International Journal of Mathematics

We develop the viewpoint that ${}^{\circ }{W}^{\ast }$, the opposite of the category of $W^{\ast }$-algebras and unital normal ∗-homomorphisms, is analogous to the category of sets and functions. For each pair of $W^{\ast }$-algebras ${ℳ}$ and ${𝒩}$, we construct their free exponential ${{ℳ}}^{\ast {𝒩}}$, which in the context of this analogy corresponds to the collection of functions from ${𝒩}$ to ${ℳ}$. We also show that every unital normal completely positive map ${ℳ}\to {𝒩}$ arises naturally from a normal state on ${{ℳ}}^{\ast {𝒩}}$.

A primitive multiple curve is a Cohen–Macaulay irreducible projective curve $Y$ that can be locally embedded in a smooth surface, and such that $C=Y_{\mathrm{\text{red}}}$ is smooth. In this case, $L={{ℐ}}_C/{{ℐ}}_C^2$ is a line bundle on $C$. If $Y$ is of multiplicity 2, i.e. if ${{ℐ}}_C^2=0$, $Y$ is called a ribbon. If $Y$ is a ribbon and $h^0(L^{-2})\ne 0$, then $Y$ can be deformed to smooth curves, but in general a coherent sheaf on $Y$ cannot be deformed in coherent sheaves on the smooth curves. It has been proved in [Reducible deformations and smoothing of primitive multiple curves, Manuscripta Math. 148 (2015) 447–469] that a ribbon with associated line bundle $L$ such that $\mathrm{\text{deg}}(L)=-d<0$ can be deformed to reduced curves having two irreducible components if $L$ can be written as

Let $E_G$ be a holomorphic principal $G$-bundle on a compact connected Riemann surface $X$, where $G$ is a connected reductive complex affine algebraic group. Fix a finite subset $D\subset X$, and for each $x\in D$ fix $w_x\in \text{ad}{(E_G)}_x$. Let $T$ be a maximal torus in the group of all holomorphic automorphisms of $E_G$. We give a necessary and sufficient condition for the existence of a $T$-invariant logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x\in D$ is $w_x$. We also give a necessary and sufficient condition for the existence of a logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x\in D$ is $w_x$, under the assumption that each $w_x$ is $T$-rigid.

In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a noncommutative ring from H-triangulations of 3-manifolds and observed that for trefoil and figure-eight knot complements the abelianization of this ring is isomorphic to the ring of regular functions on the deformation variety, Kashaev, $\delta$-groupoids in knot theory, Geom. Dedicata 150(1) (2010) 105–130; Kashaev, Noncommutative teichmüller spaces and deformation varieties of knot completeness; Kashaev, Delta-groupoids and ideal triangulation in Chern–Simons gauge theory: 20 Years After, AMS/IP Studies Advanced Mathematics, Vol. 50 (American Mathematical Society, RI, 2011). In this paper, we prove that this is true for any knot complement in a homology sphere. We also analyze some examples on other manifolds.

In this paper, we prove formulas that represent two-pointed Gromov–Witten invariant ${〈{{𝒪}}_{h^a}{{𝒪}}_{h^b}〉}_{0,d}$ of projective hypersurfaces with $d=1,2$ in terms of Chow ring of ${\overline{M}}_{0,2}({\mathbb{P}}^{N-1},d)$, the moduli spaces of stable maps from genus $0$ stable curves to projective space ${\mathbb{P}}^{N-1}$. Our formulas are based on representation of the intersection number $w{({{𝒪}}_{h^a}{{𝒪}}_{h^b})}_{0,d}$, which was introduced by Jinzenji, in terms of Chow ring of ${\widetilde{Mp}}_{0,2}(N,d)$, the moduli space of quasi maps from ${\mathbb{P}}^1$ to ${\mathbb{P}}^{N-1}$ with two marked points. In order to prove our formulas, we use the results on Chow ring of ${\overline{M}}_{0,2}({\mathbb{P}}^{N-1},d)$, that were derived by Mustaţǎ and Mustaţǎ. We also present explicit toric data of ${\widetilde{Mp}}_{0,2}(N,d)$ and prove relations of Chow ring of ${\widetilde{Mp}}_{0,2}(N,d)$.

In this paper, we provide some integral conditions to extend the Ricci flow coupled with harmonic map flow. Our results generalize the corresponding results for Ricci flow obtained by Wang [On the conditions to extend Ricci flow, Int. Math. Res. Not. 8 (2008), Articles: rnn012, 30 pp].