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 ${\mathcal O}_V$-modules. The key input is a mixed local reduction for finite Maurer--Cartan objects over the mixed de Rham--Dolbeault dga $(\wedge^\bullet V^\vee,d_V)$: a multiplicative homotopy contracts the real directions, after which Block's Dolbeault gauge theorem removes the positive transverse form degrees. For compact $X$, this gives an exact equivalence between the homotopy category of bounded finite-rank flat $V$-superconnections and $D^b_{\mathrm{coh}}(X,{\mathcal O}_V)$, interpolating between the de Rham and Dolbeault realizations. We prove locally finite \v{C}ech descent under necessary uniform amplitude and rank bounds, identify the coherent heart with equivariant coherent analytic sheaves on a transverse monodromy groupoid, and characterize descent to ordinary holonomy. For a holomorphic suspension, we identify the full superconnection category, up to Morita equivalence, with the homotopy fixed points of the Dolbeault category of the transversal and derive an equivariant Ext spectral sequence. Examples on $S^1$ and $S^2$ delimit when ordinary monodromy $1$-groupoids can recover the derived category.
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, minimality at only $\binom n2$ distinct dilation parameters forces all pair coefficients to vanish. A dimension-reduction argument then shows, without any hypothesis on the coordinate components of the normal, that the second fundamental form vanishes identically. This yields an affine characterization. Repeated-weight helicoidal examples in every dimension and a resonant quadratic cone show that curvature cancellation can survive in genuinely nonuniform families. An application gives a finite-output-level rigidity criterion and an explicit representation for weighted-homogeneous production functions with minimal isoquants.
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 compatible or to fall into a terminal class. Our main theorem states that every bounded, globally $C^{0,\alpha}$ entire viscosity solution of \[ \mathrm{Hess}_{\mathbb F}u\in\partial\mathcal{A} \] is constant if and only if $\mathcal{A}$ is Liouville admissible; thus the Liouville-type property is characterized as a geometric property of the admissible set. A central class of examples arises from polarizations of univariate G{\aa}rding polynomials satisfying the monotone root sequence condition, producing mixed elementary-symmetric admissible sets and recovering the standard $k$-Hessian equations as monomial cases. The framework also allows anisotropic constructions, including linear pullbacks and intersections of admissible sets.
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 groups. On a general equiregular manifold, convergence for every smooth test function is equivalent to convergence of the rescaled horizontal first moments of metric balls in first-kind privileged coordinates. This criterion is independent of $\mu$; when it holds, the principal symbol is determined by the normalized second-moment tensor of the tangent unit ball, and the drift satisfies an explicit change-of-measure formula. In step at most two, we verify the criterion by combining a real-analytic finite-jet reduction with tame integration, which rules out oscillation of the normalized moments. We also compute the limit on Lie groups and prove unconditional distributional convergence of the volume-weighted operators on every equiregular manifold.
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 $M_{H,S}(\mathbf{v},L)$ admits a support morphism to $|L|$ and is interpreted as a relative compactified Jacobian. Using this fibration, Lefschetz-type properties of positive linear systems, monodromy, and mixed Hodge structures, we construct a surjective homomorphism $ \pi_1(S)\twoheadrightarrow\pi_1\bigl(M_{H,S}(\mathbf{v},L)\bigr) $ and compute the first two Betti numbers of an Albanese fiber $F$, $M_{H,S}(\mathbf{v},L)$, and the distinguished component $M^\circ_{H,S}(\mathbf{v})$. Fourier-Mukai transforms and Bridgeland wall crossing extend these computations to primitive admissible Mukai vectors of positive rank. We further prove that $H^2(F,\mathbb{C})$ is of pure Hodge type $(1,1)$. If $\lambda_S=\ell(\mathbf{v})=1,$ then $F$ is a strict irreducible Calabi-Yau variety up to a finite quasi-\'etale cover. Under the additional genericity assumption on the pullback polarization, $M^\circ_{H,S}(\mathbf{v})$ is smooth and is noncanonically birational to $\operatorname{Pic}^0(S)\times\operatorname{Hilb}^{\mathbf{v}^2/2}(S)$.
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 globally solvable in Roumieu Gevrey ultradistributions, or equivalently that the corresponding top-degree cohomology vanishes. No global hypoellipticity assumption and no arithmetic condition on the periods of the defining forms are required. The proof is carried out in the physical variables and combines fiber translations, a local normal form, and a transport formula along paths in the base manifold. These tools yield propagation of Gevrey regularity, non-confinement of Gevrey singularities, and the support control needed to apply an abstract solvability criterion. The result highlights a sharp contrast with the compact setting, where compatibility conditions are unavoidable and solvability for compatible data may depend on exponential small-denominator conditions.
S. Coriasco, A. Kirilov, W. A. D. Moraes et al.· 0 citations
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