In this note, the second author's quantized volume comparison conjecture is solved: If $X$ is a $K$-semistable Fano manifold of dimension $n$, then for every integer $m\geq1$, \[ h^0(X,-mK_X)\leq h^0(\mathbb P^n,-mK_{\mathbb P^n}) =\binom{n+m(n+1)}{n}, \] and equality for one $m$ characterizes projective space. Somewhat surprisingly, the same statement actually holds whenever $T_X$ is slope semistable with respect to $-K_X$. The idea is to apply a jet-dimension counting trick to a filtration of subsheaves induced by $H^0(X,-mK_X)$. Then the slope semistability condition yields the desired dimension bound.
Let $3\leq n\leq7$, $2\leq k\leq n-1$, and $m=n-k-1$. We prove that a complete, connected, noncompact spin manifold $(M^n,g)$ with scalar curvature $R_M\geq k(k-1)$ is isometric to $\mathbb{S}^k\times\mathbb{T}^m_\Lambda\times\mathbb{R}$ if it admits a smooth proper map of nonzero degree to $\mathbb{S}^k\times\mathbb{T...
LeBrun's K\"ahler realization $g_m$ of the Taub--NUT metric on $\mathbb{C}^2$ is complete, Ricci-flat and not flat. Loi, Zedda and Zuddas proved that no multiple $\alpha g_m$ admits a K\"ahler immersion into a finite- or infinite-dimensional complex projective space when $m>\alpha/2$, and conjectured that the same hold...
If $M$ is a closed oriented manifold of even dimension, then its $n$th symmetric product $M(n)=M^n/S_n$ is a rational homology manifold for any $n \geq 0$ and hence has a canonically defined $L$-class in $H^*(M(n);\mathbb{Q})$. A complicated formula for these classes was given many years ago in the second author's thes...
Let $X$ be a smooth compact irreducible hyperk\"ahler manifold of dimension six. We prove that the second Betti number satisfies $b_2(X) \le 23$. Furthermore, we show that the third cohomology vanishes whenever $b_2(X) \ge 19$, and that all odd cohomology vanishes when $b_2(X) = 23$. If the second Chern class lies in t...
Jie Fu, Bin Wang, Shi-Hao Wang et al.· 0 citations
Let $X=G/K$ be a symmetric space of noncompact type, of dimension $n$ and rank $r$, and let $Y=\Gamma\backslash X$. Sarnak's local bound for an $L^2$-normalized spherical joint eigenfunction with regular tempered parameter of size $T$ is $\|\phi\|_\infty\ll T^{(n-r)/2}$. We prove $o(T^{(n-r)/2})$ locally uniformly on e...
Let $M$ be a closed manifold equipped with a collection of smooth real vector fields $X_1,\ldots, X_k$ and a smooth measure $\mu$. Let $P:=\sum_j X_j^\dagger X_j $ be the associated sublaplacian. Under the assumption that each pair of points of $M$ can be connected by a path obtained by joining integral curves of the v...
G. Dall'Ara· 0 citations
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