In the papers,  "Examples of Holomorphic Symplectic Manifolds which admit 
no K\"ahler Structures II, III" published earlier in 1995, 
we found compact simply connected holomorphic symplectic manifolds which are 
not K\"ahler with complex dimension 2n. Actually, we proved that on those 
manifolds, there is a quadric
on the second cohomology such that its n-th power is proportional to the 2n-th
wedge product power of the elements. The quadric has a kernel b. Therefore, b is 
the kernel for the 2 Lefschets map for any element (or symplectic structure).
That is, the Lefschets condition does not hold.

Motivated by these results M. Fernandez et al found a real eight dimensional
compact simply conected symplectic manifold which is not formal in Annals of 
Math. 167 (2008) 1045--1054.
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We found that when n=2, our original eight dimensional example is also not
formal. There are two other elements a and c in the second cohomology,
which is defined for all n, such that ab^{n-1} , b^n, , cb^{n-1} are exact.
Therefore, we can define the b^{n-1} Massey product d=(b^{n-1} : a, b, c).
We proved for n=2, d is nonzero and therefore, our original compact 
simply connected holomorphic symplectic eight dimensional nonk\"ahler manifold
is actually nonformal.
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For n large, the situation is much more complicate. However, we could have
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Conjecture: For n>2, those examples are
nonformal.
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We leave the proof of this conjeture to the future.
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(Abstract posted on Jan. 21, 2015) 
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1991 Mathematics Subject Classification: 32M12, 53C25, 14M20,
14J45, 32L07