An inclusion-exclusion principle for tautological sheaves on Hilbert schemes of points (Q6593247)

The author proves an equation for Euler characteristics of tautological sheaves on Hilbert schemes of points on double point degenerations. To state the result, let \(C\) be a smooth curve over a field \(k\) of characteristic zero and \(0 \in C\). A \textit{double point degeneration} is a proper flat morphism \(\pi: \mathcal X \to C\) of proper algebraic spaces whose fiber \(X=\pi^{-1}(p)\) is smooth for \(p \neq 0\) and whose reducible central fiber is \(\pi^{-1} (0) = Y_1 \cup Y_2\), where \(Y_1, Y_2\) are smooth with normal crossing divisors and \(D = Y_1 \cap Y_2\) is smooth. A vector bundle \(G\) on \(Y\) yields \(\Lambda_{-u} G = \sum_{i=0}^{\mathrm{rank} G} (-u)^i \wedge^i G\), a polynomial in the variable \(u\) with coefficients in the \(K\)-theory of \(Y\). For bundles \(G_1, G_2\) on \(Y\), define \(\chi (G_1, G_2) = \sum_{i=0}^\infty (-1)^i \dim_k \mathrm{Ext}^i_{\mathcal O_Y} (G_1, G_2)\). A bundle \(E\) on \(Y\) gives rise to tautological sheaves \(E^{[n]} = p_* q^* E\) on \(\mathrm{Hilb}^n (Y)\) for each \(n > 0\), where \(p: Z \to \mathrm{Hilb}^n (Y)\) is the universal family and \(q: Z \to Y\) is the projection. With this set up and vector bundles \(E, F\) on \(\mathcal X\), the author gives a formula for the infinite series \(\log (1 + \sum_{n=1}^\infty \chi (\Lambda_{-u} (E|_X^{[n]}), \Lambda_{-v} (F|_X^{[n]})) Q^n)\), namely\N\[\N\sum_{i=1}^2 \log (1 + \sum_{n=1}^\infty \chi (\Lambda_{-u} (E|_{Y_i}^{[n]}), \Lambda_{-v} (F|_{Y_i}^{[n]})) Q^n) - \log (1 + \sum_{n=1}^\infty \chi (\Lambda_{-u} (E|_P^{[n]}), \Lambda_{-v} (F|_P^{[n]})) Q^n)\N\]\Nwhere \(P = \mathbb P (N_{Y_1/D} \oplus \mathcal O_D) \to D \hookrightarrow \mathcal X\). The theorem is proved using the good degeneration technique of \textit{J. Li} and \textit{B. Wu} [Commun. Anal. Geom. 23, No. 4, 841--921 (2015; Zbl 1349.14014)].\N\NAlgebraic cobordism of vector bundles [\textit{Y. P. Lee} and \textit{R. Pandharipande}, J. Eur. Math. Soc. (JEMS) 14, No. 4, 1081--1101 (2012; Zbl 1348.14107)] is used to conclude the existence of universal polynomials \(f_{i,j,k} \in \mathbb Q [u_1, \dots, u_n, v_1, \dots, v_r, w_1, \dots, w_s]\) such that for any smooth proper algebraic space \(X\) of pure dimension \(d\), a rank \(r\)-bundle \(E\) and a rank \(s\)-bundle \(F\), there is the equation\N\[\N1 + \sum_{n=1}^\infty \chi(\Lambda_{-u} E^{[n]}, \Lambda_{-v} F^{[n]}) Q^n = \exp (\sum_{i=1}^\infty \sum_{j=1}^\infty \sum_{k=1}^\infty Q^i u^j v^k \int_X \Phi_{f_{i,j,k}} (E,F))\N\]\Nwhere the integral \(\Phi_{f_{i,j,k}} (E,F)\) is obtained by linearly extending the evaluation at a monomial \(\prod u_i^{k_i} \prod v_j^{l_j} \prod w_k^{m_k}\) by \( \int_X \prod c_i (T_X)^{k_i} \prod c_j (E)^{l_j} \prod c_k (F)^{m_k}. \) The proof uses a result of \textit{B. G. Moĭshezon} [Izv. Akad. Nauk SSSR, Ser. Mat. 31, 1385--1414 (1967; Zbl 0186.26205); translation in Math. USSR, Izv. 1(1967), 1331--1356 (1968)] to reduce to the case of projective schemes: for lack of a reference, Appendix A gives a proof following \textit{X. Ma} and \textit{G. Marinescu} [Holomorphic Morse inequalities and Bergman kernels. Basel: Birkhäuser (2007; Zbl 1135.32001)]. When \(X\) is a surface, the existence of universal polynomials was shown by \textit{G. Ellingsrud} et al. [J. Algebr. Geom. 10, No. 1, 81--100 (2001; Zbl 0976.14002)]. In higher dimensions there is a similar result of \textit{J. V. Rennemo} [Geom. Topol. 21, No. 1, 253--314 (2017; Zbl 1387.14029)].\N\NA conjecture of \textit{Z. Wang} and \textit{J. Zhou} [J. Algebr. Geom. 23, No. 4, 669--692 (2014; Zbl 1327.14028)] says that if \(X\) is a smooth projective \(k\) scheme of dimension \(d \geq 2\) and \(K,L\) are line bundles on \(X\), then\N\[\N1 + \sum_{n=1}^\infty \chi(\Lambda_{-u} E^{[n]}, \Lambda_{-v} F^{[n]}) Q^n = \exp (\sum_{r=1}^\infty \chi (\Lambda_{-u^r} K, \Lambda_{-v^r} L) \frac{Q^r}{r}.\N\]\NThe author applies the existence of universal coefficients to reduce the conjecture to the case where \(X\) is a product of projective spaces and \(K, L\) are pullbacks of \(0, \mathcal O\), or \(\mathcal O (1)\) along one of the natural projections onto the factors. When \(\dim X = 3\), the author confirms the conjecture modulo \(Q^7\).