Abstract. We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is done using a deformation argument and the fact that the derived endomorphism algebra of the sequence is constant. Applying Kuznetsov's results on heights of exceptional sequences, we also show that the sequence on S itself is not full and its (left or right) orthogonal complement is also a phantom category.

In the last three years a new concept -the concept of wall crossing has emerged. The current situation with wall crossing phenomena, after papers of Seiberg-Witten, Gaiotto-Moore-Neitzke, Vafa-Cecoti and seminal works by Donaldson-Thomas, Joyce-Song, Maulik-Nekrasov-Okounkov-Pandharipande, Douglas, Bridgeland, and Kontsevich-Soibelman, is very similar to the situation with Higgs Bundles after the works of Higgs and Hitchin -it is clear that a general "Hodge type" of theory exists and needs to be developed. Nonabelian Hodge theory did lead to strong mathematical applications -uniformization, Langlands program to mention a few. In the wall crossing it is also clear that some "Hodge type" of theory exists -Stability Hodge Structure (SHS). This theory needs to be developed in order to reap some mathematical benefits -solve long standing problems in algebraic geometry. In this paper we look at SHS from the perspective of Landau-Ginzburg models and we look at some applications. We consider simple examples and explain some conjectures these examples suggest.