The period-index conjecture bounds the index of a class in the Brauer group of a field in terms of its period and the field's transcendence
degree. For 
,
its period 
is the least positive integer 
with 
. Its index 
is the square root
of the vector space dimension over 
of the central division algebra
representing 
.
The period divides the index, and the two have the same prime
divisors.
For a field 
of finite transcendence
degree 
over an algebraically closed field,
the conjecture asserts
 |
Colliot-Thélène's survey records this formulation (Colliot-Thélène 2001).
Perry (2026) reported counterexamples with period 2 and index 
for every 
.
They occur over fields of rational
functions on smooth projective 
-dimensional varieties over an
algebraically closed field 
.
In characteristic 0, the construction requires
.
In characteristic 
, it requires 
, where 
is the finite field with
elements. In particular, the three-dimensional example in characteristic
0 has period 2 and index 8, whereas the conjecture would require the index to divide
4.
Perry (2026) proposes the corrected bound
 |
where 
is a factorial and 
is the exponent of the prime 
in the prime factorization of 
. The correction disappears when the period is relatively
prime to 
.
The paper proves the Hodge-theoretic analogue of the corrected bound on smooth proper
complex varieties. This analogue concerns obstructions
from integral Hodge cycles and does not establish
the proposed bound for the algebraic index.
Perry (2026) credits ChatGPT with a flawed example that inspired the eventual construction, together with literature searches and lattice theory calculations. The author found the working construction and wrote the paper. As of Sep. 18, 2026, independent specialist verification and external peer review of the complete new results had not been reported.
