International Journal of Mathematics

We establish bounds for the gradient of solutions to the forward conjugate heat equation of differential forms on a Riemannian manifold with the metric evolves under the Ricci flow.

Given a tame differential calculus over a noncommutative algebra ${𝒜}$ and an ${𝒜}$-bilinear metric $g_0,$ consider the conformal deformation $g=k.g_0,$ $k$ being an invertible element of ${𝒜}.$ We prove that there exists a unique connection $\nabla$ on the bimodule of one-forms of the differential calculus which is torsionless and compatible with $g.$ We derive a concrete formula connecting $\nabla$ and the Levi-Civita connection for the metric $g_0.$ As an application, we compute the Ricci and scalar curvatures for a general conformal perturbation of the canonical metric on the noncommutative $2$-torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.

For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+A{x}^m+B$ with non-squarefree discriminant, many of which have small Galois group. Moreover, in certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce asymptotics on the number of such trinomials with $A\le X$.

We say that a CR singular submanifold $M$ has a removable CR singularity if the CR structure at the CR points of $M$ extends through the singularity as an abstract CR structure on $M$. We study such real-analytic submanifolds, in which case removability is equivalent to $M$ being the image of a generic real-analytic submanifold $N$ under a holomorphic map that is a diffeomorphism of $N$ onto $M$, what we call a CR image. We study the stability of the CR singularity under perturbation, the associated quadratic invariants, and conditions for removability of a CR singularity. A lemma is also proved about perturbing away the zeros of holomorphic functions on CR submanifolds, which could be of independent interest.

Let ${{𝒮}}_N$ be symmetric group of degree $N$. In the representation of ${{𝒮}}_N$ corresponding to trapeziform partition (precisely $\tau =(m^{2k},a,b)$, $0\le b\le a<m$), we propose nonsymmetric Jack polynomials in $2mk+a+b$ variables which are singular at $\frac{1}{m+2}$. Moreover those nonsymmetric Jack polynomials span a module of ${{𝒮}}_N$ which is isomorphic to the representation of type $\sigma =(mk+a,mk+b)$.

We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well-known result of Gabriélov, Lazzeri and Lê for hypersurfaces. We use A’Campo’s theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety.

We show that any two bridge positions of a handlebody-knot are stably equivalent.

Extending a result of the first author and Katsura, we prove that for every UHF algebra $A$ of infinite type, in every uncountable cardinality $\kappa$ there are $2^{\kappa }$ nonisomorphic approximately matricial C*-algebras with the same $K_0$ group as $A$. These algebras are group C*-algebras “twisted” by prescribed canonical commutation relations (CCR), and they can also be considered as nonseparable generalizations of noncommutative tori.

In this paper, we prove some local rigidity theorems of holomorphic curves in a complex Grassmann manifold $G(3,6)$ by moving frames. By applying our rigidity theorems, we also give a characterization of all homogeneous holomorphic two-spheres in $G(3,6)$ classified by the second author.

Given a compact connected Riemann surface $\mathrm{\Sigma }$ of genus $g_{\mathrm{\Sigma }}\ge 2$, and an effective divisor $D={\sum }_i{n}_i{x}_i$ on $\mathrm{\Sigma }$ with $\text{degree}(D)<2(g_{\mathrm{\Sigma }}-1)$, there is a unique cone metric on $\mathrm{\Sigma }$ of constant negative curvature $-4$ such that the cone angle at each point $x_i$ is $2\pi n_i$ [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on $\mathrm{\Sigma }$ corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on $\mathrm{\Sigma }$ parametrized by a nonempty open subset of $H^0(\mathrm{\Sigma },K_{\mathrm{\Sigma }}^{\otimes 2}\otimes {{𝒪}}_{\mathrm{\Sigma }}(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case $D=0$.

Minimal algebraic surfaces of general type $X$ such that $K_X^2=2\chi ({{𝒪}}_X)-6$ are called Horikawa surfaces. In this note, ${\mathbb{Z}}_3$-actions on Horikawa surfaces are studied. The main result states that given an admissible pair $(K^2,\chi )$ such that $K^2=2\chi -6$, all the connected components of Gieseker’s moduli space ${𝔐}_{K^2,\chi }$ contain surfaces admitting a ${\mathbb{Z}}_3$-action. On the other hand, the examples considered allow us to produce normal stable surfaces that do not admit a $\mathbb{Q}$-Gorenstein smoothing. This is illustrated by constructing non-smoothable normal surfaces in the KSBA-compactification ${\overline{𝔐}}_{K^2,\chi }$ of Gieseker’s moduli space ${𝔐}_{K^2,\chi }$ for every admissible pair $(K^2,\chi )$ such that $K^2=2\chi -5$. Furthermore, the surfaces constructed belong to connected components of ${\overline{𝔐}}_{K^2,\chi }$ without canonical models.

For a smooth projective variety $X$, we study analogs of Quot schemes using hearts of non-standard $t$-structures of $D^b(\mathrm{\text{Coh}}(X))$. The technical framework uses families of $t$-structures as studied in A. Bayer, M. Lahoz, E. Macrì, H. Nuer, A. Perry and P. Stellari, Stability conditions in families, preprint (2019), arXiv:1902.08184. We provide several examples and suggest possible directions of further investigation, as we reinterpret moduli spaces of stable pairs, in the sense of M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117(2) (1994) 317–353; D. Huybrechts and M. Lehn, Stable pairs on curves and surfaces, J. Algebraic Geom. 4(1) (1995) 67–104, as instances of Quot schemes.