More Like this

1. The moduli spaces of sheaves on surfaces, pathologies and brill-noether problems

   https://doi.org/10.1007/978-3-319-94881-2_4  

   Cosun, I.; Huizenga, J. (November 2018, Abel symposia)

   Brill-Noether Theorems play a central role in the birational geometry of moduli spaces of sheaves on surfaces. This paper surveys recent work on the Brill-Noether problem for rational surfaces. In order to highlight some of the difficulties for more general surfaces, we show that moduli spaces of rank 2 sheaves on very general hypersurfaces of degree d in P3 can have arbitrarily many irreducible components as d tends to infinity. 

   more » « less
2. Hodge sheaves underlying flat projective families

   https://doi.org/10.1007/s00209-023-03219-4  

   Kovács, Sándor J.; Taji, Behrouz (March 2023, Mathematische Zeitschrift)

   Abstract We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms for smooth maps and their functorial properties. 

   more » « less
3. The six functors for Zariski-constructible sheaves in rigid geometry

   https://doi.org/10.1112/S0010437X22007291  

   Bhatt, Bhargav; Hansen, David (February 2022, Compositio Mathematica)

   We prove a generic smoothness result in rigid analytic geometry over a characteristic zero non-archimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well adapted to ‘spreading out’ arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six-functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves. 

   more » « less
4. Conformal Blocks on Smoothings via Mode Transition Algebras

   https://doi.org/10.1007/s00220-025-05237-1  

   Damiolini, Chiara; Gibney, Angela; Krashen, Daniel (June 2025, Communications in Mathematical Physics)

   Here we introduce a series of associative algebras attached to a vertex operator algebra V of CFT type, called mode transition algebras, and show they reflect both algebraic properties of V and geometric constructions on moduli of curves. Pointed and coordinatized curves, labeled by admissible V-modules, give rise to sheaves of coinvariants. We show that if the mode transition algebras admit multiplicative identities satisfying certain natural properties (called strong identity elements), these sheaves deform as wanted on families of curves with nodes. This provides new contexts in which coherent sheaves of coinvariants form vector bundles. We also show that mode transition algebras carry information about higher level Zhu algebras and generalized Verma modules. To illustrate, we explicitly describe the higher level Zhu algebras of the Heisenberg vertex operator algebra, proving a conjecture of Addabbo–Barron. 

   more » « less
5. Duality and Kernels in Microlocal Geometry

   https://doi.org/10.1093/imrn/rnaf070  

   Kuo, Christopher; Li, Wenyuan (March 2025, International Mathematics Research Notices)

   Abstract We study the dualizability of sheaves on manifolds with isotropic singular supports $$\operatorname{Sh}_\Lambda (M)$$ and microsheaves with isotropic supports $$\operatorname{\mu sh}_\Lambda (\Lambda )$$ and obtain a classification result of colimit-preserving functors by convolutions of sheaf kernels. Moreover, for sheaves with isotropic singular supports and compact supports $$\operatorname{Sh}_\Lambda ^{b}(M)_{0}$$, the standard categorical duality and Verdier duality are related by the wrap-once functor, which is the inverse Serre functor in proper objects, and we thus show that the Verdier duality extends naturally to all compact objects $$\operatorname{Sh}_\Lambda ^{c}(M)_{0}$$ when the wrap-once functor is an equivalence, for instance, when $$\Lambda $$ is a full Legendrian stop or a swappable Legendrian stop. 

   more » « less