The purpose of this paper is to present a mathematical theory of the half-twisted $(0, 2)$ gauged linear sigma model and its correlation functions that agrees with and extends results from physics. The theory is associated to a smooth projective toric variety $X$ and a deformation $\mathcal{E}$ of its tangent bundle $T_X$. It gives a quantum deformation of the cohomology ring of the exterior algebra of $\mathcal{E}*$. We prove that in the general case, the correlation functions are independent of "nonlinear" deformations. We derive quantum sheaf cohomology relations that correctly specialize to the ordinary quantum cohomology relations described by Batyrev in the special case $\mathcal{E} = T_X$.

In this paper we introduce a global geometric invariant $\alpha(M)$ related to injectivity radius to complete non-compact Riemannian manifolds and prove: If $\alpha(M^n) \gt 1$, then $M^n$ is isometric to $\mathbb{R}^n$ when Ricci curvature is non-negative, and is diffeomorphic to $\mathbb{R}^n$ for $n \neq 4$ and homeomorphic to $\mathbb{R}^4$ for $n = 4$ if without any curved assumption.

Let $(M^m, T)$ be a smooth involution on a closed smooth $m$-dimensional manifold and $F = {\bigcup}^n_{j=0} F^j (n \lt m)$ its fixed point set, where $F^j$ denotes the union of those components of $F$ having dimension $j$. The famous Five Halves Theorem of J. Boardman, announced in 1967, establishes that, if $F$ is nonbounding, then $m \leq \frac{5}{2} n$; further, this estimative is best possible. In this paper we obtain improvements of this theorem, taking into account certain natural numbers which we call the decomposability degrees $\ell(F^j)$ of the nonbounding components $F^j$ of $F$ (see the definition in Section 1). Also, these improvements are obtained under assumptions on the set of dimensions occurring in $F$, which we denote $\pi_0(F)$. The main result of this paper is: suppose the involution $(M^m, T)$ has $\pi_0(F) = \{ 0, 1, \dots, j, n \}$, where $2 \leq j \lt n \lt m$ and $F^j$ is nonbounding. Write $\mathcal{M}(n - j)$ for the function of $n - j$ defined in the following way: writing $n - j = 2^p q$, where $q \geq 1$ is odd and $p \geq 0$, $M(n - j) = 2n + p - q + 1$ if $p \leq q$ and $M(n - j) = 2n + 2^{p-q}$ if $p \geq q$. Then $m \leq \mathcal{M}(n - j) + 2j + \ell (F^j)$. In addition, we develop a method to construct involutions $(M^m, T)$ with $\pi_0(F)$ as above, in some special situations, which in some cases will show that the above bound is best possible. This will provide some improvements of the general Five Halves Theorem $(\pi_0(F) = \{ i / 0 \leq i \leq n \} )$, by considering the particular case $j = n - 1$.

We make an observation which enables one to deduce the existence of an algebraic stack of logarithmic maps for all generalized Deligne-Faltings logarithmic structures (in particular simple normal crossings divisors) from the simplest case with characteristic generated by $\mathbb{N}$ (essentially the smooth divisor case).

Suppose that $\Sigma = \partial M$ is the $n$-dimensional boundary of a connected compact Riemannian spin manifold $(M, \langle , \rangle)$ with non-negative scalar curvature, and that the (inward) mean curvature $H$ of $\Sigma$ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric $\langle , \rangle {}_H = H^2 \langle , \rangle$ is at least $n/2$ and equality holds if and only if there exists a non-trivial parallel spinor field on $M$. As a consequence, if $\Sigma$ admits an isometric and isospin immersion $F$ with mean curvature $H_0$ as a hypersurface into another spin Riemannian manifold $M_0$ admitting a parallel spinor field, then$$\int_{\Sigma} H{ } d\Sigma \leq \int_{\Sigma} \frac{H^2_0}{H} { } d\Sigma$$where $H$ is the mean curvature of $\Sigma$ as the boundary of $M$ and $H_0$ stands for the mean curvature of the immersion $F$ of $\Sigma$ into $\mathbb{R}^{n+1}$. Equality holds if and only if $\Sigma$ is connected, $M$ is a Euclidean domain and the embedding of $\Sigma$ in $M$ and its immersion in $\mathbb{R}^{n+1}$ are congruent.

Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine precise formulas, up to and including constant terms, for the asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces. As a corollary we state precise asymptotic formulas for the beta-invariant introduced around 2000 by R. Hain and D. Reed. These formulas are a refinement of a result Hain and Reed prove in their paper. We illustrate our results with some explicit calculations on degenerating genus two surfaces.

Let $X, Y \subset k^m(k = \mathbb{R},\mathbb{C})$ be smooth manifolds. We investigate the central symmetry of the configuration of $X$ and $Y$. For $p \in k^m$ we introduce a number $\mu(p)$ of pairs of points $x \in X$ and $y \in Y$ such that $p$ is the center of the interval $\overline{xy}$. We show that if $X, Y$ (including the case $X = Y$ ) are algebraic manifolds in a general position, then there is a closed (semi-algebraic) set $B \subset k^m$, called symmetry defect set of the $X$ and $Y$ configuration, such that the function $\mu$ is locally constant and not identically zero outside $B$. If $k = \mathbb{C}$, we estimate the number $\mu$ (in fact we compute it in many cases) and show that the symmetry defect is an algebraic hypersurface and consequently the function $\mu$ is constant and positive outside $B$. We also show that in the generic case the topological type of the symmetry defect set of a plane curve is constant, i.e. the symmetry defect sets for two generic curves of the same degree are homeomorphic (by the same method we can prove similar statement for any irreducible family of smooth varieties $Z^n \subset \mathbb{C}^{2n}$). Moreover, for $k = \mathbb{R}$, we estimate the number of connected components of the set $U = k^m \backslash B$. In the last section we give an algorithm to compute the symmetry defect set for complex smooth affine varieties in general position.

This article studies the Lie algebra $\mathrm{Der}(\mathrm{k}\Gamma)$ of derivations on the path algebra $\mathrm{k}\Gamma$ of a quiver $\Gamma$ and the Lie algebra on the first Hochschild cohomology group $HH1(\mathrm{k}\Gamma)$. We relate these Lie algebras to the algebraic and combinatorial properties of the path algebra. Characterizations of derivations on a path algebra are obtained, leading to a canonical basis of $\mathrm{Der}(\mathrm{k}\Gamma)$ and its Lie algebra properties. Special derivations are associated to the vertices, arrows and faces of a quiver, and the concepts of a connection matrix and boundary matrix are introduced to study the relations among these derivations, giving rise to an interpretation of Euler's polyhedron formula in terms of derivations. By taking dimensions, this relation among spaces of derivations recovers Euler's polyhedron formula. This relation also leads to a combinatorial construction of a canonical basis of the Lie algebra $HH1(\mathrm{k}\Gamma)$, together with a new semidirect sum decomposition of $HH1(\mathrm{k}\Gamma)$.