MATH 7375 Topics in Topology

Cohomology Jump Loci

Spring 2024

Course Information

Course	MATH 7375 · Topics in Topology
Instructor	Alex Suciu
Course Web Site	https://suciu.sites.northeastern.edu/courses/math7375-spring2024/
Time and Place	Tuesday & Friday 5:50pm – 7:30pm, 304 Kariotis Hall
Office	435 LA – Lake Hall
Email	a.suciu@northeastern.edu
Office Hours	Tuesday & Friday 4:30pm – 5:30pm
Prerequisites:	MATH 7221 – Topology 2

Course Description

The cohomology jumping loci of a space X come in two basic flavors: the characteristic varieties (the jump loci for cohomology with coefficients in rank 1 local systems), and the resonance varieties (the jump loci for the homology of the cochain complexes arising from multiplication by degree 1 classes in the cohomology ring of X). The interplay between these varieties leads to new obstructions to formality and (quasi-) projectivity, and informs on the homological finiteness properties of various spaces and groups.

Bibliography

• M. Aprodu, G. Farkas, C. Raicu, A. Sammartano, A. Suciu, Higher resonance schemes and Koszul modules of simplicial complexes, arXiv:2309.00609
• M. Aprodu, G. Farkas, C. Raicu, A. Suciu, Reduced resonance schemes and Chen ranks, arXiv:2303.07855
• G. Denham, A. Suciu, S. Yuzvinsky, Abelian duality and propagation of resonance, Selecta Mathematica 23 (2017), no. 4, 2331–2367.  
• A. Dimca, A. Suciu, Which 3-manifold groups are Kähler groups?, J. Eur. Math. Soc. 11 (2009), no. 3, 521–528.
• A. Dimca, S. Papadima, A. Suciu, Topology and geometry of cohomology jump loci, Duke Math. J. 148 (2009), no. 3, 405–457.
• ––––, Non-finiteness properties of fundamental groups of smooth projective varieties, J. Reine Angew. Math 629 (2009), 89–105.
• D. Matei, A. Suciu, Cohomology rings and nilpotent quotients of real and complex arrangements, Arrangements—Tokyo 1998, 185–215, Adv. Stud. Pure Math., vol. 27, Kinokuniya, Tokyo, 2000.
• ––––, Hall invariants, homology of subgroups, and characteristic varieties, Int. Math. Res. Not. 2002 (2002), no. 9, 465–503. 
• S. Papadima, A. Suciu, Algebraic invariants for right-angled Artin groups, Math. Ann. 334 (2006), no. 3, 533–555. 
• ––––, Toric complexes and Artin kernels, Adv. Math. 220 (2009), no. 2, 441–477.
• ––––, The spectral sequence of an equivariant chain complex and homology with local coefficients,Trans. Amer. Math. Soc. 362 (2010), no. 5, 2685–2721.
• ––––, Bieri-Neumann-Strebel-Renz invariants and homology jumping loci, Proc. London Math. Society 100 (2010), no. 3, 795–834.
• ––––, Algebraic monodromy and obstructions to formality, Forum Math. 22 (2010), no. 5, 973–983.
• ––––, Jump loci in the equivariant spectral sequence, Math. Res. Lett. 21 (2014), no. 4, 863–883. 
• ––––, Vanishing resonance and representations of Lie algebras, J. Reine Angew. Math. 706 (2015), 83–101. 
• A. Suciu, Fundamental groups, Alexander invariants, and cohomology jumping loci, Topology of algebraic varieties and singularities, 179–223, Contemp. Math., vol. 538, Amer. Math. Soc., Providence, RI, 2011.
• ––––, Resonance varieties and Dwyer-Fried invariants, Arrangements of Hyperplanes (Sapporo 2009), 359–398, Advanced Studies in Pure Mathematics, vol. 62, Kinokuniya, Tokyo, 2012.
• ––––, Geometric and homological finiteness in free abelian covers, Configuration Spaces: Geometry, Combinatorics and Topology (Centro De Giorgi, 2010), 461–501, Publications of the Scuola Normale Superiore, vol. 14, Edizioni della Normale, Pisa, 2012. 
• ––––, Characteristic varieties and Betti numbers of free abelian covers, Int. Math. Res. Not. IMRN 2014 (2014), no. 4, 1063–1124. 
• ––––, Around the tangent cone theorem, Configuration Spaces: Geometry, Topology and Representation Theory, 1–39, Springer INdAM series, vol. 14, Springer, Cham, 2016. 
• ––––, Poincaré duality and resonance varieties, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 6, 3001–3027. 
• ––––, Sigma-invariants and tropical varieties, Math. Ann. 380 (2021), no. 3-4, 1427–1463. 
• ––––, Cohomology jump loci of 3-manifolds, Manuscripta Math. 67 (2022), no. 1-2, 89–123. 
• ––––, Cohomology, Bocksteins, and resonance varieties in characteristic 2, Compactifications, Configurations, and Cohomology, 131–157, Contemp. Math., vol. 790, Amer. Math. Soc., Providence, RI, 2023.
• ––––, Alexander invariants and cohomology jump loci in group extensions, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (2024). arXiv:2107.05148 
• ––––, Koszul modules, holonomy Lie algebras, and resonance varieties of commutative differential graded algebras, in preparation.
• A. Suciu, H. Wang, Pure virtual braids, resonance, and formality, Math. Z. 286 (2017), no. 3-4, 1495–1524.
• ––––, Chen ranks and resonance varieties of the upper McCool groups, Adv. in Appl. Math. 110 (2019), 197–234.
• A. Suciu, Y. Yang, G. Zhao, Homological finiteness of abelian covers, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), no. 1, 101–153.