Study new invariants in complex geometry using Bott-Chern hypercohomology.
problem Understanding geometry through Bott-Chern hypercohomology and bimeromorphic invariants.
method Construct new invariants involving sheaf cohomology, establish blow-up formula and canonical morphism.
result Compute invariants for specific complex threefolds like Iwasawa manifolds and quintic threefolds.
Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.
problem Cohomology of Lie algebroids over algebraic spaces.
method Express hypercohomology as a derived functor, simplify via Čech cohomology, define Hochschild hypercohomology, present Hochschild-Kostant-Rosenberg theorem.
result Presented a version of Hochschild-Kostant-Rosenberg theorem for locally free Lie algebroids.
Investigates conditions for spectral sequence degeneracy in holomorphic Poisson structures.
problem Conditions for spectral sequence degeneracy in holomorphic Poisson structures.
method Uses Lie bi-algebroids, generalized complex structures, and hypercohomology of bi-complexes.
result Investigates conditions for spectral sequence degeneracy on the first page.
Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.
problem Index theory and analytic torsion of nonlinear PDEs.
method Microlocal sheaf theory, factorization algebras, Spencer hypercohomology.
result Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.
Revisits Vafa-Witten theory, deriving new invariants and homologies.
problem Understanding Vafa-Witten equations and their invariants.
method Physical derivation and mathematical analysis of Vafa-Witten equations.
result Novel invariants and Floer homologies derived from Vafa-Witten equations.
The paper establishes isomorphisms between different versions of intersection homology duality and products.
problem Comparing and relating different versions of intersection homology duality and products.
method Sheaf-theoretic and singular chain methods, PL pseudomanifolds, and de Rham isomorphism.
result Isomorphisms between sheaf-theoretic and singular intersection homology duality and products.
Study of Hermitian metrics on Lie algebroids over complex spaces.
problem Exploring Hermitian metrics on Lie algebroids over analytic spaces.
method Introduced Hermitian metrics on holomorphic Lie algebroids and examined associated characteristics foliations.
result Extended equivariant de Rham cohomology to Hermitian Lie algebroids.
This paper deals with moduli spaces of framed principal bundles with connections with irregular singularities over a compact Riemann surface. These spaces have been constructed by Boalch by means of an infinite-dimensional symplectic reduction. It is proved that the symplectic structure induced from the Atiyah--Bott fo…
Constructs Morse homology for complex algebraic varieties.
problem Homology of vanishing cycles in complex algebraic varieties.
method Morse homology groups associated with a perturbed function on a compactified variety.
result Morse homology groups are isomorphic to the homology of vanishing cycles.
We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of 3-manifolds into the plane. Furthermore, we show that certain cohomology classes associat…
Authors find a Hodge-type decomposition for holomorphic Poisson cohomology on nilmanifolds.
problem Investigating conditions for spectral sequence degeneration in holomorphic Poisson cohomology.
method Analyzing spectral sequences associated with bi-complexes on nilmanifolds.
result A Hodge-type decomposition of holomorphic Poisson cohomology is established for a specific class of structures.
Identifies holonomy of affine surfaces via meromorphic connections.
problem Holonomy of complex affine surfaces.
method Moduli space of complex affine surfaces and meromorphic connections.
result Holonomy map is a submersion for non-finite-area translation surfaces.
A sheaf-theoretic model connects SL(2,C) Floer homology to 3-manifold invariants.
problem Modeling SL(2,C) Floer homology using sheaf theory.
method Using SL(2,C) character varieties and Lagrangians, Joyce's perverse sheaves are associated to critical loci.
result The perverse sheaf is an invariant of the 3-manifold, providing a model for Floer homology.
Derived Poisson structures from Lie pairs are studied and their algebraic properties are explored.
problem Exploring derived Poisson structures from Lie pairs.
method Algebraic and homotopy transfer theorems for derived Poisson algebras.
result Derived Poisson algebra structure on totΩA∙(Λ∙(L/A)) is unique up to isomorphism. The paper develops a theory of Ehresmann structures in positive characteristic.
problem Developing a theory for Ehresmann structures in positive characteristic.
method Comparing Frobenius-Ehresmann structures with Cartan geometries and studying their equivalence.
result Formulating and proving the Ehresmann-Weil-Thurston principle for Frobenius-Ehresmann structures.