Hodge Theory for Polymatroids

We construct a Leray model for a discrete polymatroid with arbitrary building set and we prove a generalized GoreskyMacPherson formula. The first row of the model is the Chow ring of the polymatroid; we prove Poincaré duality, Hard Lefschetz, and Hodge-Riemann theorems for the Chow ring. Furthermore we provide a relative Lefschetz decomposition with respect to the deletion of an element.

[1]  Geoff Whittle.  Characteristic Polynomials of Weighted Lattices , 1993 .

[2]  The Hard Lefschetz Theorem and the topology of semismall maps , 2000, math/0006187.

[3]  C. Procesi,et al.  Wonderful models of subspace arrangements , 1995 .

[4]  G. Ziegler,et al.  Equivariant topology of configuration spaces , 2012, 1207.2852.

[5]  M. Vladoiu,et al.  Discrete Polymatroids , 2003 .

[6]  Sergey Yuzvinsky.  Small rational model of subspace complement , 1998 .

[7]  J. Morgan,et al.  The algebraic topology of smooth algebraic varieties , 1986 .

[8]  G. Rota.  On the Foundations of Combinatorial Theory , 2009 .

[9]  Jacob P. Matherne,et al.  A semi-small decomposition of the Chow ring of a matroid , 2020, Advances in Mathematics.

[10]  E. Feichtner,et al.  A Leray model for the Orlik-Solomon algebra , 2019, 1910.10001.

[11]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[12]  Alberto Ravagnani,et al.  Rank-metric codes and q-polymatroids , 2018, Journal of Algebraic Combinatorics.

[13]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[14]  Marcelo Aguiar,et al.  Hopf Monoids and Generalized Permutahedra , 2017, Memoirs of the American Mathematical Society.

[15]  G. Rota.  On the foundations of combinatorial theory I. Theory of Möbius Functions , 1964 .

[16]  Dmitry N. Kozlov,et al.  Incidence combinatorics of resolutions , 2003 .

[17]  M. Goresky,et al.  Stratified Morse theory , 1988 .

[18]  Chow rings of toric varieties defined by atomic lattices , 2003, math/0305142.

[19]  Volkmar Welker,et al.  Minimal Resolutions Via Algebraic Discrete Morse Theory , 2009 .

[20]  Priyavrat Deshpande.  The Goresky-MacPherson formula for toric arrangements , 2018, 1808.06054.

[21]  James G. Oxley,et al.  Matroid theory , 1992 .

[22]  Karim A. Adiprasito,et al.  Hodge theory for combinatorial geometries , 2015, Annals of Mathematics.

[23]  S. Yuzvinsky.  Rational Model of Subspace Complement on Atomic Complex , 1999 .

[24]  L. Migliorini,et al.  The decomposition theorem, perverse sheaves and the topology of algebraic maps , 2007, 0712.0349.

[25]  June Huh,et al.  Log-concavity of characteristic polynomials and the Bergman fan of matroids , 2011, 1104.2519.

[26]  Matthias Lenz,et al.  The f-vector of a representable-matroid complex is log-concave , 2013, Adv. Appl. Math..

[27]  June Huh,et al.  Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs , 2010, 1008.4749.

[28]  Botong Wang,et al.  Enumeration of points, lines, planes, etc , 2016, 1609.05484.

[29]  Carsten Schultz,et al.  The cohomology rings of complements of subspace arrangements , 2001 .