Aspherical Kähler manifolds with solvable fundamental group

We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of $\mathbb{C}^{n}We survey recent developments which led to the proof of the Benson-Gordon conjecture on K?hler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a K?hler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical K?hler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of \mathbbCⁿ\mathbb{C}^{n} by a discrete group of complex isometries.     

On the Zero Defect Conjecture

Brlek et al., conjectured in 2008 that any fixed point of a primitive morphism with finite palindromic defect is either periodic or its palindromic defect is zero. Bucci and Vaslet disproved this conjecture in 2012 by a counterexample over ternary alphabet. We prove that the conjecture is valid on binary alphabet. We also describe a class of morphisms over multiliteral alphabet for which the conjecture still holds. The proof is based on properties of extension graphs.     

A new family of surfaces of general type with K^2=7 and p_g=0

We construct a new family of smooth minimal surfaces of general type with $K^2=7$ and $p_g=0$ . We show that a surface in this family has ample canonical divisor and birational bicanonical morphism. We also prove that these surfaces satisfy Bloch’s conjecture.     

Toroidalization of locally toroidal morphisms from N-folds to surfaces

The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a toroidal morphism. Following a suggestion by Dale Cutkosky, we define the notion of locally toroidal morphisms and ask whether any locally toroidal morphism can be modified into a toroidal morphism. In this paper, we answer the question in the affirmative when the morphism is between any arbitrary variety and a surface.     

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U⊂X be an open set whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent intermediate extension functor.Under suitable hypotheses, we introduce a construction (called “S₂-extension”) in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical “S₂-ification” of appropriate X. The construction also has applications to the “Macaulayfication” problem, and it is particularly well-behaved when X is Gorenstein.Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S₂-extension to give a uniform construction of the desired variety.     

A Connection Between the Reticulation of a Ring of Quotients and the Localization Lattice of the Reticulation of a Commutative Ring

Through this article, R denotes a commutative ring with identity. The aim of this article is to construct a morphism of lattices between the reticulation of a ring of quotients (of a commutative ring with respect to a Gabriel topology) and the localization lattice of the reticulation of the initial ring. The article is structured as follows: The first section introduces all the necessary notions required to simplify the reading of the article. The second section presents some properties of the topology induced by a Gabriel topology on the reticulation L(R). In the third section the morphism δ, which achieves the intended link, is constructed. The last section studies the defined morphism in some particular cases.     

Periodic points, linearizing maps, and the dynamical Mordell-Lang problem

Under suitable hypotheses, we prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism . We also prove a version of the Mordell-Lang conjecture that holds for any endomorphism of a semiabelian variety. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics.     

On the Stanley ring of cubical complex

We investigate the properties of the Stanley ring of a cubical complex, a cubical analogue of the Stanley-Reisner ring of a simplicial complex. We compute its Hilbert series in terms of thef-vector, and prove that by taking the initial ideal of the defining relations, with respect to the reverse lexicographic order, we obtain the defining relations of the Stanley-Reisner ring of the triangulation via “pulling the vertices” of the cubical complex. Applying an old idea of Hochster we see that this ring is Cohen-Macaulay when the complex is shellable, and we show that its defining ideal is generated by quadrics when the complex is also a subcomplex of the boundary complex of a convex cubical polytope. We present a cubical analogue of balanced Cohen-Macaulay simplicial complexes: the class of edge-orientable shellable cubical complexes. Using Stanley's results about balanced Cohen-Macaulay simplicial complexes and the degree two homogeneous generating system of the defining ideal, we obtain an infinite set of examples for a conjecture of Eisenbud, Green, and Harris. This conjecture says that theh-vector of a polynomial ring inn variables modulo an ideal which has ann-element homogeneous system of parameters of degree two, is thef-vector of a simplicial complex.     

Linear balls and the multiplicity conjecture

A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley–Reisner ring has a linear resolution. It turns out that the Stanley–Reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture. A class of shellable spheres arising naturally from commutative algebra whose Stanley–Reisner rings satisfy the multiplicity conjecture will be presented.     

The Multiplicity Conjecture for Barycentric Subdivisions

For a simplicial complex Δ we study the effect of barycentric subdivision on ring theoretic invariants of its Stanley–Reisner ring. In particular, for Stanley–Reisner rings of barycentric subdivisions we verify a conjecture by Huneke and Herzog and Srinivasan, that relates the multiplicity of a standard graded k-algebra to the product of the maximal and minimal shifts in its minimal free resolution up to the height. On the way to proving the conjecture, we develop new and list well-known results on behavior of dimension, Hilbert series, multiplicity, local cohomology, depth, and regularity when passing from the Stanley–Reisner ring of Δ to the one of its barycentric subdivision.     

The Fusion Algebra of Bimodule Categories

We establish an algebra-isomorphism between the complexified Grothendieck ring of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of . As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category.      

