Let $(E,h)$ be a singular Hermitian vector bundle on a complex manifold $X$, and let \[ \mathcal E(h)_x=\{F\in\mathcal O(E)_x:|F|_h^2\in L^1_{\mathrm{loc},x}\} \] be its multiplier submodule sheaf. We introduce a finite plurisubharmonic Gram scalarization condition under which the higher-rank integrability problem reduces to finitely many scalar multiplier ideals. This reduction yields coherence and strong openness without imposing a general positivity hypothesis. When the scalar weights and the varying weight have analytic singularities, it also gives a theory of module jumping numbers, including a simultaneous-residue criterion for actual jumps. Finally, we study the induced Skoda filtration: Artin--Rees yields eventual periodicity, Tor controls whether periodicity starts at the scalar threshold, and a direct-image quotient measures the obstruction to descent.
Let $X$ carry a transversely holomorphic foliation, equivalently an elliptic involutive structure $V\subset T_{\mathbb C}X$, and let ${\mathcal O}_V$ be its sheaf of leafwise-constant, transversely holomorphic functions. We construct a finite superconnection model for the derived category of coherent ${\mathcal O}_V$-m...
Let $n\ge3$ and $D_t=\operatorname{diag}(t^{g_1},\ldots,t^{g_n})$ be a positive diagonal dilation family. We study connected embedded Euclidean hypersurfaces whose diagonal images are minimal. The level-set minimality operator splits into coefficients indexed by the pair sums $g_i+g_j$. Under pair-sum nonresonance, min...
We prove Liouville rigidity theorems for translation-invariant real and complex Hessian equations in the viscosity sense, where the PDE is encoded by an admissible set $\mathcal{A}$. The main structural notion is Liouville admissibility, a recursive geometric condition requiring each quotient set to be either boundary...
Let $(M,\mathcal{D},g)$ be a smooth equiregular sub-Riemannian manifold equipped with a smooth positive measure $\mu$. We study the small-scale limit of the metric-ball mean-value operator \[ A_hf(x)=\frac{1}{h^{2}\mu(B(x,h)) }\int_{B(x,h)}(f(q)-f(x))\, d\mu(q). \] Exact homogeneity yields the pointwise limit on Carnot...
Fabrice Baudoin, Jonathan Junné, David Tewodrose· 0 citations
We study the topology and low-degree Hodge theory of moduli spaces of semistable sheaves on bielliptic surfaces. For primitive rank-zero Mukai vectors $\mathbf{v}=(0,c_1(L),\chi)$ satisfying the positivity condition $\operatorname{nt}(L)\geq3$, the distinguished fixed-determinant component $M_{H,S}(\mathbf{v},L)$ admit...
Let $s>1$, let $M$ be a connected, non-compact, oriented real-analytic manifold, and let $\omega_1,\ldots,\omega_m$ be real-valued closed $1$-forms of Gevrey order $s$ on $M$. We study the differential complex naturally associated with this family on $M\times\mathbb{T}^m$. We prove that its top-degree operator is globa...
S. Coriasco, A. Kirilov, W. A. D. Moraes et al.· 0 citations
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