scholarly journals Summands of theta divisors on Jacobians

2020 ◽  
Vol 156 (7) ◽  
pp. 1457-1475
Author(s):  
Thomas Krämer
Keyword(s):  
Representation Theory ◽  
Decomposition Theorem ◽  
Perverse Sheaves ◽  
Theta Divisor

We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.

scholarly journals Purity and decomposition theorems for staggered sheaves

2012 ◽  
Vol 11 (4) ◽  
pp. 695-745
Author(s):  
Pramod N. Achar ◽  
David Treumann
Keyword(s):  
Decomposition Theorem ◽  
Derived Category ◽  
Perverse Sheaves ◽  
Perverse Sheaf ◽  
Coherent Sheaves ◽  
Direct Sum ◽  
Constructible Sheaves ◽  
Decomposition Theorems ◽  
Pure Complex

AbstractTwo major results in the theory of ℓ-adic mixed constructible sheaves are the purity theorem (every simple perverse sheaf is pure) and the decomposition theorem (every pure object in the derived category is a direct sum of shifts of simple perverse sheaves). In this paper, we prove analogues of these results for coherent sheaves. Specifically, we work with staggered sheaves, which form the heart of a certain t-structure on the derived category of equivariant coherent sheaves. We prove, under some reasonable hypotheses, that every simple staggered sheaf is pure, and that every pure complex of coherent sheaves is a direct sum of shifts of simple staggered sheaves.

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