Related papers: On the Witten Rigidity Theorem for String$^c$ Mani…
Consider a Hamiltonian action by a compact Lie group on a possibly noncompact symplectic manifold. We give a short proof of a geometric formula for decomposition into irreducible representations of the equivariant index of a Spin$^c$-Dirac…
We prove a homological stability theorem for families of discrete groups (e.g. mapping class groups, automorphism groups of free groups, braid groups) with coefficients in a sequence of irreducible algebraic representations of arithmetic…
In this paper, we prove two Kastler-Kalau-Walze type theorems for conformal perturbations of twisted Dirac operators and conformal perturbations of twisted signature operators on four-dimensional manifolds with (resp. without) boundary.
We obtain a vanishing theorem for the half-kernel of a transverse ${\rm Spin}\sp c$ Dirac operator on a compact manifold endowed with a transversely almost complex Riemannian foliation twisted by a sufficiently large power of a line bundle,…
Given a spin rational homology sphere $Y$ equipped with a $\mathbb{Z}/m$-action preserving the spin structure, we use the Seiberg--Witten equations to define equivariant refinements of the invariant $\kappa(Y)$ from \cite{Man14}, which take…
Kawakubo and Uchida showed that, if a closed oriented $4k$-dimensional manifold $M$ admits a semi-free circle action such that the dimension of the fixed point set is less than $2k$, then the signature of $M$ vanishes. In this note, by…
In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac…
We prove that Kellendonk's $C^*$-algebra of an aperiodic and repetitive tiling with finite local complexity is classifiable by the Elliott invariant. Our result follows from showing that tiling $C^*$-algebras are $\mathcal{Z}$-stable, and…
We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical Q-divisors. First, we propose a condition in terms of certain anticanonical Q-divisors of given Fano…
We obtain some rigidity results for metrics whose Schouten tensor is bounded from below after conformal transformations. Liang Cheng recently proved that a complete, nonflat, locally conformally flat manifold with Ricci pinching condition…
In this paper, a vanishing theorem is stated and proved. If a 4-manifold $M$ admits a smooth action by a cyclic group $\mathbb{Z}_r$, then given an $\mathbb{Z}_r$-equivariant $Spin^c$-structure $\mathcal{C}$ on $M$, the Seiberg-Witten…
The rigidity theorem of Witten-Bott-Taubes-Hirzebruch tells us that, if the circle group acts on a closed almost complex (or more generally unitary) manifold whose first Chern class is divisible by a positive integer N greater than 1, then…
We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over a field of characteristic zero. In particular, we prove a vanishing result for higher…
Two consistency conditions for partition functions established by Akemann and Dam-gaard in their studies of the fermionic mass dependence of the QCD partition function at low energy ({\it a la} Leutwiller-Smilga-Verbaarschot) are…
We introduce a gauge-theoretic integer lift of the Rohlin invariant of a smooth 4-manifold X with the homology of $S^1 \times S^3$. The invariant has two terms; one is a count of solutions to the Seiberg-Witten equations on X, and the other…
We prove a criterion for K-stability of a $\mathbb{Q}$-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with…
Based on a construction by Kashiwara and Rouquier, we present an analogue of the Beilinson- Bernstein localization theorem for hypertoric varieties. In this case, sheaves of differential operators are replaced by sheaves of W-algebras. As a…
We prove that the homotopy algebraic K-theory of tame quasi-DM stacks satisfies cdh-descent. We apply this descent result to prove that if X is a Noetherian tame quasi-DM stack and i < -dim(X), then K_i(X)[1/n] = 0 (resp. K_i(X, Z/n) = 0)…
Rozansky and Witten proposed in 1996 a family of new three-dimensional topological quantum field theories, indexed by compact (or asymptotically flat) hyperkaehler manifolds. As a byproduct they proved that hyperkaehler manifolds also give…
We consider a family of quintic threefolds specializing to a certain reducible threefold. We describe the space of genus zero stable morphisms to the central fiber (as defined by J. Li). As an elementary application of an extension of the…