the flow of all vector fields. We compute this quotient explicitly in many of these cases. The proofs involve constructing a natural D-module representing the invariants under the flow of the vector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicity we study in more detail). We give many counterexamples to naive generalizations of our results. These examples have been a source of motivation for us.
Let X be an analytic vector field on a real or complex 2-manifold, and K a compact set of zeros of X whose fixed point index is not zero. Let A denote the Lie algebra of analytic vector fields Y on M such that at every point of M the values of X and [X,Y] are linearly dependent. Then the vector fields in A have a common zero in K. Application: Let G be a connected Lie group having a 1-dimensional normal subgroup. Then every action of G on M has a fixed point.
On a real ($mathbb F=mathbb R$) or complex ($mathbb F=mathbb C$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {it tracks} $X$ if $[Y,X]=fX$ for some continuous function $fcolon Mrightarrowmathbb F$. Let $K$ be a compact subset of the zero set ${mathsf Z}(X)$ such that ${mathsf Z}(X)-K$ is closed, with nonzero Poincare-Hopf index (for example $K={mathsf Z}(X)$ when $M$ is compact and $chi(M) eq 0$) and let $mathcal G$ be a finite-dimensional Lie algebra of analytic vector fields on $M$. smallskip {bf Theorem.} Let $X$ be analytic and nontrivial. If every element of $mathcal G$ tracks $X$ and, in the complex case when ${mathsf i}_K (X)$ is positive and even no quotient of $mathcal G$ is isomorphic to ${mathfrak {s}}{mathfrak {l}} (2,mathbb C)$, then $mathcal G$ has some zero in $K$. smallskip {bf Corollary.} If $Y$ tracks a nontrivial vector field $X$, both of them analytic, then $Y$ vanishes somewhere in $K$. smallskip Besides fixed point theorems for certain types of transformation groups are proved. Several illustrative examples are given.
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