International Journal of Mathematics

In this note, relying on a variant of Grauert–Remmert extension theorem for plurisubharmonic functions on locally irreducible complex spaces, we show that if $\pi :Y\to X$ is a holomorphic surjection from an irreducible complex space $Y$ to a locally irreducible complex space $X$ and $u:X\to ℝ\cup \{-\infty \}$ is an upper semicontinuous function on $X$, then $u\in \text{Psh}(X)$ if and only if $u\circ \pi \in \text{Psh}(Y)$. Moreover, by concrete examples, we also present that both of the irreducibility assumptions on $X$ and $Y$ are sharp.

The notion of general weighted Ricci curvatures appears naturally in many problems. The N-Ricci curvature and the projective Ricci curvature are just two special ones with totally different geometric meanings. In this paper, we study general weighted Ricci curvatures. We find that Randers metrics of certain isotropic weighted Ricci curvature must have isotropic S-curvature. Then we classify them via their navigation expressions. We also find equations that characterize Randers metrics of almost isotropic weighted Ricci curvature.

This paper is devoted to branching problems for holomorphic discrete series representations of a conformal group $G$ of a tube domain $T_{\mathrm{\Omega }}$ over a symmetric cone $\mathrm{\Omega }$ restricted to the conformal group $G^{\prime }$ of a tube domain $T_{{\mathrm{\Omega }}^{\prime }}$ holomorphically embedded in $T_{\mathrm{\Omega }}$. The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. We answer this program with the introduction of another functional model for holomorphic discrete series representations. This model leads to a geometrical interpretation and a close relation to the theory of orthogonal polynomials for such branching problems. This program is illustrated by three cases. First, we consider the $n$-fold tensor product of holomorphic discrete series of the universal covering of $S{L}_2(ℝ)$. Then, it is tested on the restrictions of a member of the scalar-valued holomorphic discrete series of the conformal group $S{O}_0(2,n)$ to the subgroup $S{O}_0(2,n-p)$, and finally to the subgroup $S{O}_0(2,n-p)\times SO(p)$.

On the symmetrized bidisk $G_2$ with the Bergman metric, the holomorphic sectional curvature is negatively pinched and the holomorphic bisectional curvature is not. The consequences of invariant metrics are provided.

We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in $ℂ{\text{P}}^d$ of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer–Zehnder conjecture in the nondegenerate case: every Hamiltonian diffeomorphism of $ℂ{\text{P}}^d$ that has at least $d+2$ nondegenerate periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of $J$-holomorphic curves. An appendix written by Shelukhin contains a new proof of the Smith-type inequality for barcodes of Hamiltonian diffeomorphisms that arise from Floer theory, which lends itself to adaptation to the setting of generating functions.

Starting with a vertex-weighted pointed graph $(\mathrm{\Gamma },\mu ,v_0)$, we form the free loop algebra $S_0$ defined in Hartglass–Penneys’ article on canonical ${\mathrm{\text{C}}}^{\ast }$-algebras associated to a planar algebra. Under mild conditions, ${{𝒮}}_0$ is a non-nuclear simple ${\mathrm{\text{C}}}^{\ast }$-algebra with unique tracial state. There is a canonical polynomial subalgebra $A\subset {{𝒮}}_0$ together with a Dirac number operator $N$ such that $(A,L^{2}A,N)$ is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify $({{𝒮}}_0,A,N)$ yields a compact quantum metric space in the sense of Rieffel.

We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our ${\mathrm{\text{C}}}^{\ast }$-algebras are non-nuclear, we adjust the Lip-norm coming from $N$ to utilize the finite dimensional filtration of $A$. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces.

As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) ${\mathrm{\text{C}}}^{\ast }$-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS ${\mathrm{\text{C}}}^{\ast }$-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance.

We introduce the notion of strongly asymptotically log del Pezzo flags, and classify such flags under the assumption that their zero-dimensional part lies in the boundary. We use this result to give a new and conceptual proof of the classification of strongly asymptotically log del Pezzo surfaces, originally due to Cheltsov and Rubinstein.

We extend the Gordon–Litherland pairing to links in thickened surfaces, and use it to define signature, determinant and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the $S^{\ast }$-equivalence class of the spanning surface. We prove a duality result relating the invariants from one $S^{\ast }$-equivalence class of spanning surfaces to the restricted invariants of the other. Using Kuperberg’s theorem, these invariants give rise to well-defined invariants of checkerboard colorable virtual links. The determinants can be applied to determine the minimal support genus of a checkerboard colorable virtual link. The duality result leads to a simple algorithm for computing the invariants from the Tait graph associated to a checkerboard coloring. We show these invariants simultaneously generalize the combinatorial invariants defined by Im, Lee and Lee, and those defined by Boden, Chrisman and Gaudreau for almost classical links. We examine the behavior of the invariants under orientation reversal, mirror symmetry and crossing change. We give a 4-dimensional interpretation of the Gordon–Litherland pairing by relating it to the intersection form on the relative homology of certain double branched covers. This correspondence is made explicit through the use of virtual linking matrices associated to (virtual) spanning surfaces and their associated (virtual) Kirby diagrams.

In this paper, we study the Gromov–Witten theory of the Hilbert scheme $X^{[2]}$ of two points on an elliptic surface $X$. Assume that $\left|K_X\right|$ contains an element supported on the smooth fibers of $X$. By analyzing the degeneracy locus and localized virtual cycle arising from the cosection localization theory of Kiem and Li [Y. Kiem and J. Li, Gromov–Witten invariants of varieties with holomorphic 2-forms, preprint; Y. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025–1050], we determine the $1$-point genus-$0$ Gromov–Witten invariant ${〈w〉}_{0,d({\beta }_f-2{\beta }_2)}^{X[2]}$ up to some rational number $m(d,X)$ depending only on $d$ and $X$, where $w\in H^4(X^{[2]},ℂ)$, $d\ge 1$, $f$ is a smooth fiber of $X$, ${\beta }_f=x_0+f\subset X^{[2]}$ with $x_0\in X-f$ being a fixed point, and ${\beta }_2=\{\xi \in X^{[2]}|\mathrm{\text{Supp}}(\xi )=\{x_0\}\}$. Moreover, we propose a conjecture regarding $m(d,X)$, and prove that the conjecture is true for $X=C\times E$ where $E$ is an elliptic curve and $C$ is a smooth curve.