We find necessary conditions for the projectivization of a rank two vector bundle $\mathcal{E}$ on $\mathbb{P}^3_{\mathbb{C}}$ to be related to a smooth Fano 4-fold by a small $\mathbb{Q}$-factorial modification (SQM). Furthermore, we fully describe the birational geometry of $\mathbb{P}_{\mathbb{P}^3}(\mathcal{E})$ when $\mathcal{E}$ is stable, with $c_1=-1$, $c_2=2$. In this case, we prove that there exists a flip from $X^+$ to $\mathbb{P}_{\mathbb{P}^3}(\mathcal{E})$, where $X^+$ is smooth and Fano. We also determine the birational geometry and numerical invariants of $X^+$.
We construct explicit $U(n)$-equivariant isometric embeddings $T^*\mathbb{CP}^{n-1} \hookrightarrow \mathbb{R}^{n^4+2n^2-1}$, $n\ge 2$, with respect to Calabi's hyperk\"{a}hler metric on $T^*\mathbb{CP}^{n-1}$. To achieve this, we first construct a diffeomorphism from $T^*\mathbb{CP}^{n-1}$ onto the manifold of rank-1...
For any prime number $p$, the classic Hopkins-Mahowald theorem asserts that $H\mathbb{Z}/p$ is an $\mathbb{E}_2$-Thom spectrum over the $p$-complete sphere spectrum. Kitchloo extended this result to show that $H\mathbb{Z}/p^k$ is an $\mathbb{E}_2$-Thom spectrum over the $p$-local sphere spectrum, except for the case wh...
Let $p>3$ and let $\Gamma$ be a non-elementary Gromov hyperbolic group with $\partial_\infty\Gamma\cong\mathbb S^{p-1}$. We prove that every $\mathbb{H}^{p,2}$-convex cocompact representation $$ \rho:\Gamma\longrightarrow \mathrm{PO}_{p,3} $$ whose proximal limit set $\Lambda_\rho\subset\partial_\infty\mathbb{H}^{p,2}$...
Let $u:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ be a smooth nonconstant harmonic map. We prove that its Morse index is equal to $ 2n+2$ if and only if \begin{equation*} u = h \circ \pi \circ \Xi, \end{equation*} where $\Xi:\mathbb{S}^{2n+1}\to\mathbb{S}^{2n+1}$ is an isometric transformation, $\pi:\mathbb{S}^{2n+1}\to\mathbb...
We prove that for every $N\geq 3$, every solution to the $Q$-curvature-type problem $$ \alpha P_N u + (N-1)!\left(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw}\right)=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ \alpha\ge\frac{1}{2}$ and $\alpha \not =1$. The proof consists of a spectral gap est...
Chang-Feng Gui, Ye-Yao Hu, Tuo Li et al.· 0 citations
We introduce a topological abelian group $\mathrm{K}_1^\Delta(A)$ for every $\mathrm{C}^\ast$-algebra $A$, which is functorial in $A$ and invariant under stabilisations. $\mathrm{K}_1^\Delta$ can distinguish group actions indistinguishable by equivariant (K)K-theory and traces --- we illustrate this on the irrational r...
James Gabe, Alistair Miller· 0 citations
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