The Jacobian Conjecture is established : If f1, · · · , fn be elements in a polynomial ring k[X1, · · · , Xn] over a field k of characteristic zero such that det(∂fi/∂Xj) is a nonzero constant, then k[f1, · · · , fn] = k[X1, · · · , Xn]. Let k be an algebraically closed field, let k be an affine space of dimension n over k and let f : k −→ k be a morphism of algebraic varieties. Then f is given...

Abstract We give an explicit positive answer, in the case of reduced curve singularities, to a question B. Teissier about existence toric embedded resolution after reembedding. In singularity ( C , O ) contained non singular surface S such reembedding may be defined terms sequence maximal contact curves minimal . prove that there exists modification, reembedding, which provides use properties s...

We prove that the Seidel morphism of (M × M , ω ⊕ ω) is naturally related to the Seidel morphisms of (M,ω) and (M , ω), when these manifolds are monotone. We deduce a condition for loops of Hamiltonian diffeomorphisms of the product to be homotopically non trivial. This result was inspired by and extends results obtained by Pedroza [P]. All the symplectic manifolds we consider in this note are ...

Given a proper morphism X → S, we show that a large class of objects in the derived category of X naturally form an Artin stack locally of finite presentation over S. This class includes S-flat coherent sheaves and, more generally, contains the collection of all S-flat objects which can appear in the heart of a reasonable sheaf of t-structures on X. In this sense, this is the Mother of all Modu...

In this paper we prove another pairing theorem for bordered Floer homology. Unlike the original pairing theorem, this one is stated in terms of homomorphisms, not tensor products. The present formulation is closer in spirit to the usual TQFT framework, and allows a more direct comparison with Fukaya-categorical constructions. The result also leads to various dualities in bordered Floer homology.

Let X be a smooth projective variety over an algebraically field k with char(k) = p > 0 and F : X → X1 be the relative Frobenius morphism. When dim(X) = 1, we prove that F∗W is a stable bundle for any stable bundle W (Theorem 2.3). As a step to study the question for higher dimensional X , we generalize the canonical filtration (defined by Joshi-Ramanan-Xia-Yu for curves) to higher dimensional ...