P=W Conjecture

The conjecture asserts that the perverse Leray filtration of a Hitchin fibration agrees, after doubling its index, with the mixed-Hodge weight filtration on the corresponding character variety under the nonabelian Hodge correspondence. In the smooth fixed-determinant trace-free setting, let be a smooth projective complex curve of genus , , and a line bundle of degree coprime to . Put and . The twisted Betti space is

$X^B={(A_1,B_1,...,A_g,B_g) in SL_n(C)^(2g):product_(j)[A_j,B_j]=exp(2piid/n)I_n}//SL_n(C).$

(1)

The double slash denotes the affine invariant-theory quotient by simultaneous conjugation, and . For the Hitchin map , the normalization is

$P_kH^m(X,Q)=im{H^(m-R)(A,^ptau_(<=k)Rh_*Q_X[R])->H^m(X,Q)}.$

(2)

Transport to using the nonabelian Hodge correspondence. The predicted equality is

(3)

See also

Character Variety, Higgs Bundle, Hitchin Fibration, Mixed Hodge Structure, Nonabelian Hodge Correspondence, Perverse Leray Filtration

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References

de Cataldo, M. A.; Hausel, T.; and Migliorini, L. "Topology of Hitchin Systems and Hodge Theory of Character Varieties: The Case

." Ann. Math. 175, 1329-1407, 2012. https://doi.org/10.4007/annals.2012.175.3.7.de Cataldo, M. A.; Maulik, D.; and Shen, J. "On the

Conjecture for

." Selecta Math. 28, Article 90, 2022. https://doi.org/10.1007/s00029-022-00803-0.

Cite this as:

Weisstein, Eric W. "P=W Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/P=WConjecture.html

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