Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero.We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approximately the same motivic class of Borel reductions as the degree of Borel reduction tends to −∞.

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Abstract. In "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld introduced the "very good" property for a smooth complex equidimensional stack. They prove that for a semisimple group G over C, the moduli stack Bun G (X) of G-bundles over a smooth complex projective curve X is "very good", as long as X has genus g > 1. In the case of the projective line, when g = 0, this is not the case. However, the result can sometimes be extended to the projective line by introducing additional parabolic structure at a collection of marked points and slightly modifying the definition of a "very good" stack. We provide a sufficient condition for the moduli stack of parabolic vector bundles over P 1 to be very good. We then use this property to study the space of solutions to the Deligne-Simpson problem.

In "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld introduced the "very good" property for a smooth complex equidimensional stack. They prove that for a semisimple group G over C, the moduli stack Bun G (X) of G-bundles over a smooth complex projective curve X is "very good", as long as X has genus g > 1. In the case of the projective line, when g = 0, this is not the case. However, the result can sometimes be extended to the projective line by introducing additional parabolic structure at a collection of marked points and slightly modifying the definition of a "very good" stack. We provide a sufficient condition for the moduli stack of parabolic vector bundles over P 1 to be very good. We then use this property to study the space of solutions to the Deligne-Simpson problem.This work was partially supported by NSF grant DMS 1101558. 1 2 ALEXANDER SOIBELMAN 4. Moduli of Parabolic Bundles 23 4.1. Outline 23 4.2. Generalities on Parabolic Bundles 24 4.3. The moduli stack of parabolic bundles over P 1 24 4.4. Proof of Theorem 1.2.1 26 5. Stability for Parabolic Bundles 31 5.1. Outline 31 5.2. Definitions 31 5.3. Semistability and the Very Good property 32 5.4. Stability for Quiver Representations and Stability for Parabolic Bundles 34 6. Quivers and Parabolic Bundles 36 6.1. Outline 36 6.2. Moduli Functor: parabolic bundles and flag bundles 37 6.3. Moduli functor: parabolic bundles and squids 42 6.4. The very good property for trivial bundles 44 7. Application to the Deligne-Simpson Problem 46 7.1. Outline 46 7.2. Logarithmic Connections and Squid Representations 46 7.3. The very good property and the additive Deligne-Simpson problem 52 7.4. The very good property and the multiplicative Deligne-Simpson problem 53 References 54

Let X be a smooth projective curve over a field of characteristic zero and let D be a non-empty set of rational points of X. We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on (X, D) and motivic classes of moduli stacks of semistable parabolic Higgs bundles on (X, D). As a by-product we give a criteria for non-emptiness of these moduli stacks, which can be viewed as a version of the Deligne-Simpson problem.