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Register a new userDivisorial Zariski decomposition and algebraic Morse inequalities
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Authors
Stefano Trapani
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In this note we use the divisorial Zariski decomposition to give a more intrinsic version of the algebraic Morse inequalities.
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Let $alpha$ be a big class on a compact Kahler manifold. We prove that a decomposition $alpha=alpha_1+alpha_2$ into the sum of a modified nef class $alpha_1$ and a pseudoeffective class $alpha_2$ is the divisorial Zariski decomposition of $alpha$ if and only if $operatorname{vol}(alpha)=operatorname{vol}(alpha_1)$. We deduce from this result some properties of full mass currents.
In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of paths which are not only critical paths of the transitive closure but also allowed elementary paths of the digraph, and prove that the homology of the new chain complex is isomorphic to the path homology. On the basis of the above results, we give the Morse inequalities on digraphs.
In this short Note we show that the direct image sheaf R 1 $pi$ * (O X) associated to an analytic family of compact complex manifolds $pi$ : X $rightarrow$ S parametrized by a reduced complex space S is a locally free (coherent) sheaf of O S --modules. This result allows to improve a semi-continuity type result for the algebraic dimension of compact complex manifolds in an analytic family given in [B.15]. AMS Classification 2010. 32G05-32A20-32J10.
Let $f:M to mathbb{R}$ be a Morse-Bott function on a compact smooth finite dimensional manifold $M$. The polynomial Morse inequalities and an explicit perturbation of $f$ defined using Morse functions $f_j$ on the critical submanifolds $C_j$ of $f$ show immediately that $MB_t(f) = P_t(M) + (1+t)R(t)$, where $MB_t(f)$ is the Morse-Bott polynomial of $f$ and $P_t(M)$ is the Poincare polynomial of $M$. We prove that $R(t)$ is a polynomial with nonnegative integer coefficients by showing that the number of gradient flow lines of the perturbation of $f$ between two critical points $p,q in C_j$ coincides with the number of gradient flow lines between $p$ and $q$ of the Morse function $f_j$. This leads to a relationship between the kernels of the Morse-Smale-Witten boundary operators associated to the Morse functions $f_j$ and the perturbation of $f$. This method works when $M$ and all the critical submanifolds are oriented or when $mathbb{Z}_2$ coefficients are used.
Let G be a complex reductive group acting on a finite-dimensional complex vector space H. Let B be a Borel subgroup of G and let T be the associated torus. The Mumford cone is the polyhedral cone generated by the T-weights of the polynomial functions on H which are semi-invariant under the Borel subgroup. In this article, we determine the inequalities of the Mumford cone in the case of the linear representation associated to a quiver and a dimension vector n=(n_x). We give these inequalities in terms of filtered dimension vectors, and we directly adapt Schofields argument to inductively determine the dimension vectors of general subrepresentations in the filtered context. In particular, this gives one further proof of the Horn inequalities for tensor products.
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