For a simply connected solvable Lie group G with a cocompact discrete subgroup 􀀀, we consider the space of differential forms on the solvmanifold G/􀀀 with values in a certain flat bundle so that this space has a structure of a differential graded algebra (DGA). We construct Sullivan’s minimal model of this DGA. This result is an extension of Nomizu’s theorem for ordinary coefficients in the nilpotent case. By using this result, we refine Hasegawa’s result of formality of nilmanifolds and Benson-Gordon’s result of hard Lefschetz properties of nilmanifolds.

We prove the factoriality of a nodal hypersurface in P4 of degree d that has at most 2(d ¡ 1)2=3 singular points, and we prove the factoriality of a double cover of P3 branched over a nodal surface of degree 2r having less than (2r ¡ 1)r singular points.

The Wess-Zumino-Witten term was first introduced in the low energy σ-model which describes pions, the Goldstone bosons for the broken flavor symmetry in quantum chromodynamics. We introduce a new definition of this term in arbitrary gravitational backgrounds. It matches several features of the fundamental gauge theory, including the presence of fermionic states and the anomaly of the flavor symmetry. To achieve this matching, we use a certain generalized differential cohomology theory. We also prove a formula for the determinant line bundle of special families of Dirac operators on 4-manifolds in terms of this cohomology theory. One consequence is that there are no global anomalies in the Standard Model (in arbitrary gravitational backgrounds).

We study the spectral geometric properties of the scalar Laplace–Beltrami operator associated to the Weil–Petersson metric $g_{WP}$ on $\mathcal{M}_{\gamma}$ , the Riemann moduli space of surfaces of genus  $\gamma>1$. This space has a singular compactification with respect to $g_{WP}$, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete.

On considère une variété riemannienne (M,g) non compacte, complète, à géométrie bornée et courbure de Ricci parallèle. Nous montrons que certains opérateurs “affines” en la courbure de Ricci sont localement inversibles, dans des espaces de Sobolev classiques, au voisinage de g.