We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension two and prove a Hodge to de Rham degeneration theorem for such log spaces. We show that new developments of Bogomolov-Tian-Todorov theory can be applied to obtain smoothings. The theory relates to recent work in mirror symmetry and the construction of Frobenius manifold structures. It has potential applications to the classification of Fano fourfolds. Contents1. Introduction 1 2. Generically Log Smooth Families 8 3. Elementary Log Toroidal Families 10 4. Log Toroidal Families 14 5. Log Structures and Infinitesimal Deformations 15 6. Toroidal Crossing Spaces as Log Toroidal Families 18 7. Differentials for Elementary Log Toroidal Families 21 8. Base Change of Differentials for Log Toroidal Families 28 9. Spreading Out Log Toroidal Families 29 10. The Cartier Isomorphism 30 11. The Decomposition of F * W • X 0 /S 0 32 12. The Hodge-to-de-Rham Spectral Sequence 36 13. Smoothings via Maurer-Cartan Solutions 39 References 43

We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is to study log structures that are incoherent on a subspace of codimension 2 and prove a Hodge–de Rham degeneration theorem for such log spaces that also settles a conjecture by Danilov. We show that the homotopy equivalence between Maurer–Cartan solutions and deformations combined with Batalin–Vilkovisky theory can be used to obtain smoothings. The construction of new Calabi–Yau and Fano manifolds as well as Frobenius manifold structures on moduli spaces provides potential applications.

For an affine toric variety Spec(A), we give a convex geometric description of the Hodge decomposition of its Hochschild cohomology. Under certain assumptions we compute the dimensions of the Hodge summands T 1 (i) (A), generalizing the existing results about the André-Quillen cohomology group T 1(1) (A). We prove that every Poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization.

No abstract