why did he kill her
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Well, yes but indirectly. King Leontes is all paranoid that Hermione is pregnant with the child of his buddy King Polixenes of Bohemia. There is of course nothing to this, Shakespeare's leading men are often plagued with false imaginings of being a cuckold (wife cheating on him); just ask Othello! After attempting to poison his friend Leontes accuses his wife of being a hoar and sends Hermione and their unborn child to prison. Once the child is born Leontes orders her to be abandoned. Later Leontes finds out his son Mamillius has died over anxiety over his mother's treatment. By this time Hermione has really had enough. She dies of her heartbreak. Although Leontes did not physically kill Hermione, he was most certainly responsible for her demise.
Although I am inclined to agree with the comment prior to my own, I would suggest that Leontes' behaviour has had an impact on Hermione's physical health, not only her "heart-break". It is necessary also to appreciate that after having just given birth, and being kept in such horrendous conditions i.e. prison, there is always going to be an apparent lack of energy, and in this case, so much so that it has caused her death. The contributing factors to her death are an accumulation of Leontes' treatment. Child birth and lack of nourishment are conveyed through many stage productions wherreby Hermione is displayed in filthy rags with milk stained onto her bosom. Her heart-break is what contributes to her weakness but it is essential to recognise the reality of her situation not only the romantic metaphor that Shakespeare uses.
The Winter's Tale. Shakespeare