Verlinde-type formulas for rational surfaces (Q2302424)

In this paper, the author studies \(K\)-theoretic Donaldson invariants which are holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank-\(2\) sheaves on surfaces. Let \((X, \omega)\) be a pair consisting of a rational surface \(X\) and an ample line bundle \(\omega\) on \(X\). Let \(M^X_\omega(c_1, d)\) be the moduli space of \(\omega\)-semistable torsion-free coherent sheaves of rank-\(2\) on \(X\) with Chern classes \(c_1 \in H^2(X, \mathbb Z)\) and \(c_2 \in \mathbb Z\) such that \(d = 4c_2 - c_1^2\). Associated to a line bundle \(L\) on \(X\), there is a determinant line bundle \(\mu(L)\) on \(M^X_\omega(c_1, d)\). Let \(\Lambda\) be a formal variable. The goal of the paper is to study the generating function \[ \chi_{c_1}^{X, \omega}(L)= \sum_{d > 0} \chi\big (M^X_\omega(c_1, d), \mu(L) \big ) \Lambda^d \] of the holomorphic Euler characteristics \(\chi\big (M^X_\omega(c_1, d), \mu(L) \big )\). Assume that \(\omega \cdot K_X < 0\) where \(K_X\) is the canonical divisor of \(X\), and that \(\omega = H - a_1E_1 - \ldots - a_n E_n\) with each \(a_i < 1/\sqrt{n}\) when \(X\) is the blown-up of the projective plane \(\mathbb P^2\) at \(n\) points with exceptional divisors \(E_1, \ldots, E_n\) and \(H\) is a line in \(\mathbb P^2\). The main theorem of the paper states that if \(X\) is a rational surface, then there exist a polynomial \(P_{c_1, L}^X(\Lambda) \in \Lambda^{-c_1^2} \mathbb Q[\Lambda^{\pm 4}]\) and a non-negative integer \(l_{c_1, L}^X\) such that \[ \chi_{c_1}^{X, \omega}(L)\equiv \frac{P_{c_1, L}^X(\Lambda)}{(1 - \Lambda^4)^{l_{c_1, L}^X}} \] where for two Laurent series \(P(\Lambda) = \sum_n a_n \Lambda^n, Q(\Lambda) = \sum_n b_n \Lambda^n \in \mathbb Q[\Lambda^{-1}][[\Lambda]]\), define \(P(\Lambda) \equiv Q(\Lambda)\) if there exists an integer \(n_0\) such that \(a_n = b_n\) for all \(n \ge n_0\). Based on some explicit calculations on \(\mathbb P^2\) and \(\mathbb P^1 \times \mathbb P^1\), the author proposed several interesting conjectures regarding \(P_{c_1, L}^X(\Lambda)\) and \(l_{c_1, L}^X\). These results are analogue to the Verlinde formula for algebraic curves, and related to Le Potier's strange duality conjecture. The main ideas of the proof are to use the wall-crossing formula and blown-up formula for \(\chi_{c_1}^{X, \omega}(L)\), and to analyze the \(K\)-theoretic Donaldson invariants with point class. Section~2 is devoted to background materials such as determinant line bundles, walls and chambers, and \(K\)-theoretic Donaldson invariants. Section~3 reviews the strange duality conjecture for surfaces, and interprets the main results and conjectures in view of strange duality. Section~4 recalls Theta functions, modular forms and the wall-crossing formula. For the \(K\)-theoretic Donaldson invariants with point class, the polynomiality and vanishing of the wall-crossing formula are investigated. In Section~5, the author studies the K-theoretic Donaldson invariants for polarizations on the boundary of the ample cone. Section~6 applies blowup polynomials, blowup formulas and blowdown formulas to the present paper. Recursion formulas for rational ruled surfaces are proved in Section~7. Computations of the invariants for \(\mathbb P^2\) and \(\mathbb P^1 \times \mathbb P^1\) are carried out in Section~8 and Section~9 respectively.