Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Generating functions for $ K$-theoretic Donaldson invariants and Le Potier's strange duality


Authors: Lothar Göttsche and Yao Yuan
Journal: J. Algebraic Geom. 28 (2019), 43-98
DOI: https://doi.org/10.1090/jag/703
Published electronically: September 4, 2018
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Abstract: For a projective algebraic surface $ X$ with an ample line bundle $ H$, let $ M_H^X(c)$ be the moduli space $ H$-semistable sheaves $ \mathcal {E}$ of class $ c$ in the Grothendieck group $ K(X)$. We write $ c=(r,c_1,c_2)$ or $ c=(r,c_1,\chi )$ with $ r$ the rank, $ c_1,c_2$ the Chern classes, and $ \chi $ the holomorphic Euler characteristic. We also write $ M_H^X(2,c_1,c_2)=M_X^X(c_1,d)$, with $ d=4c_2-c_1^2$. The $ K$-theoretic Donaldson invariants are the holomorphic Euler characteristics $ \chi (M_H^X(c_1,d),\mu (L))$, where $ \mu (L)$ is the determinant line bundle associated to a line bundle on $ X$. More generally for suitable classes $ c^*\in K(X)$ there is a determinant line bundle $ \mathcal {D}_{c,c^*}$ on $ M^X_H(c)$. We first compute some generating functions for $ K$-theoretic Donaldson invariants on $ \mathbb{P}^2$ and rational ruled surfaces, using the wallcrossing formula of [Pure Appl. Math. Q. 5 (2009), pp. 1029-1111].

Then we show that Le Potier's strange duality conjecture relating $ H^0(M^X_H(c),\mathcal {D}_{c,c^*})$ and $ H^0(M^X_H(c^*),\mathcal {D}_{c^*,c})$ holds for the cases $ c=(2,c_1=0,c_2>2)$ and $ c^{*}=(0,L,\chi =0)$ with $ L=-K_X$ on $ \mathbb{P}^2$, and $ L=-K_X$ or $ -K_X+F$ on $ \mathbb{P}^1\times \mathbb{P}^1$ and $ \widehat {\mathbb{P}^2}$ with $ F$ the fibre class of the ruling, and also the case $ c=(2,H,c_2)$ and $ c^*=(0,2H,\chi =-1)$ on $ \mathbb{P}^2$.


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Additional Information

Lothar Göttsche
Affiliation: International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy
Email: gottsche@ictp.it

Yao Yuan
Affiliation: Yau Mathematical Sciences Center, Tsinghua University, 100084, Beijing, People’s Republic of China
Email: yyuan@mail.tsinghua.edu.cn

DOI: https://doi.org/10.1090/jag/703
Received by editor(s): October 3, 2016
Received by editor(s) in revised form: February 20, 2017, and April 5, 2017
Published electronically: September 4, 2018
Additional Notes: The second-named author was supported by NSFC grant 11301292.
Article copyright: © Copyright 2018 University Press, Inc.

American Mathematical Society