Proprietes et applications de la notion de contenu

The notion of content is used to solve certain problems. In the first part, we show that the structural morphism of a content algebra (see the paper of D.E. Rush [22] ) is spectrally open, under mild hypothesis. We show also that a flat module is universally content if and only if it is a Mittag-Leffler’s module in the sense of f2ll . In the second part, using content, we exhibit a kind of localization of a commutative ring A, attached to eyery subset X of Spec(A) i.e. a flat morphism A→ X(A). We can thus show that every quasi-compact, stable under generization subset of a spectra is a spectral image under a flat morphism, in a canonical way. We can also give in certain cases an elementary construction of the maximal flat injective epimor-phism of a ring. Suppose that A is a Noetherian ring and consider pro-perties of Noetherian rings such as factoriality, normality and so on.

Let X be the set of prime ideals of A at which A has the property. If X is stable under generization, the flat morphism A→ X(A) verifies hin general the ring X(A) has lornlly the property and a prime ideal P of A has a prime ideal lying over in X(A) if and only if pthe ring has the property at P.     

A Conjecture on the Hall Topology for the Free Group

The Hall topology for the free group is the coarsest topologysuch that every group morphism from the free group onto a finitediscrete group is continuous. It was shoen by M.Hall Jr thatevery finitely generated subgroup of the free group is closedfor this topology. We conjecture that if H₁, H₂,...,H_n are finitelygenerated subgroups of the free group, then the product H₁ H₂...H_n is closed. We discuss some consequences of this conjecture.First, it would give a nice and simple algorithm to computethe closure of a given rational subset of the free group. Next,it implies a similar conjecture for the free monoid, which inturn is equivalent to a deep conjecture on finite semigroupsfor the solution of which J. Rhodes has offered $100. We hopethat our new conjecture will shed some light on Rhodes' conjecture.     

Index theory for quasi-crystals I. Computation of the gap-label group

In this paper, we give a complete solution to the gap labelling conjecture for quasi-crystals. The method adopted relies on the index theory for laminations, and the main tools are the Connes-Skandalis longitudinal K-theory index morphism together with the measured index formula.     

Configuration spaces of labeled particles and finite Eilenberg-MacLane complexes

For any Coxeter system (W, S), the group W acts naturally on the complement of the associated complex hyperplane arrangement. By the well-known conjecture, the orbit space of this action is the classifying space of the corresponding Artin group. We describe some properties of configuration spaces of particles labeled by elements of a partial monoid and use them to prove that the orbit space mentioned in the conjecture is the classifying space of the positive Artin monoid. In particular, the conjecture reduces to a problem concerning the group completion of this monoid.     

Version kählérienne d'une conjecture de Robert J. Zimmer

Let M be a compact Kähler manifold. Let G be a connected simple real Lie group. Let Γ be a lattice in G. We prove the following: if the R-rank of G is strictly larger than the complex dimension of M any morphism from Γ to the group of holomorphic diffeomorphisms of M has finite image. This is a particular case in a conjecture of Robert J. Zimmer     

On highly potential words

We introduce a class of infinite words, called highly potential words because of their seemingly high potential of being a good supply of examples and counterexamples regarding various problems on words. We prove that they are all aperiodic words of finite positive defect, and having their set of factors closed under reversal, thus giving examples Brlek and Reutenauer were looking for. We prove that they indeed satisfy the Brlek–Reutenauer conjecture. We observe that each highly potential word is recurrent, but not uniformly recurrent. Considering a theorem from the paper of Balková, Pelantová and Starosta, later found to be incorrect, we show that highly potential words constitute an infinite family of counterexamples to that theorem. Finally, we construct a highly potential word which is a fixed point of a nonidentical morphism, thus showing that a stronger version of a conjecture by Blondin-Massé et al., as stated by Brlek and Reutenauer, is false.     

On the multigraded Hilbert and Poincaré-Betti series and the Golod property of monomial rings

In this paper we study the multigraded Hilbert and Poincaré-Betti series of A=S/a, where S is the ring of polynomials in n indeterminates divided by the monomial ideal a. There is a conjecture about the multigraded Poincaré-Betti series by Charalambous and Reeves which they proved in the case where the Taylor resolution is minimal. We introduce a conjecture about the minimal A-free resolution of the residue class field and show that this conjecture implies the conjecture of Charalambous and Reeves and, in addition, gives a formula for the Hilbert series. Using Algebraic Discrete Morse theory, we prove that the homology of the Koszul complex of A with respect to x₁,…,x_n is isomorphic to a graded commutative ring of polynomials over certain sets in the Taylor resolution divided by an ideal r of relations. This leads to a proof of our conjecture for some classes of algebras A. We also give an approach for the proof of our conjecture via Algebraic Discrete Morse theory in the general case.The conjecture implies that A is Golod if and only if the product (i.e. the first Massey operation) on the Koszul homology is trivial. Under the assumption of the conjecture we finally prove that a very simple purely combinatorial condition on the minimal monomial generating system of a implies Golodness for A.     

Towards the classification of weak Fano threefolds with ρ = 2

In this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases is proven.