Nero Budur

Cohomology jump loci of differential graded Lie algebras

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

On the local zeta functions and the b-functions of certain hyperplane arrangements

Conjectures of Igusa for p-adic local zeta functions and of Denef and Loeser for topological local zeta functions assert that (the real part of) the poles of these local zeta functions are roots of the Bernstein–Sato polynomials (that is, the b -functions). We prove these conjectures for certain hyperplane arrangements, including the case of reduced hyperplane arrangements in three-dimensional affine space.

Intersection spaces, perverse sheaves and type IIB string theory

The method of intersection spaces associates rational Poincare complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA the- ory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the inter- section space complex, on the hypersurface. Moreover, the intersection space complex underlies a mixed Hodge module, so its hypercohomology groups carry canonical mixed Hodge structures. For a large class of singularities, e.g., weighted homogeneous ones, global Poincare duality is induced by a more refined Verdier self-duality isomorphism for this perverse sheaf. For such singularities, we prove furthermore that the pushforward of the constant sheaf of a nearby smooth deformation under the specialization map to the singular space splits off the intersection space complex as a direct summand. The complementary summand is the contribution of the singularity. Thus, we obtain for such hypersurfaces a mirror statement of the Beilinson-Bernstein-Deligne decomposition of the pushforward of the constant sheaf under an algebraic resolution map into the intersection sheaf plus contributions from the singularities.

Jumping Numbers of Hyperplane Arrangements

Saito [8] proved that the jumping numbers of a hyperplane arrangement depend only on the combinatorics of the arrangement. However, a formula in terms of the combinatorial data was still missing. In this note, we give a formula and a different proof of the fact that the jumping numbers of a hyperplane arrangement depend only on the combinatorics. We also give a combinatorial formula for part of the Hodge spectrum and for the inner jumping multiplicities.

Log canonical thresholds of quasi-ordinary hypersurface singularities.

The log canonical thresholds of irreducible quasi-ordinary hypersurface singularities are computed, using an explicit list of pole candidates for the motivic zeta function found by the last two authors.