Johnson's Quot-scheme integral conjecture

Let XX be a smooth projective surface with χ(OX)=1\chi(\mathcal{O}_X)=1, let ff be the invariants of an ideal sheaf of kk points, let VV be a vector bundle such that Quot(V,k)\operatorname{Quot}(V,k) has expected dimension zero, and let ee^* denote the invariants of the kernels of the parametrized surjections. Johnson's Quot-scheme integral conjecture.

X[k]c2k(V[k])=χ(X[k],OX[k](Θe)).\int_{X^{[k]}} c_{2k}({V^*}^{[k]})=\chi(X^{[k]},\mathcal{O}_{X^{[k]}}(\Theta_e)).

The expected dimension is zero exactly when χ(ef)=0\chi(e\cdot f)=0. The conjecture relates a top Chern-class integral counting finite Quot schemes to the Euler characteristic of a determinant line bundle; the source does not state a general proof.

Sources & referencesView supporting material

Primary source

Thomas Goller and Yinbang Lin, “Rank-one sheaves and stable pairs on surfaces”, arXiv:1907.05180 (2019).

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